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„µ¦ª„ - ¨‡¨³„´œ—oª¥ª·›¸
2. œ´„Á¦¸¥œšÎµ„·‹„¦¦¤˜µ¤˜´ª°¥nµŠ ¨³Â f„®´—
3. œ´„Á¦¸¥œž¦³Á¤·œ¡´•œµ„µ¦…°Š˜œÁ°Š
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1. Á°„­µ¦„µ¦­°œ
2.  f„ž’·´˜·
3. Á‡¦ºÉ°ŠŒµ¥…oµ¤«¸¦¬³
ž¦³Á¤·œŸ¨
ž¦³Á¤·œŸ¨‹µ„ f„®´— ¨³„µ¦š—­°
2
„µ¦ª„‹Îµœªœ®¨µ¥‹Îµœªœ
Á¦ºÉ°Šš¸É 1 „µ¦ª„‹Îµœªœ®¨µ¥‹Îµœªœ˜µ¤Âœª˜´ŠÊ
Ĝ„µ¦ª„‹Îµœªœ®¨µ¥Ç ‹Îµœªœ ×¥„µ¦˜´ÊŠª„„´œœ´œÊ ž{®µš¸ÉšÎµÄ®oŸ·—¡¨µ—Å—oŠnµ¥ „ȇº° „µ¦
š—¨³„µ¦ª„Á¨…Ĝċš¸É¤˜¸ ´ªÁ¨…¤µ„„ªnµ 1 ®¨´„ ­Îµ®¦´ª·›¸„µ¦ª„š¸É‹³Âœ³œÎµœ¸Ê Áž}œª·›¸„µ¦ª„Äœ
Áªš‡–·˜ Ž¹ÉŠŠnµ¥„ªnµª·›¸„µ¦ª„š´ªÉ Ç Åž Á¡¦µ³‹³‡·—ĜċÁŒ¡µ³„µ¦ª„Á¨…×—Ášnµœ´Êœ (Á¨…×— ŗo„n
0, 1, 2, 3, 4, 5, 6, 7, 8, 9) ¨³‹³Á…¸¥œŸ¨¨´¡›rÁŒ¡µ³Á¨…×— ™oµŸ¨¨´¡›rÁ„·œ 9 ‹³Äo x šœ˜´ªš—
—´Š˜´ª°¥nµŠ ˜n°Åžœ¸Ê
˜´ª°¥nµŠ „µ¦ª„Á¨…×—
x
x
x
x
x
x
x
x
x
x
x
x
1 + 4 = 5 2 + 3 = 5 8 + 2 = 0 ( 0 ༡ 10) 4 + 9 = 3 ( 3 ༡ 13)
3 + 5 = 8 8 + 1 = 9 5 + 7 = 2 ( 2 ༡ 12) 7 + 8 = 5 ( 5 ༡ 15)
6 + 3 = 9 0 + 7 = 7 9 + 8 = 7 ( 7 ༡ 17) 9 + 9 = 8 ( 8 ༡ 18)
Ĝ„µ¦ª„Á¨…Áž}œÂ™ª®¨µ¥Ç ™ª ‹³œÎµ®¨´„„µ¦ª„Á¨…×—…oµŠ˜oœ¤µÄo×¥‹³ª„š¸¨³®¨´„
Á¦·É¤˜oœ‹µ„®¨´„®œnª¥ ®¨´„­· ®¨´„¦o°¥ ŞÁ¦ºÉ°¥Ç ‹œ®¤—®¨´„ „µ¦ª„‹³ª„Á¨…ÁŒ¡µ³Á¨…×—‹µ„
™ªœ¨ŠÂ™ª¨nµŠ ™oµ¤¸„µ¦š—‹³Á…¸¥œ x šœ„µ¦š—š¸˜É ´ªª„ ®¨´Š‹µ„œ´Êœ‹³œÎµÁ¨…×—…°ŠŸ¨ª„„´
Á¨…×—Äœ®¨´„®œnª¥…°ŠÂ™ªš¸É°¥¼n™´—Åž…oµŠ¨nµŠ šÎµÁnœœ¸Ê‹œ®¤—™ª Á­¦È‹Â¨oª‹¹Šª„Á¨…×—Äœ®¨´„
­· ˜n„n°œšÎµ„µ¦ª„Á¨…×—Äœ®¨´„­·‹³˜o°Š˜¦ª‹­°„n°œªnµ¤¸ x Ĝ®¨´„®œnª¥°¥¼nÁšnµÄ— Á¤ºÉ°œ´ x
Ĝ®¨´„®œnª¥Å—o¨oªªnµ¤¸‹ÎµœªœÁšnµÄ—„ÈÄ®o™º°ªnµ‹Îµœªœ x š¸Éœ´Å—oÁž}œ˜´ªš—Åž¥´Š®¨´„­· ˜n™oµÅ¤n¤¸ x
Ĝ®¨´„®œnª¥Á¨¥ ™º°ªnµÅ¤n¤˜¸ ´ªš—Åž¥´Š®¨´„­· Ĝ„¦–¸š¸É¤¸˜´ªš—Ä®oœÎµ˜´ªš—Åžª„„´˜´ªÁ¨…Äœ®¨´„
­·…°ŠÂ™ªÂ¦„¨oªšÎµ„µ¦ª„¨Š¤µÁnœÁ—¸¥ª„´„µ¦ª„Äœ®¨´„®œnª¥ ­Îµ®¦´„µ¦ª„Äœ®¨´„°ºœÉ Ç „È
„¦³šÎµÁnœÁ—¸¥ª„´œ
˜´ª°¥nµŠš¸É 1.1.1
„µ¦ª„›¦¦¤—µ
1
2
š—
6
8
3
„µ¦ª„Á…¸¥œ x šœ„µ¦š—
4
2
šœ„µ¦š—9
+ ‹»‹»——šœ„µ¦š—
6
3
4
x
8
+
š·«šµŠ„µ¦ª„
3
„µ¦ª„›¦¦¤—µÄœ®¨´„®œnª¥ 6 + 8 = 14 ‹¹ŠÁ…¸¥œ 4 Áž}œ Ÿ¨¨´¡›rĜ®¨´„®œnª¥
¨³š— 1 ŪoÁ®œº° 2 Ž¹ÉŠÁž}œÁ¨…Äœ®¨´„­·…°Š˜´ª˜´ÊŠ ¦ª¤˜´ª˜´ÊŠ„´˜´ªš—Å—o 1 + 2 = 3 ‹¹ŠÁ…¸¥œŸ¨¨´¡›r 3
š¸É®¨´„­·…°ŠŸ¨¨´¡›r ‹¹ŠÅ—o 26 + 8 = 34
„µ¦ª„Á…¸¥œ x šœ„µ¦š— ‹³Á…¸¥œ x „ε„´ÅªoÁ®œº°˜´ªª„ Á¤º°É „µ¦ª„œ´ÊœÅ—oŸ¨¨´¡›r
˜´ÊŠÂ˜n 10 …¹ÊœÅž
¡·‹µ¦–µ„µ¦ª„Äœ®¨´„®œnª¥
6
x
8
4
6 + 8 = 4x ‹¹ŠÄ­n x Ūoš¸É 8 Ž¹ÉŠÁž}œ˜´ªª„
š·«šµŠ„µ¦ª„ ‹µ„œ¨Š¨nµŠ) Ä­n 4 š¸Éŗo‹µ„
14 ŪoĜŸ¨¨´¡›r
+
­Îµ®¦´ x Ĝ®¨´„®œnª¥ œ´œÊ ‡º° 1 Ĝ®¨´„­·š¸ÉÁž}œ˜´ªš—œ´ÉœÁ°Š
®¨´Š‹µ„œ´œÊ œÎµ x š¸Éšœ 1 Ĝ®¨´„­·Åžª„„´ 2 Ĝ®¨´„­·…°Š˜´ª˜´ÊŠÅ—oÁž}œ 3 Ÿ¨¨´¡›r ‡º° 34
˜´ª°¥nµŠš¸É 1.1.2 ‹Š®µŸ¨ª„˜n°Åžœ¸Ê
8
3
5
4
8
9
6
9
2
+
„
œª‡·— „µ¦ª„Äœ®¨´„®œnª¥
®¨´„®œnª¥
x
8
3
5
5 + 9 = 4 œÎµ x ŞĭnŪoš¸É 9 Ž¹ÉŠÁž}œ˜´ªª„ œÎµŸ¨¨´¡›r
4
8
x
9 +
x
4 ޝª„„´ 2 Ĝ¦¦š´—š¸É 3 ŗoŸ¨¨´¡›rÁž}œ 6 Ä­n 6 ˜¦Š„´
6
9
x
2
®¨´„®œnª¥Äœ¦¦š´—š¸É 4 š¸ÉÁž}œ¦¦š´—Ÿ¨¨´¡›r
6
4
„µ¦ª„Äœ®¨´„­·
®¨´„­·
8
3
5
®œ¹ÉŠ x Ĝ®¨´„®œnª¥š—Áž}œ 1 Ĝ®¨´„­· ‹¹ŠœÎµ 1 ޝª„
4
8
x
9 +
x
„´ 3 Ĝ®¨´„­·…°ŠÂ™ªÂ¦„Å—oÁž}œ 4 œÎµ 4 ޝª„„´ 8
6
9
x
2
‹³Å—o 4 + 8 = 2
1
6
x
‹¹ŠÁ…¸¥œ x Ūoš¸É 8 Ž¹ÉŠÁž}œ˜´ªª„¨oªœÎµ 2 ޝª„„´ 9 ŗoŸ¨¨´¡›r
x
Áž}œ 1 ‹¹ŠÁ…¸¥œ x Ūoš¸É 9 Ž¹ÉŠÁž}œ˜´ªª„ Á…¸¥œŸ¨¨´¡›r 1 ˜¦Š„´
®¨´„­·Äœ¦¦š´—š¸É 4 š¸ÉÁž}œ¦¦š´— Ÿ¨¨´¡›r
„µ¦ª„Äœ®¨´„¦o°¥
®¨´„¦o°¥
8
x
3
5
4
8
x
9 + ®¨´„¦o°¥…°ŠÂ™ªÂ¦„ 2 + 8 = 0 ‹¹ŠÄ­n x š¸É 8 Ž¹ÉŠÁž}œ˜´ªª„ œÎµ 0
6
x
9
x
2
ޝª„„´ 4 ŗo 4 œÎµ 4 ޝª„„´ 6 Ĝ¦¦š´—š¸É 3 4 + 6 = 0 ‹¹Š
0
1
6
Ä­n x š¸É 6 š¸ÉÁž}œ˜´ªª„ ¨³Á…¸¥œ 0 ¨ŠÄœ®¨´„¦o°¥…°Š¦¦š´—Ÿ¨¨´¡›r
8
x
3
5
ÁœºÉ°Š‹µ„Äœ®¨´„¦o°¥¤¸­°Š x š—Áž}œ 2 Ş®¨´„¡´œ ˜n®¨´„¡´œÅ¤n¤¸
4
8
x
9 + „µ¦ª„‹¹ŠÄ­n 2 ŪoĜ®¨´„¡´œ…°Š¦¦š´—Ÿ¨¨´¡›r
x
9
x
2
x
­°Š x Ĝ®¨´„­·š—Áž}œ 2 Ĝ®¨´„¦o°¥ ‹¹ŠœÎµ 2 ޝª„„´ 8 Ĝ
x
x
‡nµš—Åž®¨´„¡´œ
6
2
0
1
6
‹¹ŠÅ—oŸ¨ª„ ‡º° 2 0 1 6
5
˜´ª°¥nµŠš¸É 1.1.3 ˜´ª°¥nµŠ˜n°Åžœ¸Ê‹³Â­—Š…´Êœ˜°œ„µ¦ª„š¸¨³®¨´„ „µ¦ª„Á¦·É¤˜´ÊŠÂ˜n®¨´„®œnª¥ Áž}œ
…´Êœš¸É 1 „µ¦ª„®¨´„­·Áž}œ…´Êœš¸É 2 ŞÁ¦ºÉ°¥Ç
7
8
x
9
x
2
4
2
x
7
2
7
x
2
x
9
x
9
x
9
9
x
9
7
2
6
7
x
2
7
8
8
6
7
+
œª‡·—
…´Êœš¸É 1 4 + 2 = 6 o 6 + 9 = 15 (Ä­n x Á®œº° 9) o 5 + 2 = 7 Ä­n 7 Áž}œŸ¨¨´¡›rĜ®¨´„®œnª¥
¨³¤¸˜´ªš—Áž}œ 1
…´Êœš¸É 2 œÎµ 1 (˜´ªš—) ¤µª„Äœ®¨´„­· 1 + 2 = 3 o 3 + 7 = 10 (Ä­n x Á®œº° 7)
o 0 + 9 = 9 o 9 + 7 = 16 (Ä­n x Á®œº° 7) Ä­n 6 Áž}œŸ¨¨´¡›rĜ®¨´„­· ¨³¤¸˜´ªš—Áž}œ 2
…´Êœš¸É 3 œÎµ 2 (˜´ªš—) ¤µª„Äœ®¨´„¦o°¥ 2 + 9 = 11 (Ä­n x Á®œº° 9) o 1 + 2 = 3 o 3 + 9 = 12
(Ä­n x Á®œº° 9) o 2 + 6 = 8 Ä­n 8 Áž}œŸ¨¨´¡›rĜ®¨´„¦o°¥ ¨³¤¸˜ª´ š—Áž}œ 2
…´Êœš¸É 4 œÎµ 2 (˜´ªš—) ¤µª„Äœ®¨´„¡´œ 2 + 8 = 10 (Ä­n x Á®œº° 8) o 0 + 7 = 7 o 7 + 9 = 16
(Ä­n x Á®œº° 9) o 6 + 2 = 8 Ä­n 8 Áž}œŸ¨¨´¡›rĜ®¨´„¡´œ ¨³¤¸˜ª´ š—Áž}œ 2
…´Êœš¸É 5 œÎµ 2 (˜´ªš—) ¤µª„Äœ®¨´„®¤ºœÉ 2 + 7 = 9 o 9 + 2 = 11 (Ä­n x Á®œº° 2) 1 + 9 = 10
(Ä­n x Á®œº° 9) o 0 + 7 = 7 Ä­n 7 Áž}œŸ¨¨´¡›r Ĝ®¨´„®¤ºÉœ ¨³¤¸˜ª´ š—Áž}œ 2 ®¤— „µ¦ª„ ‹¹ŠÁ…¸¥œ˜´ª
š— 2 Áž}œ 2 Ĝ®¨´„­œ „µ¦ª„Áž}œ—´Šœ¸Ê
2
7
8
x
9
x
2
4
2
x
7
2
7
x
2 +
9
x
9
x
9
x
9
9
7
2
6
7
x
2
7
8
8
6
7
x
6
˜´ª°¥nµŠš¸É 1.1.4 ‹Š®µŸ¨ª„˜n°Åžœ¸Ê
2
9
7
9
6
6
5
7
2
8
2
3
9
9
9
5
4
3
2
1
+
6
œª‡·—
x
x
…´Êœš¸É 1 6 + 8 = 14 o 4 + 9 = 13 o 3 + 1 = (4) ( 2 ‹»— š— 2 Ş…´Êœš¸É 2)
x
x
…´Êœš¸É 2 2 + 9 = 11 o 1 + 2 = 3 o 3 + 9 = 12 o 2 + 2 = (4) ( 2 ‹»— š— 2 Ş…´Êœš¸É 3)
x
x
…´Êœš¸É 3 2 + 7 = 9 o 9 + 7 = 16 o 6 + 9 = 15 o 5 + 3 = (8) ( 2 ‹»— š— 2 Ş…´Êœš¸É 4)
x
x
…´Êœš¸É 4 2 + 9 = 11 o 1 + 5 = 6 ---> 6 + 3 = 9 o 9 + 4 = 13(3) ( 2 ‹»— š— 2 Ş…´Êœš¸É 5)
x
…´Êœš¸É 5 2 + 2 = 4 o 4 + 6 = 10 ---> 0 + 2 = 2o2 + 5 = (7) ( 1 ‹»— š—1Ş…´Êœ®¨´„­œ) ‹³Å—o
1
2
9
x
7
9
x
6
6
x
5
7
x
2
8
2
3
9
x
9
x
9
5
4
x
3
2
1
7
3
8
4
4
x
+
x
7
˜´ª°¥nµŠš¸É 1.1.5 ‹Š®µŸ¨ª„…°Š
4
1
7
5
4
1
7
3
8
2
6+
3
8
x
2
6+
4
7
5
3
4
x
7
5
3
2
1
0
4
2
1
0
4
4
8
5
8
œª‡·—
’ 0
1
x
5
x
Á¡ºÉ°Áž}œ„µ¦ f„®´—„µ¦ª„Á¨…×— ˜µ¦µŠ˜n°Åžœ¸Ê­—Š˜µ¦µŠ„µ¦ª„Á¨…×— Ž¹ÉŠ­µ¤µ¦™
Á…oµÄ‹Å—o×¥Šnµ¥
+
0
1
2
3
4
5
6
7
8
9
®¤µ¥Á®˜»
0
0
1
2
3
4
5
6
7
8
9
0 = 10
1
1
2
3
4
5
6
7
8
9
0
x
1 = 11
2
2
3
4
5
6
7
8
9
0
x
1
x
2 = 12
3
3
4
5
6
7
8
9
0
x
1
x
2
x
3 = 13
4
4
5
6
7
8
9
0
x
1
x
2
x
3
x
4 = 14
5
5
6
7
8
9
0
x
1
x
2
x
3
x
4
x
5 = 15
6
6
7
8
9
0
x
1
x
2
x
3
x
4
x
5
x
6 = 16
7
7
8
9
0
x
1
x
2
x
3
x
4
x
5
x
6
x
7 = 17
8
8
9
0
x
1
x
2
x
3
x
4
x
5
x
6
x
7
x
8 = 18
9
9
0
x
1
x
2
x
3
x
4
x
5
x
6
x
7
x
8
x
9 = 19
x
x
x
x
x
x
x
x
x
x
8
 f„®´—š¸É 1
1. ‹Š®µŸ¨ª„˜n°Åžœ¸Ê
(1) 8 1 3
(2) 6 6 6
256+
945+
749
132
„
„
(4) 3 1 3 8
4 575+
3649
7322
(5) 2
9
3
2
129
754+
681
134
„
(7) 5 9 4 1 5
7248
489+
82116
38414
(8) 6 7 5 4 1
836
4795+
38583
87654
„
(10) 4 8 5 9 5
4728
316
8427
52983
125+
6666
12345
37621
(11) 4 1 2 8 1
5239
93 527
38092
4216
55766+
3338
1485
23117
„
(3) 1 4 8
987+
456
®
(6) 8 1 4 2
6754+
9999
3333
®
(9) 1 4 1 5 1
87878
2649+
3555
78214
®
9
2. ‹Š®µŸ¨ª„…°Š
31465 + 47474 + 38641 + 27264 + 38886 =
„
48753 + 99486 + 10238 + 47655 + 95384 =
„
­œ»„„´˜´ªÁ¨… (1)
‡¼nÁ®¤º°œ
­µ¤˜´ªÁ®¤º°œ
­¸É˜´ªÁ®¤º°œ
11 u 1 = 11
37 u 3 = 111
101 u 11 = 1111
11 u 2 = 22
37 u 6 = 222
101 u 22 = 2222
11 u 3 = 33
37 u 9 = 333
101 u 33 = 3333
11 u 4 = 44
37 u 12 = 444
101 u 44 = 4444
11 u 5 = 55
37 u 15 = 555
101 u 55 = 5555
11 u 6 = 66
37 u 18 = 666
101 u 66 = 6666
11 u 7 = 77
37 u 21 = 777
101 u 77 = 7777
11 u 8 = 88
37 u 24 = 888
101 u 88 = 8888
11 u 9 = 99
37 u 27 = 999
101 u 99 = 9999
10
‹ÎµœªœÄœ¦³Á‡¦ºÉ°Š®¤µ¥‡¨³
Á¦ºÉ°Šš¸É 2 ‹ÎµœªœÄœ¦³“µœ­·„´‹ÎµœªœÄœ¦³Á‡¦º°É Š®¤µ¥‡¨³
Ĝ¦³˜´ªÁ¨…“µœ­·Ž¹ÉŠ˜´ªÁ¨…š¸Éčoŗo„n ˜´ªÁ¨… 0 Ş‹œ™¹Š 9 ­Îµ®¦´Äœ¦³Á‡¦º°É Š®¤µ¥
‡¨³ œ·¥¤Áž¨¸¥É œ¦¼ž˜´ªÁ¨…š¸É¤µ„„ªnµ 5 Ä®oÁž}œ˜´ªÁ¨…š¸Éœ°o ¥„ªnµ 5 ¨oªÁ…¸¥œ­´¨´„¬–r - (…¸—œ) œ
˜´ªÁ¨…Á®¨nµœ´œÊ °´œ‹³šÎµÄ®o„µ¦‡Îµœª–Šnµ¥…¹Êœ Á¡¦µ³„µ¦‡Îµœª–čo‡nµ˜´ªÁ¨…Á¡¸¥Š 0 ™¹Š 5 ¥n°¤Šnµ¥Â¨³Á¦Èª
„ªnµÄo‡nµ˜´ªÁ¨…˜´ÊŠÂ˜n 0 ™¹Š 9 „µ¦Á…¸¥œ­´¨´„¬–ršœ˜´ªÁ¨…š¸É¤µ„„ªnµ 5 ­—Š—´Š˜´ª°¥nµŠ˜n°Åžœ¸Ê
_
˜´ª°¥nµŠš¸É 1.1.6 ¡·‹µ¦–µ 9 ‹³Á®Èœªnµ 9 = 10 - 1 = 10 + (-1) ™oµÁ…¸¥œ -1 Áž}œ 1
_
_
_
‹³Å—o 10 + (-1) = 10 + 1 Ž¹ÉŠ‹³Á…¸¥œÁž}œ 1 1 —´Šœ´Êœ 9 = 1 1 °›·µ¥Å—o—´Šœ¸Ê
_
Ĝ¦³˜´ªÁ¨…“µœ­· ™oµÁ…¸¥œ ab ‹³®¤µ¥™¹Š a u (10) + b šÎµœ°ŠÁ—¸¥ª„´œ 1 1 ‹³®¤µ¥™¹Š
_
_
1 u (10) + 1 = 10 + 1
_
œ´Éœ‡º° 1 1
_
=
_
1 u (10) + 1 = 10+ 1 = 10 - 1 = 9
˜´ª°¥nµŠš¸É 1.1.7 29 = 30 – 1 = 30 + (-1) = 3 u (10) + (-1)
_
_
= 30 + 1 = 3 1
_
œ´Éœ‡º° 29 = 3 1
­Îµ®¦´‹ÎµœªœÁ˜È¤ m ė Ç „ε®œ— m = – m
‹µ„„µ¦„ε®œ— m = – m ­Îµ®¦´‹ÎµœªœÁ˜È¤ m Ä—Ç ‹³Å—o‡»–­¤´˜·˜n°Åžœ¸Ê
­Îµ®¦´‹ÎµœªœÁ˜È¤ m, n ė Ç
____
1. m n = m + n
2. m
=m
____
____
3. m n = n m = m + n = n + m
­—ŠÅ—o—´Šœ¸Ê
____
1. m n
2. m
____
3. m n
=
– (m + n) = (– m) + (– n) = m + n
=
– m = – (– m) = m
=
– (m + n ) = (– m) – n = (– m) – (–n)
=
(– m) + n = m + n
11
____
nm
=
– ( n + m) = (– n ) + (– m) = – (– n) + (– m)
=
n + (– m) = n + m
œ°„‹µ„œ¸Ê ­Îµ®¦´Äœ¦³˜´ªÁ¨…“µœ­· ‹³¤¸‡»–­¤´˜·—´Š˜n°Åžœ¸Ê
­Îµ®¦´Á¨…×— an , an - 1 , an - 2 ,..., a2 , a1 , a0
___________________
__ ____ ____ __ __ __
4. a n a n 1a n 2 ... a2 a1 a0 = an an1 an2 ... a2 a1 a0
‡»–­¤´˜·…o° 4. ‹³Â­—Šž¦³„°—oª¥˜´ª°¥nµŠ —´Šœ¸Ê
__
__
23 = 2 3
_____
__
__ _
(ÁœºÉ°Š‹µ„ 23 = 20 + 3 —´Šœ´Êœ 23 = 20 3 = 20 + 3
__ _
_ _
×¥„µ¦Á…¸¥œÂ¤¸‡nµž¦³‹Îµ˜ÎµÂ®œnŠ ‹³Å—o 20 + 3 = 2 3
_ _
„µ¦Á…¸¥œ‹ÎµœªœÄœ¦³Á‡¦ºÉ°Š®¤µ¥‡¨³œ´Êœ ‹³¡‹Îµœªœš¸ÉÁ…¸¥œ—´Š˜´ª°¥nµŠ˜n°Åžœ¸Ê
_
_
_
_
1 , 2 , 3 , 0 , 4 , 2 , 1 , 5 , 6 , 1 , 0 , 2 , 1 ²¨² Ž¹ÉŠ®¨´„Ä—¤¸…¸—œ°¥¼nÁ®œº°‹Îµœªœ®¨´„œ´ÊœÇ ‹³¤¸‡nµÁž}œ
‹Îµœªœ¨š¸É¤¸…œµ—Ášnµ„´…œµ—…°Š‡nµ˜´ªÁ¨…Äœ®¨´„œ´œÊ
˜´ª°¥nµŠ˜n°Åžœ¸ÊÁž}œ˜´ª°¥nµŠ„µ¦Á…¸¥œ˜´ªÁ¨…Äœ¦³“µœ­· Áž}œ˜´ªÁ¨…Äœ¦³Á‡¦ºÉ°Š®¤µ¥
‡¨³Ã—¥¤¸…¸—œ –
_
˜´ª°¥nµŠš¸É 1.1.8 79 = 80 + (–1) = 80 + 1
_
_
Á…¸¥œ 80 + 1 Ĝ¦¼ž‡nµž¦³‹Îµ˜ÎµÂ®œnŠ‹³Å—o 8 1
_
—´Šœ´Êœ
79 = 8 1
œ°„‹µ„œ¸Ê
80 = 100 – 20 = 100 + 2 0 = 100 + 2 0 = 1 2 0
—´Šœ´Êœ
79 = 80 + 1
_
_
_
_
(Á¡¦µ³ 0 + 1 = 1 )
œ´Éœ‡º°
__
__
__ _
__
__
= 1 20 + 1 = 1 2 1
__
79 = 1 2 1
®¤µ¥Á®˜» „µ¦Á…¸¥œ­´¨´„¬–ršœ 79 °µ‹šÎµÅ—o—´Šœ¸Ê
__
__
_
__
79 = 100 – 21 = 100 + 2 1 = 100 + 2 0 + 1 = 1 2 1
12
Ĝ„µ¦Á…¸¥œ‹ÎµœªœÄœ¦³˜´ªÁ¨…“µœ­·Ä®oÁž}œ˜´ªÁ¨…š¸ÉčoĜ¦³Á‡¦ºÉ°Š®¤µ¥‡¨³ ץčo
Á¨…×—ŤnÁ„·œ 5 ­µ¤µ¦™šÎµÅ—oץčo‹Îµœªœš­· ¨³‹ÎµœªœšÁ„oµ Ž¹ÉŠÄœ„µ¦°›·µ¥‡ªµ¤®¤µ¥…°Š
‹ÎµœªœšÁ„oµ ®¦º°‹Îµœªœš­·…°ŠÁ¨…×—Ä—Ç °›·µ¥Å—o—´Šœ¸Ê
‹ÎµœªœšÁ„oµ ­Îµ®¦´Á¨…×— a Ä—Ç ‹ÎµœªœšÁ„oµ…°Š a
‡º°Á¨…×— b Ž¹ÉŠ a + b = 9
ÁœºÉ°Š‹µ„­Îµ®¦´Á¨…×— a, b Ä—Ç a + b = b + a —´Šœ´Êœ ™oµ a + b = 9 ‹³Å—o b + a = 9
‹¹Š„¨nµªÅ—oªnµ b Áž}œ‹ÎµœªœšÁ„oµ…°Š a ¨³ a Áž}œ‹ÎµœªœšÁ„oµ…°Š b ®¦º°„¨nµªªnµ a ¨³ b
Áž}œ‹ÎµœªœšÁ„oµŽ¹ÉŠ„´œÂ¨³„´œ
‡¼n‹ÎµœªœšÁ„oµ…°ŠÁ¨…×— 0 ™¹Š 9 ¤¸—´Šœ¸Ê
0 ¨³ 9 Áž}œ‹ÎµœªœšÁ„oµŽ¹ÉŠ„´œÂ¨³„´œÁ¡¦µ³ 0 + 9 = 9 + 0 = 9
1 ¨³ 8 Áž}œ‹ÎµœªœšÁ„oµŽ¹ÉŠ„´œÂ¨³„´œÁ¡¦µ³ 1 + 8 = 8 + 1 = 9
2 ¨³ 7 Áž}œ‹ÎµœªœšÁ„oµŽ¹ÉŠ„´œÂ¨³„´œÁ¡¦µ³ 2 + 7 = 7 + 2 = 9
3 ¨³ 6 Áž}œ‹ÎµœªœšÁ„oµŽ¹ÉŠ„´œÂ¨³„´œÁ¡¦µ³ 3 + 6 = 6 + 3 = 9
4 ¨³ 5 Áž}œ‹ÎµœªœšÁ„oµŽ¹ÉŠ„´œÂ¨³„´œÁ¡¦µ³ 4 + 5 = 5 + 4 = 9
‹Îµœªœš­· ­Îµ®¦´Á¨…×— a Ä—Ç ‹Îµœªœš­·…°Š a
‡º°Á¨…×— b Ž¹ÉŠ a + b = 10
šÎµœ°ŠÁ—¸¥ª„´œ„´‹ÎµœªœšÁ„oµ ­Îµ®¦´Á¨…×— a ¨³ b Ž¹ÉŠ a + b = 10 „¨nµªªnµ a ¨³ b Áž}œ
‹Îµœªœš­·Ž¹ÉŠ„´œÂ¨³„´œ
‡¼n‹Îµœªœš­·…°ŠÁ¨…×— 0 ™¹Š 9 ¤¸—´Šœ¸Ê
1 ¨³ 9 Áž}œ‹Îµœªœš­·Ž¹ÉŠ„´œÂ¨³„´œÁ¡¦µ³ 1 + 9 = 9 + 1 = 10
2 ¨³ 8 Áž}œ‹Îµœªœš­·Ž¹ÉŠ„´œÂ¨³„´œÁ¡¦µ³ 2 + 8 = 8 + 2 = 10
3 ¨³ 7 Áž}œ‹Îµœªœš­·Ž¹ÉŠ„´œÂ¨³„´œÁ¡¦µ³ 3 + 7 = 7 + 3 = 10
4 ¨³ 6 Áž}œ‹Îµœªœš­·Ž¹ÉŠ„´œÂ¨³„´œÁ¡¦µ³ 4 + 6 = 6 + 4 = 10
5 Áž}œ‹Îµœªœš­·…°Š˜´ªÁ°ŠÁ¡¦µ³ 5 + 5 = 10
13
˜´ª°¥nµŠš¸É 1.1.9 ‹ŠÁ…¸¥œ 2789 ץčo˜ª´ Á¨…ŤnÁ„·œ 5
œª‡·— 2 7 8 9
=
3 0 0 0–2 1 1
=
3 0 0 0 + 211
=
3 211
=
3 211
___
___
___
œ°„‹µ„ª·›¸‡·—˜µ¤Âœª‡·—…oµŠ˜oœÂ¨oª °µ‹Äoª·›¸‡·—¨´—Å—o—´Šœ¸Ê
¡·‹µ¦–µ 789 ˜´ªÁ¨…š¸É˜o°ŠÁž¨¸É¥œÂž¨Š‡º° 9 Ĝ®¨´„®œnª¥ 8 Ĝ®¨´„­· ¨³ 7 Ĝ®¨´„¦o°¥
_
˜´ªÁ¨… …ªµ­»— (®¨´„®œnª¥) ‡º° 9 ¤¸‹Îµœªœš­· ‡º° 1 Ä­n…¸—œÅ—oÁž}œ 1
_
˜´ªÁ¨… ®¨´„­· ‡º° 8 ¤¸‹ÎµœªœšÁ„oµ ‡º° 1 Ä­n…¸—œÅ—oÁž}œ 1
_
˜´ªÁ¨… Žoµ¥­»— (®¨´„¦o°¥) ‡º° 7 ¤¸‹ÎµœªœšÁ„oµ ‡º° 2 Ä­n…¸—œÅ—oÁž}œ 2
Ĝ„µ¦Á…¸¥œ 2789 ×¥Á…¸¥œÁŒ¡µ³˜´ªÁ¨…š¸Éœo°¥„ªnµ 5 ¨³Äo …¸—œšÎµÅ—o—´Šœ¸Ê Ä®oÁ…¸¥œ˜´ªÁ¨…
‹µ„…ªµÅžŽoµ¥Ã—¥Á…¸¥œ‹Îµœªœš­·…°Š 9 ¡¦o°¤…¸—œ™´—¤µÁž}œ‹ÎµœªœšÁ„oµ…°Š 8 ¡¦o°¤…¸—œ
¨³‹ÎµœªœšÁ„oµ…°Š 7 ¡¦o°¤…¸—œ ¨³˜´ªÁ¨…Žoµ¥­»—Á…¸¥œ‡nµ…°Š 2 š¸ÉÁ¡·É¤…¹Êœ 1 œ´Éœ‡º°Á…¸¥œ 3 Žoµ¥­»—
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‹³Å—o 3 2 1 1
Á…¸¥œÁž}œŸ´ŠÅ—o—´Šœ¸Ê (Á¦·É¤‹µ„ …ªµÅžŽoµ¥)
2
7
8
9
Á¡·É¤‡nµ…¹Êœ 1
‹Îµœªœš
‹Îµœªœš
‹Îµœªœš
Á„oµ
Á„oµ
­·
…°Š 7
…°Š 8
…°Š 9
2
1
1
Ä­n…¸—œ
Ä­n…¸—œ
Ä­n…¸—œ
3
_
2
___
_
1
_
1
œ´Éœ‡º° 2789 Áž¨¸É¥œÁž}œ 3 2 1 1
14
…o°­´ŠÁ„˜ ­Îµ®¦´‹ÎµœªœÁ˜È¤ m Ä—Ç ™oµ¤¸»—…°ŠÁ¨…×—˜·—„´œ®¨µ¥˜´ª ×¥š¸ÉÁ¨…×—˜n¨³˜´ªš¸É˜·—„´œ
œ´Êœ¤¸‡nµÁ„·œ 5 Ĝ„µ¦Âž¨ŠÁ¨…×—Á®¨nµœ´ÊœšÎµÅ—o×¥˜´ªÁ¨……ªµ­»—…°Š»—ž¨ŠÁž}œ‹Îµœªœš­·…°ŠÁ¨…
חœ´Êœ ¨oªÁ…¸¥œ…¸—œÅªoš¸É‹Îµœªœš­·œ´Êœ ­Îµ®¦´Á¨…×—™´—Ç ¤µšµŠŽoµ¥Äœ»—œ´Êœ ž¨ŠÁž}œ‹Îµœªœ
šÁ„oµÂ¨oªÁ…¸¥œ…¸—œÅªoš¸É‹ÎµœªœšÁ„oµÁ®¨nµœ´Êœ ¨³Á¨…×—š¸É™´—ÅžšµŠŽoµ¥š¸É¤¸‡nµœo°¥„ªnµ 5 Ä®oÁ¡·É¤‡nµ
…¹Êœ 1 „È‹³Áž}œ„µ¦Á…¸¥œ‹ÎµœªœÁ˜È¤ m š¸É¤¸Á¨…×—¤¸‡nµÅ¤nÁ„·œ 5
˜´ª°¥nµŠš¸É 1.1.10 ‹ŠÁ…¸¥œ 928716 ץčo˜´ªÁ¨…ŤnÁ„·œ 5 (œ·…·¨´¤­¼˜¦)
09 2 8 7 1 6
œª‡·—
_
_ _
_
11 31 3 2 4
_
4 Áž}œ ‹Îµœªœš­·…°Š 6 ‹¹ŠÁ…¸¥œ 4
Á¡·É¤‡nµ 1 …¹Êœ°¸„ 1 Áž}œ 2
_
3 Áž}œ‹Îµœªœš­·…°Š 7 ‹¹ŠÁ…¸¥œ 3
_
1 Áž}œ‹ÎµœªœšÁ„oµ…°Š 8 ‹¹ŠÁ…¸¥œ 1
Á¡·É¤‡nµ 2 …¹Êœ°¸„ 1 Áž}œ 3
_
1 Áž}œ‹Îµœªœš­·…°Š 9 ‹¹ŠÁ…¸¥œ 1
ÁœºÉ°Š‹µ„Äœ®¨´„…°Š 9 ‹³˜o°ŠÂž¨Š‡nµ 9
_
Áž}œ 1 —´Šœ´Êœ
‹¹Š‹ÎµÁž}œ˜o°ŠÁ…¸¥œ 0 ®œoµ 9 Á¡ºÉ°‹³˜o°ŠÁ¡·¤É ‡nµ…¹Êœ 1 ‹¹ŠÁ…¸¥œ 1 Á¡·É¤…¹Êœ¤µÅªo®œoµ­»—…°Š˜´ªÁ¨…š¸É
ž¨ŠÂ¨oª
_
_ _
_
œ´Éœ‡º° 928716 = 1 1 3 1 3 2 4
15
˜´ª°¥nµŠš¸É 1.1.11 ‹ŠÂž¨Š 257182 Áž}œ˜´ªÁ¨…œ·…·¨´¤­¼˜¦
œª‡·—
25 7 1 8 2
_
_
œª‡·— 2 6 3 2 2 2
Ťn¤¸„µ¦Áž¨¸É¥œÂž¨Š
‹Îµœªœš­·
1+1
Á¦·Á¦·É¤É¤»»——Ä®¤n
‹Îµœªœš­·
5+1
Ťn¤¸„µ¦Áž¨¸É¥œÂž¨Š
_
_
—´Šœ´Êœ 2 5 7 1 8 2 = 2 6 3 2 2 2
®¨´Š‹µ„„µ¦Âž¨Š‡¦´ÊŠÂ¦„¨oª¥´Š¤¸Á¨…×— 6 š¸É¤¸‡nµÁ„·œ 5 ‹¹ŠÂž¨Š˜n°Åž
_
_
_ _
_
26 3 2 2 2
3 4 3 22 2
Ťn¤¸„µ¦Áž¨¸É¥œÂž¨Š
‹Îµœªœš­·
_
_
2+1
_ _
_
_ _
_
—´Šœ´Êœ 2 6 3 2 2 2 =
34 3222
œ´Éœ‡º° 2 5 7 1 8 2 =
34 3222
˜´ª°¥nµŠš¸É 1.1.12 ‹ŠÂž¨Š 353782 Áž}œ˜´ªÁ¨…œ·…·¨´¤­¼˜¦
œª‡·—
35 37 8 2
_ _
35 42 2 2
16
_ _
—´Šœ´Êœ 3 5 3 7 8 2 = 3 5 4 2 2 2
Ĝ„µ¦Âž¨Š˜´ªÁ¨…Äœœ·…·¨´¤­¼˜¦Ä®oÁž}œ˜´ªÁ¨…Äœ¦³“µœ­· ­µ¤µ¦™šÎµÅ—oץčo„¦³ªœ„µ¦
¥o°œ„¨´ ‹µ„„µ¦Âž¨Š˜´ªÁ¨…Äœ¦³“µœ­·Áž}œ˜´ªÁ¨…Äœœ·…·¨´¤­¼˜¦
¡·‹µ¦–µ„µ¦Âž¨Š˜´ªÁ¨…¦³“µœ­·Áž}œ˜´ªÁ¨…Äœœ·…¨· ´¤­¼˜¦
37 1 6 9
_
_ _
4 32 3 1
_
‹Îµœªœš­·…°Š 9 ‡º° 1 ¨oªÄ­n…¸—œÁž}œ 1
_
‹ÎµœªœšÁ„oµ…°Š 6 ‡º° 3 ¨oªÄ­n…¸—œÁž}œ 3
‹µ„ 1 Á¡·É¤‡nµ…¹Êœ°¸„ 1 Áž}œ 2
Á¦·É¤˜oœÄ®¤n ‹Îµœªœš­·…°Š 7 ‡º° 3 ¨oªÄ­n…¸—œÁž}œ 3_
‹µ„ 3 Á¡·É¤‡nµ…¹Êœ°¸„ 1 Áž}œ 4
„µ¦Âž¨Š˜´ªÁ¨…Äœœ·…·¨´¤­¼˜¦ Áž}œ˜´ªÁ¨…¦³“µœ­·Äo„¦³ªœ„µ¦¥o°œ„¨´ —´Šœ¸Ê
_
_ _
4 3 2 31
3 7 1 6 9
_
Á°µ…¸—œ°°„‹µ„ 1 ŗo 1 ®µ‹Îµœªœš­·
…°Š 1 ŗoÁž}œ 9
_
Á°µ…¸—œ°°„‹µ„ 3 ŗo 3 ®µ‹ÎµœªœšÁ„oµ
…°Š 3 ŗoÁž}œ 6
2 ¤¸‡nµ¨—¨Š 1 ŗoÁž}œ 1
_
Ä®¤n
Á¦·Á¦·É¤˜oɤœ˜oœÄ®¤n
Á°µ…¸—œ°°„‹µ„ 3 ŗo 3 ®µ‹Îµœªœš­·
…°Š 3 ŗoÁž}œ 7
4 ¤¸‡nµ¨—¨Š 1 Áž}œ 3
œ´Éœ‡º°
_
_ _
432 3 1 =37169
17
_
_
˜´ª°¥nµŠš¸É 1.1.13 ‹ŠÂž¨Š 3 3 4 4 Áž}œ˜´ªÁ¨…Äœ¦³“µœ 10
œª‡·—
_
_
3 3 4 4
2 7 3 6
_
Á°µ °°„‹µ„ 4 ŗoÁž}œ 4 Ž¹ÉŠ‹Îµœªœš­·
…°Š 4 ‡º° 6
¨—‡nµ 4 ¨Š 1 Áž}œ 3
Á¦·É¤˜oœ Á°µ °°„‹µ„ 3_ ŗoÁž}œ 3 Ž¹ÉŠ‹Îµœªœš­·…°Š
3 ༡ 7
_
¨—‡nµ 3 ¨Š 1 Áž}œ 2
_
œ´Éœ‡º° 3 3 4 4 = 2 7 3 6
_ _ _
˜´ª°¥nµŠš¸É 1.1.14 ‹ŠÂž¨Š 3 3 4 2 Áž}œ˜´ªÁ¨…Äœ¦³“µœ­·
œª‡·—
_ _ _
3 3 4 2
2 6 5 8
š­·
šÁ„oµ
_ _ _
¨—‡nµ¨Š 1
—´Šœ´Êœ 3 3 4 2 = 2658
18
_
_ _
˜´ª°¥nµŠš¸É 1.1.15 ‹ŠÂž¨Š 3 6 2 2 7 Áž}œ˜´ªÁ¨…Äœ¦³“µœ­·
œª‡·—
_
_ _
3 6 2 2 7
2 4 1 7 3
š­·
šÁ„oµ
¨—‡nµ¨Š 1
Á¦·É¤˜oœ
š­·
_
_ _
¨—‡nµ¨Š 1
—´Šœ´Êœ 3 6 2 2 7 = 24173
19
Á¦ºÉ°Šš¸É 3 ‹Îµœªœ˜¦Š…oµ¤
_ _
˜n°Åž‹³„¨nµª™¹Š‹Îµœªœ˜¦Š…oµ¤Äœ¦³Á‡¦ºÉ°Š®¤µ¥‡¨³ Ánœ –3 6 2 1 ¤¸‡nµÁž}œÁšnµÄ— Ĝ„µ¦
__ ____ ____ __ __ __
¡·‹µ¦–µ‡nµ‹Îµœªœ˜¦Š…oµ¤ ‹³˜o°ŠÄo‡–
» ­¤´˜· – an an-1 ...a2 a1 a0 = an an1 an2 ... a2 a1 a0 ¨³Äœ
„¦–¸…°Šœ·…·¨¤´ ­¼˜¦‹³˜o°ŠÂž¨Š˜´ªÁ¨…š¸É¤¸‡nµ¤µ„„ªnµ 5 Ä®oÁž}œ˜´ªÁ¨…š¸Éœo°¥„ªnµ 5 ¨³¤¸…¸—œÁ­¸¥„n°œ ‹¹Š
‹³šÎµ„µ¦®µ‹Îµœªœ˜¦Š…oµ¤—´Š‹³Â­—ŠÃ—¥˜´ª°¥nµŠ˜n°Åžœ¸Ê
_____
˜´ª°¥nµŠš¸É 1.1.16 –27489 = 2 7 4 8 9
Ĝœ·…·¨´¤­¼˜¦‹³Å¤nÁ…¸¥œ˜´ªÁ¨…š¸ÉÁ„·œ 5 ‹¹Š‹³˜o°ŠÂž¨Š 27489 Áž}œ˜´ªÁ¨…Äœœ·…·¨´¤­¼˜¦„n°œ
—´Šœ¸Ê
27489
‹³Å—o – (27489)
œ´Éœ‡º° –27489
_
_ _
=
3 3 5 1 1
=
– (3 3 5 1 1 )
=
3 3 5 1 1
=
3 3 5 1 1
=
3 3 5 1 1
_
_ _
_
_
_
_
_
_
˜´ª°¥nµŠš¸É 1.1.17 – 80379
œ´Éœ‡º° – 80379
=
– (80379)
=
– (1 2 0 4 2 1 )
=
1 2 0 4 2 1
=
1 20 4 21
=
1 20 4 21
_
_
_ _
_
_
_
_
_ _
Ĝ˜´ª°¥nµŠš¸É 1.1.18 ¨³ 1.1.19 Áž}œ„µ¦Âž¨Š‹Îµœªœ¨Äœ“µœ³“µœ­·Ä®oÁž}œ‹ÎµœªœÄœ
¦³Á‡¦ºÉ°Š®¤µ¥‡¨³ ĜšµŠ„¨´„´œ‹ÎµœªœÄœ¦³Á‡¦ºÉ°Š®¤µ¥‡¨³ Á¤ºÉ°Âž¨ŠÂ¨oª°µ‹‹³Áž}œ‹Îµœªœ
¨Äœ¦³“µœ­·Å—o—´Š˜´ª°¥nµŠ˜n°Åžœ¸Ê
_
_
˜´ª°¥nµŠš¸É 1.1.18 ‹ŠÂž¨Š 3 7 8 2 1 Áž}œ‹ÎµœªœÄœ¦³“µœ­·
œª‡·— ª·›¸š¸É 1 čo M = – (– M) ¨³ – N = N ¨oªÄo‹Îµœªœš­·Â¨³šÁ„oµ
20
_
_
3 78 2 1
_
_
—´Šœ´Êœ 3 7 8 2 1
_
_
_ _
_
=
– (– ( 3 7 8 2 1))
=
– (3 7 8 2 1 )
=
– (2 2 2 1 9)
=
–22219
=
–22219
ª·›¸š¸É 2 čo‹Îµœªœš­·Â¨³šÁ„oµÃ—¥˜¦ŠÃ—¥Ä®o˜´ªÁ¨…š»„˜´ª¤¸…¸—œ
_
_
3 7 8 2 1
_ _ _ _ _
2 2 2 1 9
_
‹Îµœªœš­·…°Š 1 ‡º° 9 ¨oªÄ­n…¸—œÁž}œ 9
_
_
2 Á¡·É¤‡nµ…¹Êœ 1 Áž}œ 1
_
‹Îµœªœš­·…°Š 8 ‡º° 2 ¨oªÄ­n…¸—œÁž}œ 2
_
‹ÎµœªœšÁ„oµ…°Š 7 ‡º° 2 ¨oªÄ­n…¸—œÁž}œ 2
_
_
_
_
3 Á¡·É¤‡nµ…¹Êœ 1 Áž}œ 2
_ _ _ _ _
—´Šœ´Êœ 3 7 8 2 1 = 2 2 2 1 9
= – 22219
ª·›¸š¸É 3 Áž¨¸É¥œ˜´ªÁ¨…š¸É¤¸…¸—œÁž}œ˜´ªÁ¨…š¸É¤¸‡nµÁž}œª„
_
_
0 3 7 8 2 1
_
1 7 7 7 8 1
1 ¤¸‡nµ‡ŠÁ—·¤
_
Á°µ…¸—œ°°„‹µ„ 2 ŗo 2 ¤¸‹Îµœªœš­·Áž}œ 8
¨—‡nµ 8 ¨Š 1
7 ¤¸‡nµ‡ŠÁ—·¤
Á¦·É¤˜oœÄ®¤n
_
Á°µ…¸—œ°°„‹µ„ 3 ŗo 3 ¤¸‹Îµœªœš­·Áž}œ 7
_
¨—‡nµ 0 ¨Š 1 Áž}œ 1
21
_
_
œÎµ 1 7 7 7 8 1 ¤µÂž¨Š°¸„‡¦´ÊŠ®œ¹ÉŠÁ¡¦µ³ 1 ¥´ŠÅ¤nčo˜ª´ Á¨…š¸É¤¸‡nµ Áž}œª„
_
17 778 1
_
_
œ´Éœ‡º° 3 7 8 2 1
=
– 100000 + 77781
=
– 22219
=
– 22219
®¤µ¥Á®˜» – 100000 + 77781 ®µ‡nµÅ—o‹µ„
Á…¸¥œ‹ÎµœªœšÁ„oµ…°Š 7778 ˜n¨³˜´ª˜µ¤¨Îµ—´ ¨³Á…¸¥œ‹Îµœªœš­·…°Š 1 ¨oªÄ­n
Á‡¦ºÉ°Š®¤µ¥¨ – …oµŠ®œoµ ‹³Å—oŸ¨¨´¡›r˜µ¤˜o°Š„µ¦
7 7 7 8 1
2 2 2 1 9
m ‹ÎµœªœšÁ„oµÂ¨³š­·
Ä­n – …oµŠ®œoµ o – 222197
…o°­´ŠÁ„˜
__
1. ­Îµ®¦´Á¨… an an+1 ...a2 a1 a0 ™oµÁ…¸¥œ…¸—œ¨Šœ an Áž}œ an ¨³™¹ŠÂ¤oªnµ ai ˜´ª°ºÉœ Ç
_
_
( i = n – 1, ..., 0) ‹³¤¸…¸—œ®¦º°Å¤n ‹³¤¸‡nµÁž}œ‹Îµœªœ¨Á¤ºÉ° an z 0 Ánœ 3 7 8 2 1 = – 22219
2. Ĝ¦³˜´ªÁ¨…“µœ­·‡nµž¦³‹Îµ®¨´„œ´˜´ÊŠÂ˜n®¨´„­·…¹ÊœÅž‹³¤¸‡µn Áž}œ 10, 100, 1000,
10000, ..., 10n ,... Á¤ºÉ° n Áž}œ‹ÎµœªœÁ˜È¤š¸É¤µ„„ªnµ®¦º°Ášnµ„´ 1 ™oµ m Áž}œ‹ÎµœªœÁ˜È¤ ¨³ m < 10n
­Îµ®¦´µŠ n ‹³¤¸‹ÎµœªœÁ˜È¤ p Ž¹ÉŠ m + p = 10n ‹³Á¦¸¥„ p ªnµ‹Îµœªœš 10n …°Š m ®µÅ—o×¥®µ
‹Îµœªœš­·…°ŠÁ¨…×—…ªµ­»—…°Š m ¨oª®µ‹ÎµœªœšÁ„oµ…°ŠÁ¨…×—…°Š m š¸É™´—ÅžšµŠŽoµ¥
—´Š˜´ª°¥nµŠ˜n°Åžœ¸Ê
˜´ª°¥nµŠš¸É 1.1.19 ‹Š®µ‹Îµœªœš 10n …°Š 728 Á¤ºÉ° 10n ¤¸‡nµ—´Šœ¸Ê
„. 1000
…. 10000
‡. 100000
22
„. ÁœºÉ°Š‹µ„ 1000 Áž}œÁ¨…š¸É¤¸ 0 °¥¼n 3 ˜´ª ¨³ 728 Áž}œ‹Îµœªœš¸É¤¸ 3 ®¨´„ ‹¹ŠÁ…¸¥œ‹Îµœªœš­·
¨³šÁ„oµ˜µ¤…o°­´ŠÁ„˜—´Šœ¸Ê
7 2 8
2 7 2
—´Šœ´Êœ ‹Îµœªœš 1000 …°Š 728 ‡º° 272
…. ÁœºÉ°Š‹µ„ 10000 Áž}œÁ¨…š¸É¤¸ 0 °¥¼n 4 ˜´ª ¨³ 728 Áž}œ‹Îµœªœš¸É¤¸ 3 ®¨´„ ‹¹ŠÁ¡·É¤ 0 …oµŠ®œoµ
Á¡ºÉ°Ä®oÁž}œ‹Îµœªœš¸É¤¸ 4 ®¨´„ —´Šœ¸Ê 0728 ®¨´Š‹µ„œ¸ÊÁ…¸¥œš­·Â¨³šÁ„oµ —´Šœ¸Ê
0 7 2 8
9 2 7 2
—´Šœ´Êœ‹Îµœªœš 10000 …°Š 728 ‡º° 9272
‡. šÎµœ°ŠÁ—¸¥ª„´œ‹Îµœªœš 100000 …°Š 728 ‡º° 99272
š­¦»ž „µ¦Âž¨ŠÁ¨…×—š¸É¤¸‡nµÁ„·œ 5 Ä®oÁž}œ˜´ªÁ¨…¦³Á‡¦ºÉ°Š®¤µ¥‡¨³ Ž¹ÉŠÄo˜´ªÁ¨…ŤnÁ„·œ 5 ‹³šÎµ
Ä®o„µ¦‡·—‡Îµœª–Šnµ¥…¹Êœ
23
 f„®´—š¸É 2
1. ‹ŠÁ…¸¥œ‹Îµœªœ˜n°Åžœ¸ÊÁ‡¦ºÉ°Š®¤µ¥‡¨³
(1.1) 2 9 8
(1.2) 6 3
(1.3) 9 7 1
(1.4) 1 6 1 8
(1.5) 2 9 8 7 0 5
(1.6) 1 9 1 8 1 6
(1.7) 4 8 9 9 0 2 9 1
(1.8) 9 0 8 3 5 2 8 1
(1.9) 9 9 9 8 0 8 9 2
(1.10) 9 0 9 1 2 5 4 6 8 3 2
(1.11) 9 2 3 4 6 7 1
(1.12) 9 8 3 x 0 1 2
(1.13) 2 x 1 0 3 5
(1.14) – 3 x 5 2 4 1
2. ‹ŠÁ…¸¥œ‹Îµœªœ˜n°Åžœ¸ÊĜ˜´ªÁ¨…¦³“µœ 10
_
_ _ _
(2.1) 2 4 1
(2.2) 3 1 2 8
_ _ _
_ _ _ _
(2.3) 3 4 1 3
(2.4) 1 2 3 1 1 1 1
_ _ _ _
_ _ _ _ _
(2.5) 2 3 2 1 4 5 3
(2.6) 3 3 4 6 0 1 3 4
_ _ _ _ _
_ _ _ _
(2.7) 5 1 3 2 3 3 2 1
(2.8) 5 1 2 0 1 2 0 2
_ _ _ _
_ _ _ _
(2.9) 5 4 0 3 2 2 1 1 0
(2.10) 5 1 3 2 3 3 5 4 3
_
_ _ _ _
__
(2.11) 5 0 0 3 4 2 0 4
(2.12) 3 0 1 x 2 3 4 5
_ _
_
_ _
_
(2.13) 3 0 1 x 4 7 3 2
(2.14) 1 1 1 x 1 1 1 1
24
„µ¦ª„ - ¨‡¨³„´œ®¨µ¥‹Îµœªœ
Á¦ºÉ°Šš¸É 4 „µ¦¨š¸É¤¸„µ¦…°¥º¤
Ĝ„µ¦¨‹Îµœªœ­°Š‹Îµœªœ ™oµÁ¨…×—Ĝ˜n¨³®¨´„…°Š˜´ª˜´ÊŠ¤¸‡µn ¤µ„„ªnµ®¦º°Ášnµ„´‡nµ…°Š
Á¨…×—…°Š˜´ª¨Äœ®¨´„œ´œÊ Ç Â¨oª‹³šÎµ„µ¦¨Å—oŠµn ¥Ã—¥Å¤n˜o°Š…°¥º¤ ˜n™oµÄœ®¨´„Ä—š¸É˜ª´ ¨¤¸‡nµ
¤µ„„ªnµ˜´ª˜´ÊŠ‹³˜o°Š¤¸„µ¦…°¥º¤‡nµÄœ®¨´„™´—…¹ÊœÅž…°Š˜´ª˜´ÊŠŽ¹ÉŠ‹³šÎµÄ®o„µ¦¨Ž´Žo°œ…¹Êœ
Á¦µ°µ‹
®¨¸„Á¨¸É¥Š„µ¦…°¥º¤Â¨³Äoª›· ¸„µ¦šÁ„oµ®¦º°š­·Á…oµnª¥ œ°„‹µ„œ¸ÂÊ šœš¸É‹³šÎµ„µ¦¨„´œÄœ®¨´„š¸É˜´ª
¨¤¸‡nµ¤µ„„ªnµ˜´ª˜´ÊŠÂ¨oª‹³šÎµ„µ¦ª„˜´ª˜´ÊŠ—oª¥‹Îµœªœš¸ÉÁ„·—‹µ„„µ¦šÁ„oµ®¦º°š­·°¸„—oª¥
¡·‹µ¦–µ„µ¦¨˜n°Åžœ¸Ê 3 – 8 = –5 Á®ÈœÅ—o´—ªnµ ˜´ª˜´ÊŠ‡º° 3 ˜´ª¨ ‡º° 8 Ž¹ÉŠ˜´ª¨¤¸‡nµ¤µ„„ªnµ˜´ª
˜´ÊŠ ª·›¸®µŸ¨¨°µ‹šÎµÅ—o×¥Áž¨¸É¥œÁ°µ˜´ª¨Áž}œ˜´ª˜´ÊŠ ¨³Áž¨¸É¥œ˜´ª˜´ÊŠÁž}œ˜´ª¨Å—oŸ¨¨´¡›rÁž}œÁšnµÄ—
Ä®oÄ­nÁ‡¦ºÉ°Š®¤µ¥¨ (–) ®œoµŸ¨¨´¡›rœ´Êœ „È‹³Å—oŸ¨¨…°Š 3 – 8 ˜µ¤˜o°Š„µ¦
­Îµ®¦´„µ¦¨ 13 – 8 ™¹ŠÂ¤oªnµ˜´ª˜´ÊŠ‹³¤¸‡nµ¤µ„„ªnµ˜´ª¨Â˜nÁ¨…×—Äœ®¨´„®œnª¥…°Š˜´ª˜´ÊŠ‡º°
3 ¤¸‡nµœo°¥„ªnµÁ¨…×—Ž¹ÉŠÁž}œ˜´ª¨‡º° 8 „¦–¸œ¸Ê‹³®µŸ¨˜nµŠÁŒ¡µ³Á¨…×—„´Á¨…×—Ťnŗo‹³˜o°ŠÄoÁ¨…
חĜ˜´ª˜´ÊŠš´ŠÊ ®¨´„­·Â¨³®¨´„®œnª¥ ×¥š´ÉªÇ Ş °µ‹®µŸ¨¨´¡›r…°Š 13 – 8 ŗo—´Šœ¸Ê
Ä®o 13 – 8 = Ž¹ÉŠ­°—‡¨o°Š„´ 13 = 8 + œ´Éœ‡º° ‹³˜o°Š®µ‡nµ ªnµ ¤¸‡nµÁž}œÁšnµÄ— Ž¹ÉŠÁ¤ºÉ°
œÎµ¤µª„„´ 8 ¨oªÁšnµ„´ 13 ‹³Á®ÈœÅ—oªµn ¤¸‡nµÁšnµ„´ 5 Á¡¦µ³ 8 + 5 = 13
œ´Éœ‡º° 13 – 8 = 5
°¥nµŠÅ¦„Șµ¤ 8 + 5 Áž}œ„µ¦ª„Á¨…×—Ž¹ŠÉ Ÿ¨ª„¤¸‡nµÁ„·œ 10
œ°„‹µ„œ¸Ê™oµ‹³®µŸ¨¨´¡›r…°Š 23 – 8 °µ‹šÎµÅ—o×¥ Ä®o 23 – 8 = Ž¹ÉŠ­°—‡¨o°Š„´ 23 = 8 + Ž¹ÉŠ ¤¸‡nµÁšnµ„´ 15 Á¡¦µ³ 8 + 15 = 23
œ´Éœ‡º° 23 – 8 = 15 ‹³Á®ÈœÅ—oªnµ¥µ„…¹Êœ„ªnµ„µ¦®µŸ¨¨´¡›r…°Š 13 – 8
œª‡·—¨³…´œÊ ˜°œ„µ¦Äo‹µÎ œªœšÁ„oµ ¨³š­·nª¥Äœ„µ¦¨ ­Îµ®¦´„¦–¸š˜¸É o°Š…°¥º¤‹³
°›·µ¥—oª¥˜´ª°¥nµŠ˜n°Åžœ¸Ê
25
˜´ª°¥nµŠš¸É 1.1.20 ‹Š®µŸ¨¨…°Š 43 – 18
œª‡·— Á…¸¥œ„µ¦¨ÄœÂœª˜´ÊŠÄ®o®¨´„…°Š˜´ª˜´ÊŠ ¨³®¨´„…°Š˜´ª¨˜¦Š„´œ
4
3
–
1
8
…´Êœš¸É 1 Ĝ®¨´„®œnª¥ ˜´ª˜´ÊŠ‡º° 3 ˜´ª¨‡º° 8 Á®ÈœÅ—o´—ªnµ ˜´ª¨¤¸‡nµ¤µ„„ªnµ˜´ª˜´ÊŠ ‹³Äo‹Îµœªœ
š­·…°Š 8 œÎµÅžª„„´ 3 Ž¹ÉŠÁž}œ˜´ª˜´ÊŠ ¨oªÄ­n c Á®œº°˜´ª¨Äœ®¨´„™´—ÅžšµŠŽoµ¥ ‹Îµœªœš­·…°Š
8 ‡º° 2 œÎµ 2 ޝª„„´ 3 ŗo 5 Ä­n 5 Áž}œŸ¨¨´¡›rĜ®¨´„®œnª¥ ¨oªÄ­n c Á®œº° 1 Ž¹ÉŠÁž}œ˜´ª¨Äœ®¨´„
™´—Åž
4
3+ _ 2
1c
8
5
…´Êœš¸É 2 šÎµ„µ¦¨Äœ®¨´„­· „¦–¸œ¸Ê 1c ‡º° 2 Ž¹ÉŠ 4 – 1c ‡º° 4 – 2 = 2 ‹¹ŠÄ­n 2 Áž}œ Ÿ¨¨´¡›rĜ®¨´„
­· ‹³Å—oŸ¨¨´¡›rĜ„µ¦¨‡º° 25
4
1c
2
3+
2 –
8
5
—´Šœ´Êœ 43 – 18 = 25
ʄ
˜´ª°¥nµŠš¸É 1.1.21 ‹Š®µŸ¨¨´¡›r…°Š 432 – 257
œª‡·— Á…¸¥œ˜´ª˜´ÊŠÂ¨³˜´ª¨Ä®o®¨´„˜¦Š„´œ
4
3
2
–
2
5
7
…´Êœš¸É 1 Ĝ®¨´„®œnª¥˜´ª˜´ÊŠ‡º° 2 œo°¥„ªnµ˜´ª¨‡º° 7 ‹Îµœªœš­·…°Š 7 ‡º° 3 œÎµ 3 ª„„´ 2
ŗoÁž}œ 5 Ĝn°ŠŸ¨¨´¡›r Á…¸¥œ c Á®œº° 5 Ž¹ÉŠÁž}œ˜´ª¨Äœ®¨´„­·
4
3
2
5c
…°¥º¤
2
–
3+
7
5
26
…´Êœš¸É 2 Ĝ®¨´„­·˜´ª˜´ÊŠ‡º° 3 œo°¥„ªnµ˜´ª¨‡º° 5c Ž¹ÉŠ‡º° 6 ‹Îµœªœš­·…°Š 6 ‡º° 4 œÎµ 4 ª„
„´ 3 ŗoÁž}œ 7 Á…¸¥œ 7 Ĝn°ŠŸ¨¨´¡›r Á…¸¥œ c Á®œº° 2 Ž¹ÉŠÁž}œ˜´ª¨Äœ®¨´„¦o°¥
4
2c
…°¥º¤
3
4+
5c
7
2
3+ –
7
-
5
…´Êœš¸É 3 Ĝ®¨´„¦o°¥ ˜´ª˜´ÊŠ‡º° 4 ˜´ª¨ ‡º° 2c ®¦º°c 3 ‹³Å—o 4 – 3 = 1 Ä­n 1 Ĝn°ŠŸ¨¨´¡›r
4
2c
3
4+
5c
2
3+ –
7
1
7
5
-
—´Šœ´Êœ 432 – 257 = 175
®¤µ¥Á®˜» Ĝ…´Êœš¸É 2 ‹Îµœªœš­·…°Š 5c ‡º° ‹Îµœªœš­·…°Š 6 Ž¹ÉŠ‹Îµœªœš­·…°Š 5c ‡º° 4
°µ‹‹³„¨nµªªnµ ‹ÎµœªœšÁ„oµ…°Š 5 ‡º° 4 „Èŗo
­Îµ®¦´„µ¦¨š¸É¤¸„µ¦…°¥º¤®¨µ¥®¨´„˜·—„´œ ®¨´„š¸É¤„¸ µ¦…°¥º¤®¨´„…ªµ­»—čo‹Îµœªœš­·…°Š
˜´ª¨ ­nªœ®¨´„š¸É¤¸„µ¦…°¥º¤®¨´„°ºÉœÇ ™´—ÅžšµŠŽoµ¥Äo‹ÎµœªœšÁ„oµ…°Š˜´ª¨Åžª„„´˜´ª˜´ÊŠš¸É˜¦Š
®¨´„Á—¸¥ª„´œ„´˜´ª¨ ®¨´Š‹µ„œ´ÊœÄ­n c Á®œº°˜´ªÁ¨…®¨´„™´—ÅžšµŠŽoµ¥š¸ÉŤn¤¸„µ¦…°¥º¤Â¨oªšÎµ„µ¦¨
˜µ¤ž„˜·
‹µ„˜´ª°¥nµŠš¸É 1.1.21
4
3
2
2
5
7
–
˜´ª¨‡º° 257 ¨³¤¸„µ¦…°¥º¤Äœ®¨´„®œnª¥Â¨³®¨´„­·˜·—„´œ
­nªœ®¨´„¦o°¥Å¤n¤¸„µ¦…°¥º¤ Ĝ®¨´„®œnª¥‹Îµœªœš­·…°Š 7 ‡º° 3 Ĝ ®¨´„­·‹ÎµœªœšÁ„oµ…°Š 5 ‡º°
4 ­Îµ®¦´®¨´„¦o°¥Å¤n¤¸„µ¦…°¥º¤Á˜·¤ c œ 2 Ĝ®¨´„¦o°¥ Ž¹ÉŠ 2c ‡ºc° 3 ¨oªšÎµ„µ¦¨ž„˜· Ĝ®¨´„š¸ÉŤn¤¸
„µ¦…°¥º¤‡º° ®¨´„¦o°¥ ­nªœ®¨´„®œnª¥Â¨³®¨´„­·ª„—oª¥‹Îµœªœš­·Â¨³‹ÎµœªœšÁ„oµ…°Š˜´ª¨
˜µ¤¨Îµ—´
27
°›·µ¥„µ¦Âž¨Š˜´ª¨ 257 —´Šœ¸Ê
– 257 = (–2) u 102 + ((–5) u 10) + (–7)
= ((–2) u 102) + (–10 + 5) u 10 + (–10 + 3)
= ((–2) u 102) + ((–10) u 10) + (5 u 10) + ((–1) u 10) + 3
= ((–2) u 102) + ((–1) u 102) + ((5 – 1) u 10) + 3
= ((–3) u 102) + (4 u 10) + 3
š­·…°Š 7
šÁ„oµ…°Š 5
¡·‹µ¦–µ˜´ª°¥nµŠš¸É 1.1.21
…°¥º¤ …°¥º¤
4
3
2 –
+
+
4
3
2c
5
7
7
5
ŗo‹µ„ 3 (‹Îµœªœš­·…°Š 7)
ª„„´ 2 Ž¹ÉŠÁž}œ˜´ª˜´ÊŠ
ŗo‹µ„ 4 (‹ÎµœªœšÁ„oµ…°Š 5)
ª„„´ 3 Ž¹ÉŠÁž}œ˜´ª˜´ÊŠ
°¥nµŠÅ¦„Șµ¤ c œ 2 Ĝ®¨´„¦o°¥ ‹ÎµÁž}œ˜o°Š¤¸ÁœºÉ°Š‹µ„Äœ®¨´„¦o°¥Å¤nčn„¦–¸š¸É˜ª´ ˜´ÊŠ‡nµ¤¸œo°¥
„ªnµ˜´ª¨
˜´ª°¥nµŠš¸É 1.1.22 ‹Š®µŸ¨¨…°Š
8
2
5
4
8
7
–
„
œª‡·—
8
4c
…°¥º¤ …°¥º¤
2
5
+
1
3+ –
8
7
3
8
28
„µ¦¨œ¸Ê¤¸„µ¦…°¥º¤­°Š®¨´„˜·—„´œ ‡º° ®¨´„®œnª¥Â¨³®¨´„­· Ĝ®¨´„®œnª¥˜´ª¨‡º° 7
‹Îµœªœš­·…°Š 7 ‡º° 3 œÎµ 3 ޝª„„´ 5 Ž¹ÉŠÁž}œ˜´ª˜´ÊŠ ŗoŸ¨¨´¡›rÁž}œ 8 Ĝ®¨´„­· ˜´ª¨‡º° 8
‹ÎµœªœšÁ„oµ…°Š 8 ‡º° 1 œÎµ 1 ޝª„„´ 2 Ž¹ÉŠÁž}œ˜´ª˜´ÊŠ ŗoŸ¨¨´¡›rÁž}œ 3
8
4c
2
5
+
3+
1
8
7
3
3
8
®¨´„¦o°¥Å¤n¤„¸ µ¦…°¥º¤Ä­n c Á®œº° 4 Ž¹ÉŠÁž}œ˜´ª¨Äœ®¨´„¦o°¥ Ž¹ÉŠ 4c ‡º° 5 „µ¦¨Å¤n¤¸„µ¦
…°¥º¤‹¹ŠœÎµ 5 ި„´ 8 ŗo 3 —´Šœ´ÊœŸ¨¨´¡›r‡º° 3 3 8
˜´ª°¥nµŠš¸É 1.1.23 ‹Š®µŸ¨¨
8
4
6
8
3
6
9
2
–
„
œª‡·— „µ¦¨¤¸„µ¦…°¥º¤Äœ®¨´„­·Ž¹ÉŠ˜´ª¨‡º° 9 ¨³®¨´„¦o°¥Ž¹ÉŠ˜´ª¨‡º° 6 ‹¹ŠÄo‹Îµœªœš­·…°Š
9 ‡º° 1 ¨³‹ÎµœªœšÁ„oµ…°Š 6 ‡º° 3 ޝª„ …–³š¸É 2 Ĝ®¨´„®œnª¥šÎµ„µ¦¨˜µ¤ž„˜· ­nªœ®¨´„¡´œ
Áž¨¸É¥œ 3 Áž}œ 3c š¸ÉÁšnµ„´ 4 ¨oªšÎµ„µ¦¨ž„˜· Ĝ®¨´„¡´œ —´Šœ¸Ê
8–
3c
4
…°¥º
4 ¤ …°¥º
6 ¤ 8–
1+
3+
6
9
2
7
7
6
˜´ª°¥nµŠš¸É 1.1.24 ‹Š®µŸ¨¨…°Š
8
2
4
5
4
7–
5
1
8
6
1
9
œª‡·— …´Êœš¸É 1 Ĝ®¨´„®œnª¥ ˜´ª˜´ÊŠ‡º° 7 œo°¥„ªnµ˜´ª¨‡º° 9 œÎµ 1 ‹Îµœªœš­·…°Š 9 ª„„´ 7 ŗo
8 ‹¹ŠÄ­n 8 š¸Én°ŠŸ¨¨´¡›r œ°„‹µ„œ¸ÄÊ œ®¨´„­· ˜´ª˜´ÊŠ‡º° 4 ¤µ„„ªnµ˜´ª¨‡º° 1 ŤnÁ„·—„µ¦…°¥º¤‹¹ŠÄ­n c Ūo
Á®œº° 1 Ž¹ÉŠÁž}œ˜´ª¨Äœ®¨´„­· ‹³Å—o 4 –1c = 4 – 2 = 2 Ä­n 2 Ĝn°ŠŸ¨¨´¡›r…°Š®¨´„­·
29
…°¥º¤
7–
1+
1
8
6
9
2
8
…´Êœš¸É 2 Ĝ®¨´„¦o°¥ ¨³®¨´„¡´œ ˜´ª˜´ÊŠœo°¥„ªnµ˜´ª¨˜·—„´œ­°Š®¨´„ —´Šœ´ÊœÄœ®¨´„¦o°¥Äo„µ¦
ª„—oª¥‹Îµœªœš­·…°Š˜´ª¨ ¨³Äœ®¨´„¡´œÄo„µ¦ª„—oª¥‹ÎµœªœšÁ„oµ…°Š˜´ª¨ œ°„‹µ„œ¸Ê —¼
™´—ŞĜ®¨´„®¤ºÉœ ˜´ª˜´ÊŠ¤¸‡nµ¤µ„„ªnµ˜´ª¨ ‹¹ŠÄ­n c Á®œº° 1 Ž¹ÉŠÁž}œ˜´ª¨ ¨oªšÎµ„µ¦‡·—‡Îµœª–Å—o
Ÿ¨¨´¡›r—´Šœ¸Ê
…°¥º¤ …°¥º¤
8
2
4
5
4
7
+
+
– 1+
–
4
1
c
5c
1c
8
6
1c
9
8
5c
2
4
5
4
–
c1
0
5
9
2
8
…´Êœš¸É 3 Ĝ®¨´„­œšÎµ„µ¦¨›¦¦¤—µ Á¡¦µ³Äœ®¨´„®¤ºœÉ Ťn¤¸„µ¦…°¥º¤‹ÎµœªœÄœ®¨´„­œ ‹³
ŗo 8 – 5 = 3 Ä­n 3 Ĝn°ŠŸ¨¨´¡›r
šÁ„oµ‡º° 1
8
–
c5
2
–
c1
4
1+
8
5
4+
6
4
–
c1
3
0
5
9
2
œ´Éœ‡º° 824547 – 518619 = 305928
7
1+
9
„ š­· ‡º° 1 ¨³ 4
˜µ¤¨Îµ—´‹µ„
8
…ªµÅžŽoµ¥
˜´ª°¥nµŠš¸É 1.1.25 ‹Š®µ‡nµ…°Š 1438 - 3165
œª‡·— „¦–¸œ¸Ê˜´ª˜´ÊŠ‡º° 1438 œo°¥„ªnµ˜´ª¨‡º° 3165 ‹¹Š‹³®µ‡nµ…°Š 3165 – 1438 ŗoŸ¨¨´¡›rÁž}œ
ÁšnµÄ— ‹³Ä­nÁ‡¦ºÉ°Š®¤µ¥ – ®œoµŸ¨¨´¡›r „È‹³Å—oŸ¨¨˜µ¤˜o°Š„µ¦
c
3
-_
c
1c
1
1
6+
4
7
6
-_
c
3c
5–
2c+
8
2
7
—´Šœ´Êœ 1438 – 3165 = –1727
30
š­¦»ž Ĝ„µ¦¨›¦¦¤—µš¸É¤¸„µ¦…°¥º¤ „´„µ¦¨Ã—¥ª·›¸š­· ¨³Ä­n c œ˜´ª¨Äœ®¨´„™´—Åžœ´Êœ
­°—‡¨o°Š„´œ—´Š‹³°›·µ¥Ã—¥˜´ª°¥nµŠ „µ¦¨˜n°Åžœ¸Ê
„µ¦¨Â…°¥º¤
3
4
1
3
2
8
1
5
°›·µ¥Å—o—´Šœ¸Ê 43
=
30 + 13
43 – 28
=
(30 + 13) – 28
=
(30 + 13) – (20 + 8)
=
(30 – 20) + (13 – 8)
=
10 + 5
=
15
4
3
2
8
—´Šœ´Êœ
–
–
„µ¦¨Âš­·Â¨³Ä­n c
4
3
2
8
–
°›·µ¥Å—o—´Šœ¸Ê 28
–28
—´Šœ´Êœ 43 – 28
4
–
c2c
1
=
=
=
=
=
=
=
3
2+ –
8
5
30 – 2
– (30 – 2) = –30 + 2
43 – 30 + 2
(40 + 3) – 30 + 2
(40 – 30) + (3 + 2) .......(*)
10 + 5
15
31
Ĝ¦¦š´— (*) ‡º°
4
2c
3
2+ –
8
‹Îµœªœš­·…°Š 8 ‡º° 2 œÎµÅž
ª„„´ 3 ŗo 5
༡ 40 Р20 c= 40 Р30
š­¦»ž 1. Ĝ„¦–¸š„¸É µ¦¨¤¸„µ¦…°¥º¤ ™oµÄoª„—oª¥‹Îµœªœš­·Â¨³Ä­n c Á®œº°˜´ª¨Äœ®¨´„™´—Åž
‹³­³—ª„„ªnµ„µ¦¨Â…°¥º¤›¦¦¤—µ
2. ­Îµ®¦´Ÿ¼oš¸É™œ´—„µ¦¨Â…°¥º¤°µ‹‹³šÎµ„µ¦¨Ã—¥Å¤n˜o°ŠÄo„µ¦ª„—oª¥‹Îµœªœš­·„È
ŗo ˜nšœš¸É‹³¨—‡nµ˜´ª˜´ÊŠÄœ®¨´„™´—Åž¨Š®œ¹ÉŠ Áž¨¸É¥œÁž}œ‡Š˜´ª˜´ÊŠ Ĝ®¨´„™´—ŞŪo ¨oªÄ­n c Á®œº°˜´ª
¨Äœ®¨´„™´—Åž‹³­³—ª„„ªnµ Ánœ
4
3
2
8
–
4
2c
1
3
8
5
–
Á¦ºÉ°Šš¸É 5 „µ¦¨š¸ÉŤn˜o°Š‡Îµœ¹Š™¹Š„µ¦…°¥º¤
œ°„‹µ„„µ¦¨š¸É˜o°Š‡Îµœ¹Š™¹Š™¹Š„µ¦…°¥º¤Â¨oª°µ‹‹³ šÎµ„µ¦¨š¸ÉŤn˜o°Š‡Îµœ¹Š™¹Š„µ¦…°¥º¤„Èŗo
¤¸®¨´„„µ¦‡·——´Šœ¸Ê
Á¤ºÉ°œÎµ˜´ª¨¤µ¡·‹µ¦–µ ¤¸…œ´Ê ˜°œ„µ¦Áž¨¸¥É œ˜´ª¨—´Šœ¸Ê
…´Êœš¸É 1 ˜´ª¨Äœ®¨´„®œnª¥Áž¨¸É¥œÁž}œ‹Îµœªœš­·…°Š˜´ª¨Äœ®¨´„®œnª¥œ´œÊ
…´Êœš¸É 2 ˜´ª¨š¸ÉÁ®¨º°Áž¨¸É¥œÁž}œ‹ÎµœªœšÁ„oµ …°Š˜´ª¨®¨´„œ´ÊœÇ
_
_
…´Êœš¸É 3 Á¤ºÉ°®¤—˜´ª¨Â¨oªÁ¡·É¤˜´ªÁ¨…™´—ÅžšµŠŽoµ¥°¸„®œ¹ÉŠ®¨´„ ץĭn 1 Áž}œ˜´ªª„
( 1 ®¤µ¥™¹Š –1 Ĝ®¨´„š¸ÁÉ ¡·É¤…¹ÊœÄ®¤nœœ´Ê )
Á¤ºÉ°Áž¨¸É¥œÂž¨Š˜´ª¨Á­¦È‹Á¦¸¥¦o°¥Â¨oªœÎµÅžª„„´˜´ª˜´ÊŠ„È‹³Å—oŸ¨¨´¡›r˜µ¤˜o°Š„µ¦ ­Îµ®¦´
_
®¨´„Žoµ¥­»—œ´œÊ „µ¦ª„—oª¥ 1 „ȇº° „µ¦¨—oª¥ 1 Ĝ®¨´„Žoµ¥­»—œ´ÉœÁ°Š
„µ¦Âž¨Š˜´ª¨Ã—¥ª·›¸„µ¦œ¸°Ê ›·µ¥Å—o—´Šœ¸Ê
­¤¤˜·˜´ª¨ ‡º° 4 7 8 6 ž¨ŠÅ—o—´Šœ¸Ê
32
4 7 8 6
_
1 5 2 1 4
_
1 ‡º° –1 ‹ÎµœªœšÁ„oµ ‹Îµœªœš­·Äœ®¨´„®¤ºÉœ
_
‹³Á®Èœªnµ 4 7 8 6 ž¨ŠÁž}œ 1 5 2 1 4
¡·‹µ¦–µ˜´ª¨ 4 7 8 6 ‡º°
–4 7 8 6
=
–10000 + 5214
=
(–1) u 104 + 5214
=
1 5214
_
˜´ª°¥nµŠš¸É 1.1.26 ‹Š®µŸ¨¨…°Š
8 4 1 6 3
–
4 7 2 3 8
„
8 4 1 6 3
œª‡·—
4 7 2 3 8
–
Áž¨¸É¥œÁž}œ
1 5 2 7 6 2
„
šÎµ„µ¦ª„
˜µ¤ž„˜·Ã—¥
_
_
8 4 1 6 3
x
x
15 2 7 6 2
_
8 4 1 6 3
+
„
+
1 ®¤µ¥™¹Š –1 0 3 6 9 2 5
Ĝ®¨´„­œ
®¤µ¥Á®˜»
x
_
Á®œº° 5 Ĝ®¨´„®¤ºÉœ ®œ¹ŠÉ x „ȇº°„µ¦š— 1 Ş®¨´„­œŽ¹ÉŠ®´„¨oµŠ„´ 1 Ĝ®¨´„­œ
¡°—¸ Ÿ¨¨´¡›rĜ®¨´„­œ‹¹ŠÁž}œ 0 ‹³Å—oŸ¨¨´¡›rÁž}œ 3 6 9 2 5 œ´Éœ‡º° ŗo„µ¦¨—´Šœ¸Ê
8 4 1 6 3
4 7 2 3 8
3 6 9 2 5
–
33
˜´ª°¥nµŠš¸É 1.1.27 ‹Š®µŸ¨¨…°Š
8 6 7 3 5
4 8 2 1
1 4 7 8 6
–
8 6 7 3 5
Áž¨¸É¥œÁž}œ
–
+
8 5 21 4 +
_
_
1
1 5 1 7 9
„
_
„
šÎµ„µ¦ª„ž„˜·¥„Áªoœ 1 ‡º° –1 Ĝ®¨´„œ´œÊ
8 6 7 3 5
_
_
x
x
x
15 1 7 9
x
x
1 8 5 2 1 4
+
+
6 7 1 2 8
Ÿ¨¨´¡›r ‡º° 6 7 1 2 8
ʄ
_
x
Ĝ®¨´„¡´œ‡º° „µ¦š— 1 Ş®¨´„®¤ºœÉ ®´„¨oµŠ„´ 1 Ĝ®¨´„®¤ºÉœ œ°„‹µ„œ¸Ê x Ĝ®¨´„®¤ºÉœ‡º°
_
„µ¦š— 1 Ş®¨´„­œ ®´„¨oµŠ„´ 1 Ĝ®¨´„­œ
_
Ĝ„¦–¸š¸ÉŸ¨¨´¡›r¤¸‡nµ˜·—¨®¨´„Žoµ¥­»—…°ŠŸ¨¨´¡›r‹³°¥¼nĜ¦¼ž a Á¤ºÉ° a Áž}œÁ¨…×—š¸É
_ _
¤µ„„ªnµ 0 Ánœ 1 , 2 Áž}œ˜oœ Ĝ„µ¦Âž¨Š„¨´Áž}œ‹Îµœªœ¨„Țεŗo×¥Ÿ¨¨´¡›r®¨´„®œnª¥Áž¨¸É¥œÁž}œ
_
_
‹Îµœªœš­· ®¨´„™´—¤µšµŠŽoµ¥Áž¨¸É¥œÁž}œ‹ÎµœªœšÁ„oµÁ¤ºÉ°Áž¨¸É¥œ¤µ™¹Š 1 Ä®oÁ°µ 1 °°„¨oªÄ­n
Á‡¦ºÉ°Š®¤µ¥ – šœ Ž¹ÉŠÁ‡¦ºÉ°Š®¤µ¥ – ‹³°¥¼®n œoµ‹ÎµœªœšÁ„oµÂ¨³š­·š¸ÉÁž¨¸É¥œ¤µ„n°œÂ¨oªŸ¨š¸Éŗo
¡¦o°¤Á‡¦ºÉ°Š®¤µ¥ – „ȇº° ‹Îµœªœ¨š¸ÉÁž}œŸ¨¨´¡›rœ´ÉœÁ°Š
_
_
Ánœ Ÿ¨¨´¡›r‡º° 1 8 2 1 Áž¨¸É¥œÁž}œŸ¨¨´¡›rš¸ÉŤn¤¸ 1 —´Šœ¸Ê
_
1 8 2 1
– 1 7 9
šÁ„oµ š­·
_
(Á¡¦µ³ 1 8 2 1 ‡º° –1000 + 821 = –179)
_
_
™oµŸ¨¨´¡›r‡º° 2 8 3 7 Ä®ošÎµ‡¨oµ¥„´„µ¦Âž¨Š 1 8 2 1
_
˜„˜nµŠÁŒ¡µ³ 2 Áž¨¸É¥œÁž}œ –1 —´Šœ¸Ê
34
_
2 8 3 7
= – 2000 + 837
= – 1000 + (–1000 + 837)
= – 1000 + (–163)
= – 1163
_
2 8 3 7
–1 1 6 3
šÁ„oµ š­·
_
_
¨³™oµŸ¨¨´¡›rÁž}œ 3 7 6 4 „Èž¨Š‡¨oµ¥˜´ª°¥nµŠ …oµŠ˜oœÂ˜n 2 Áž¨¸É¥œÁž}œ –2
˜´ª°¥nµŠš¸É 1.1.28 ‹Š®µ‡nµ…°Š
3 1 4 5
3 8 1 9
–
„
ª·›¸‡·—
3 1 4 5 _ Áž¨¸É¥œÁž}œ 3 1 4 5
+
_
x
3 8 1 9
1 6 1 8 1
¡·‹µ¦–µ
„
_
_
1 9 3 2 6
1 9 3 2 6 = – 0674 = – 674
—´Šœ´ÊœŸ¨¨´¡›r‡º° – 674
®¤µ¥Á®˜»
_
1 9 3 2 6 = – 0 6 7 4 ­—Š—´ŠÂŸœŸ´Š˜n°Åžœ¸Ê
_
1 9 3 2 6
_
Áž¨¸É¥œ 1 Áž}œ – 0 6 7 4
šÁ„oµ š­·
35
˜´ª°¥nµŠš¸É 1.1.29 ‹Š®µ‡nµ…°Š
8 9 6
–
9 4 3 –
9 2 7
–
6 8 1
„
œª‡·—
8 9 6
9 4 3
9 2 7
6 8 1
Áž¨¸É¥œÁž}œ
x
x
8 9 6
–
+
+
_
1 0 7 3
_
+
„
2 3 4 5
–
_
x
1 0 5 7
–
x
x
1 3 1 9
_
_
2 3 4 5
¡·‹µ¦–µ„µ¦Âž¨Š
_
2 3 4 5
‹³Å—o
–1 6 5 5
—´Šœ´ÊœŸ¨¨´¡›r‡º° –1655
˜´ª°¥nµŠš¸É 1.1.30
8
2
9
2
‹Š®µ‡nµ…°Š
1 4 1 3
1 3 1 2+
–
2 1 4 6–
3 4 3 2
„
œª‡·— ‹µ„Ëš¥rÁž¨¸É¥œ¦¼žÁž}œ
8 1 4 1 3
+
_
1 0 7 8 5 4+
_
+
1 7 6 5 6 8
x
2 1 3 1 2
x
x
x
_
x
x
1 8 7 1 4 7 = –12853
36
Á¦ºÉ°Šš¸É 6 „µ¦®µŸ¨¨Â¦ª¥°—
¡ºÊœ“µœ„µ¦®µŸ¨¨Â¦ª¥°—œ´œÊ Áž}œ¡ºÊœ“µœ…°Š­¤„µ¦—´Šœ¸Ê ‡º° ­Îµ®¦´‹ÎµœªœÁ˜È¤ª„
a, b ¨³ c Ä—Ç ‹³¡ªnµ a – b = c „Șn°Á¤ºÉ° a = b + c Ánœ 8 – 5 = 3 „Șn°Á¤ºÉ° 8 = 5 + 3
™oµÅ¤nš¦µ‡nµ c °µ‹Á…¸¥œ šœ c —´Šœ¸Ê
a – b = „Șn°Á¤ºÉ° a = b + ‹µ„¦¼ž­¤„µ¦¨´„¬–³œ¸Ê ‹¹Š‡ª¦ f„„µ¦®µ‡nµ‹Îµœªœš¸Éšœ Á­¸¥„n°œ œ°„‹µ„œ¸Ê°µ‹¤¸„¦–¸š¸É
Ÿ¨ª„‹ÎµœªœšµŠŽoµ¥Â¨³…ªµ…°Š­¤„µ¦Å¤nÁšnµ„´œ „È‹³šÎµÄ®o„¨µ¥Áž}œ°­¤„µ¦ ™¹ŠÂ¤oªnµ¤¸˜´ªÁ¨…Äœ
®¨´„®œnª¥…°ŠŸ¨ª„Ášnµ„´œ„Șµ¤ ‹¹Š‡ª¦ f„®µ‹Îµœªœš¸Éšœ Ĝ°­¤„µ¦ ®¦º°­¤„µ¦„n°œ—´Šœ¸Ê
˜´ª°¥nµŠš¸É 1.1.31 ‹Š®µ‹ÎµœªœÁ˜È¤ª„š¸Éœo°¥š¸É­»—®¦º° 0 š¸ÉÁ˜·¤¨ŠÄœ„¦° ¨oªšÎµÄ®oŸ¨ª„
šµŠŽoµ¥Â¨³Ÿ¨ª„…ªµ¤¸‡nµ®¨´„®œnª¥Ášnµ„´œ ¨³Á…¸¥œ < , > ®¦º° = ™oµŸ¨ª„…°Š‹ÎµœªœšµŠŽoµ¥
œo°¥„ªnµ, ¤µ„„ªnµ ®¦º°Ášnµ„´ Ÿ¨ª„…°Š‹ÎµœªœšµŠ…ªµ ˜µ¤¨Îµ—´ ¨ŠÄœn°ŠªnµŠ šµŠ—oµœ…ªµ Ž¹ÉŠ˜¦Š
®´ª˜µ¦µŠÁ…¸¥œ < , > , = Ūo ¨³Á…¸¥œ x ­—Š„µ¦¤µ„„ªnµ°¥¼nÁšnµÅ¦ šµŠ—oµœš¸É¤µ„„ªnµ (x ¤µ„„ªnµ°¥¼n 10,
xx
¤µ„„ªnµ°¥¼n 20)
Žoµ¥
…ªµ
<,>,=
1+2+3
3+4+ 9
x
3+4+ 8
5+9+1
=
9+6+3+9
3+1+ 3
3+ 7 +8
4+5+9
8+8+9+9
3 +8+4+9
x
3+7+1
9+9+ 8 +5
xx
3+4+5+ 6
1+3+6+8
=
3+7+8+2
9+9+2+ 0
=
xx
!
=
!
37
˜´ª°¥nµŠš¸É 1.1.32 ‹Š®µ‡nµ…°Š
4
6
3
2
–
d
œª‡·— a :
4
6
b:
3
2
c:
1
4d
–
Įo
a
= 46
b
= 32
c
= a–b
a
= b+c
‹³Å—o
ÁœºÉ°Š‹µ„ 46 = 32 + 14
—´Šœ´Êœ
c
= 14
˜´ª°¥nµŠš¸É 1.1.33 ‹Š®µ‡nµ…°Š
3
1
8
2
4
7
–
„
œª‡·—
a:3
1
8_
b:2
4
7
b = 247
C:
7
x
1„
c = a–b
Įo
a = 318
‹³Å—o a = b + c
®¨´„®œnª¥…°Š a ‡º° 8 ®¨´„®œnª¥…°Š b ‡º° 7 ‹³Å—o®¨´„®œnª¥…°Š c ‡º° 1 ®¨´„­·…°Š a ‡º° 1
®¨´„­·…°Š b ‡º° 4 ˜o°Š®µ®¨´„­·…°Š c š¸ÉšÎµÄ®oŸ¨ª„…°Š˜´ªÁ¨…Äœ®¨´„­·…°Š b + c ¨Ššoµ¥ —oª¥ 1
Ž¹ÉŠ 1 Áž}œ˜´ªÁ¨…Äœ®¨´„­·…°Š a ‹³Á®ÈœÅ—oªnµ®¨´„­·…°Š c ˜o°ŠÁž}œ 7 ˜n 4 + 7 = 11 Ž¹ÉŠ¤µ„„ªnµ 1 °¥¼n
10 ‹¹Š¤¸„µ¦š—Ÿ¨ª„Äœ®¨´„­·…°Š b + c Ş¥´Š®¨´„¦o°¥ Ĝš¸œÉ ¸Ê‹³Á…¸¥œ x Á®œº° 7
®¨´„¦o°¥…°Š a ‡º° 3 ®¨´„¦o°¥…°Š b ‡º° 2 ‹µ„Ÿ¨ª„®¨´„­·…°Š b + c ¤¸„µ¦š— 1 Ş®¨´„¦o°¥
‹³Å—o 2 + 1 = 3 Ž¹ÉŠÁž}œ˜´ªÁ¨…Äœ®¨´„¦o°¥ …°Š a ¡°—¸ ‹¹ŠÅ¤n˜o°ŠÁ…¸¥œ˜´ªÁ¨…Äœ®¨´„¦o°¥…°Š c ‹¹ŠÅ—o c
Ášnµ„´ 71 Áž}œ‡Îµ˜°
38
˜´ª°¥nµŠš¸É 1.1.34 ‹Š®µ‡nµ…°Š
4
8
1
5
2
9
3
1
–
d
œª‡·— „a : 4
8
1
5
b:2
9
3
1
c:1
8
x
8
x
4
–
Ĝ„¦–¸š¸É a, b ¨³ c Áž}œ‹ÎµœªœÁ˜È¤ª„ ™oµ a < b ‹³Å—o a – b Áž}œ‹ÎµœªœÁ˜È¤¨°µ‹Á…¸¥œÂšœŸ¨¨
…°Š a – b —oª¥ – c œ´Éœ‡º° a – b = – c ‹³¡ªnµ a + c = b ‹¹Š„¨nµªÅ—oªnµ a – b = – c
„Șn°Á¤ºÉ° a + c = b
—´Šœ´Êœ„µ¦®µ‡nµ – c šÎµÅ—o×¥®µ‡nµ c š¸ÉœµÎ ¤µª„„´ a ¨oªÅ—oŸ¨ª„Ášnµ„´ b ®¨´Š‹µ„œ´ÊœÄ­n
Á‡¦ºÉ°Š®¤µ¥ – ®œoµ c „È‹³Å—o – c Áž}œ‡Îµ˜°
˜´ª°¥nµŠš¸É 1.1.35 ‹Š®µ‡nµ…°Š
2
9
3
1
4
8
1
5
–
d
œª‡·—„
a:2
9
3
b:4
8
1
c:1
8
x
8
x
1
-5 –
4
‡Îµ˜° ‡º° –1884
Á¦ºÉ°Šš¸É 7 „µ¦®µŸ¨ª„ – ¨‡¨³„´œÂ¦ª¥°—
­Îµ®¦´‹ÎµœªœÁ˜È¤ª„ a, b, c ¨³ d ‹³¡ªnµ
a + b – c = d „Șn°Á¤ºÉ° a + b = c + d
39
a – b + c = d „Șn°Á¤ºÉ° a + c = b + d
ÁœºÉ°Š‹µ„ d Áž}œŸ¨¨´¡›r…°Š„µ¦ª„ – ¨ ‡¨³„´œ ¨³ d Áž}œ‹Îµœªœª„ ‹¹ŠœÎµ d ޝª„¦ª¤
„´˜´ª¨š´ÊŠ®¤— Â¥„Ūo¡ª„®œ¹ÉŠ ¨oªšÎµ„µ¦®µ‡nµ d Ž¹ÉŠÁž}œŸ¨¨´¡›r˜µ¤˜o°Š„µ¦ÄœšµŠž’·´˜· ‹³®µ
Ÿ¨ª„…°Š ˜´ª˜´ÊŠÂ¨³˜´ªª„„n°œ ¨oª‹¹Š®µ‡nµ d š¸ÉšÎµÄ®oŸ¨ª„…°Š d „´˜´ª¨š´ÊŠ®¤—¤¸‡nµÁšnµ„´‡nµ
…°ŠŸ¨ª„…°Š¡ª„¦„ —´Š˜´ª°¥nµŠ ˜n°Åžœ¸Ê
˜´ª°¥nµŠš¸É 1.1.36 ‹Š®µ‡nµ…°Š
4
8
7
5
2
1
7
9
9
+
–
„„
œª‡·—
Á…¸¥œÄ®¤nÁž}œ a : 4
8
7
b:5
2
1
c:7
9
9
+
–
d:2 0 9
®µ‡nµ d š¸ÉšÎµÄ®o c + d = a + b ÁœºÉ°Š‹µ„Ÿ¨ª„®¨´„®œnª¥ …°Š a + b = 8 ‹¹Š˜o°ŠšÎµÄ®oŸ¨ª„
®¨´„®œnª¥…°Š c + d ¨Ššoµ¥—oª¥ 8 ´Š‡´ Ä®o®¨´„®œnª¥…°Š d Áž}œ 9 ˜n®¨´„®œnª¥…°Š c „ȇº° 9 Ž¹ÉŠ
9 + 9 ¤µ„„ªnµ 7 + 1 °¥¼n 10 ‹¹ŠÁ…¸¥œ‹»—œ 9 š¸É®¨´„®œnª¥…°Š d ‹»—œ´Êœ‹³š—ŞĜ®¨´„­·…°Š c + d
¡·‹µ¦–µ®¨´„­·…°Š a + b ‹³Å—o 8 + 2 = 10 ‹¹Š˜o°ŠšÎµÄ®o®¨´„­·…°Š c + d Ášnµ„´ 10 ˜n®¨´„
­·…°Š c ‡º° 9 ¨³¤¸„µ¦š—…°Š c + d ‹µ„®¨´„®œnª¥¤µ 1 ‹»— ŗo 9 + 1 = 10 —´Šœ´Êœ ®¨´„­·…°Š d
‹¹ŠÁž}œ 0
¡·‹µ¦–µ®¨´„¦o°¥…°Š a + b ‹³Å—o 4 + 5 = 9 ÁœºÉ°Š‹µ„®¨´„¦o°¥…°Š c ‡º° 7 ‹¹ŠšÎµÄ®o®¨´„¦o°¥
…°Š d Áž}œ 2 ‹³šÎµÄ®o 4 + 5 = 7 + 2
‡Îµ˜°‹¹ŠÁž}œ 209
˜´ª°¥nµŠš¸É 1.1.37 ‹Š®µ‡nµ…°Š
x
3
8
5
9–
2
9
8
7
1
2
3
5
+
„
40
œª‡·—
a :3 8 5 9
Á…¸¥œÄ®¤nÁž}œ
–
b :2 9 8 7
c :1 2 3
+
‹µ„Ëš¥rœ¸Ê˜o°Š®µ‡nµ d
š¸ÉšÎµÄ®o a + c = b + d
5
d :2 1 0 7
®¤µ¥Á®˜» ®¨´„®œnª¥…°Š d Ťn˜o°ŠÁ˜·¤‹»—Á¡¦µ³Ÿ¨ª„®¨´„®œnª¥…°Š a + c ‡º° 9 + 5 = 14 ¨³
Ÿ¨ª„®¨´„®œnª¥…°Š b + d ‡º° 7 + 7 = 14 Ž¹ÉŠ 9 + 5 = 7 + 7 ‹¹ŠÅ¤n¤¸„µ¦š—‡nµš¸É
˜nµŠ„´œÅž®¨´„­·
˜´ª°¥nµŠš¸É 1.1.38
9
7
1
5
‹Š®µ‡nµ…°Š
8 5 7 5
+
9 7 4 2–
2 3 4 5
–
6 7 8 9
—
œª‡·—
Á…¸¥œÄ®¤nÁž}œ a :
b:
c:
d:
9
x
7
1
5
8
9
2
6
5
7
3
7
7
4
4
8
5
+
2
–
5
–
9
e: 1 0 9 1 8 3
x
x
Ĝ„µ¦‡Îµœª–‡nµ ˜´ÊŠÂ˜n®¨´„®œnª¥™¹Š®¨´„¡´œ ‡¨oµ¥„´˜´ª°¥nµŠš¸É 1.1.38 „¨nµª‡º° ™oµ¤¸„µ¦Ä­n‹—»
‹³Ä­n‹—» š¸É»—š¸ÉŸ¨ª„š¸É¤µ„„ªnµ „¦–¸®¨´„®¤ºÉœ¡ªnµ®¨´„®¤ºÉœ…°Š a + b = 16 …–³š¸®É ¨´„®¤ºÉœ…°Š
c + d + e ‡º° 1 + 5 + 0 = 6 ‹³Á®Èœªnµ 9 + 7 ¤µ„„ªnµ 1 + 5 + 0 °¥¼n 10 ‹¹ŠÁ…¸¥œ‹»—š¸É 7 Ž¹ÉŠÁž}œ˜´ªª„
…°Š 9 + 7 ¨oª‹¹Šš—‹»—¨Š¤µÁž}œ 1 Ĝ®¨´„­œ…°ŠŸ¨¨´¡›r
Ĝ„¦–¸š¸É „µ¦ª„ – ¨ ‡¨³„´œ ¤¸Ÿ¨¨´¡›rÁž}œ‹Îµœªœ¨ ‹³˜o°ŠÂ¥„˜´ª¨š´ÊŠ®¤—¤µ¦ª¤Áž}œ
¡ª„Á—¸¥ª„´œ ¨³œÎµ˜´ª˜´ÊŠ ˜´ªª„¨³‡nµ­´¤¼¦–r…°ŠŸ¨¨´¡›r ¤µª„„´œÄ®o¤¸‡nµÁšnµ„´Ÿ¨¦ª¤…°Š˜´ªÁ¨…
(Ťn‡·—Á‡¦ºÉ°Š®¤µ¥®œoµŸ¨¨´¡›r) —´Š˜´ª°¥nµŠ ˜n°Åžœ¸Ê
41
˜´ª°¥nµŠš¸É 1.1.39 ‹Š®µ‡nµ…°Š
3
8
4
7
6
7
5
3
1
3
5
9
–
+
„
œª‡·—„¦–¸œ¸Ê Ÿ¨¨´¡›rÁž}œ‹Îµœªœ¨ ™oµÄ®o a = 3847, b = 6753, c = 1359 ¨³ d Áž}œ
‹ÎµœªœÁ˜È¤ª„Ž¹ÉŠ
a – b + c = – d ‹³Å—o a + c + d = b
œª‡·—
Á…¸¥œÄ®¤nÁž}œ
a:
3
8
4
7
b:
6
7
5
3
c:
1
3
5
9
d:
–
+
„
Ĝ„µ¦‡Îµœª–‡nµ a – b + c ¡ªnµ a – b + c < 0 —´Šœ´Êœ‡Îµ˜°Áž}œ‹ÎµœªœÁ˜È¤¨Ä®oÁž}œ – d ‹¹Š
œÎµ d ަª¤„´ a ¨³ c ŪoÁž}œ¡ª„®œ¹ÉŠ Â¥„ b Ūo°„¸ ¡ª„®œ¹ÉŠ‡Îµœª–‡nµ d š¸ÉšÎµÄ®o a + c + d = b
—´Šœ´Êœ
a:
b:
c:
d:
‡Îµ˜° ‡º° – d = – 1 5 4 7
˜´ª°¥nµŠš¸É 1.1.40 ‹Š®µ‡nµ…°Š
3
2
4
8
5
6
7
8
3
6
1
1
8
7
3
x
5
4
5
5
x
4
7
–
3
+
9
xx
7„
5
2
7
9
1
–
9
+
8
–
0
„
42
œª‡·—
Á…¸¥œÄ®¤nÁž}œ
a:
b:
c:
d:
e:
3
4
5
7
3
2
8
6
8
7
5
2
7
9
x
9
1
–
9
+
8
–
0
0
‡Îµ˜°‡º° – e = – 3 7 9 0
­œ»„„´˜´ªÁ¨… (2)
®oµ˜´ªÁ®¤º°œ
®„˜´ªÁ®¤º°œ
271 u 41 = 11111
3367 u 33 = 111111
271 u 82 = 22222
3367 u 66 = 222222
271 u 123 = 33333
3367 u 99 = 333333
271 u 164 = 44444
3367 u 132 = 444444
271 u 205 = 55555
3367 u 165 = 555555
271 u 246 = 66666
3367 u 198 = 666666
271 u 287 = 77777
3367 u 231 = 777777
271 u 328 = 88888
3367 u 264 = 888888
271 u 369 = 99999
3367 u 297 = 999999
43
Á¦ºÉ°Šš¸É 8 „µ¦¨š¸ÉŸ¨¨Áž}œ‹ÎµœªœÁ‡¦ºÉ°Š®¤µ¥‡¨³
Ĝ„µ¦¨°µ‹‹³Á…¸¥œ–(…¸—œ) Á®œº°˜´ªÁ¨…Äœ®¨´„š¸ÁÉ ž}œ‹Îµœªœ¨„Èŗo Ž¹ÉŠŸ¨¨š¸É‹³Å—oÁž}œ
‹ÎµœªœÁ‡¦ºÉ°Š®¤µ¥‡¨³
®¨´Š‹µ„œ´ÊœÂž¨ŠÁž}œ‹ÎµœªœÄœ¦³“µœ­·„È‹³Å—o‡Îµ˜°˜µ¤˜o°Š„µ¦
—´Š˜´ª°¥nµŠ ˜n°Åžœ¸Ê
˜´ª°¥nµŠš¸É 1.1.41 ‹Š®µ‡nµ…°Š 4328316 – 2876439
œª‡·— …´Êœš¸É 1 Á…¸¥œ„µ¦¨Áž}œ­°ŠÂ™ªÃ—¥Ä®o™ªÂ¦„Áž}œ˜´ª˜´ÊŠ ¨³Â™ªš¸É 2 Áž}œ˜´ª¨ ¨³­Îµ¦ª‹—¼
ªnµÂ˜n¨³®¨´„š¸É¨„´œ™oµ˜´ª˜´ÊŠ¤µ„„ªnµ˜´ª¨ „Țε„µ¦¨„´œÂ›¦¦¤—µ ™oµ˜´ª¨¤µ„„ªnµ˜´ª˜´ÊŠ Ä­n‡nµš¸É
˜´ª¨¤µ„„ªnµÄœn°ŠŸ¨¨´¡›r¨oªÁ…¸¥œ…¸—œ – —´Šœ¸Ê
4 3 2 8 3 1 6
2 8 7 6 4 3 9
_ _
–
_ _ _
2 5 5 2 1 2 3
…´Êœš¸É 2 ž¨Š˜´ªÁ¨…Äœœ·…·¨´¤­¼˜¦Ä®oÁž}œ˜´ªÁ¨…¦³“µœ­· ‹³Å—o
_ _
_ _ _
2 5 5 2 1 2 3
1 4 5 1 8 7 7
Ž¹ÉŠ 1 4 5 1 8 7 7 Áž}œŸ¨¨´¡›r
œ´Éœ‡º° 4328316 – 2876439 = 1451877
˜´ª°¥nµŠš¸É 1.1.42 ‹Š®µ‡nµ…°Š 673425 – 387619
œª‡·—
6 7 3 4 2 5
–
3 8 7 6 1 9
_ _ _
_
3 1 4 2 1 4
2 8 5 8 0 6
—´Šœ´Êœ 673425 – 387619 = 285806
44
˜´ª°¥nµŠš¸É 1.1.43 ‹Š®µ‡nµ…°Š 341659 – 459283
œª‡·—
ª·›¸š¸É 1 3 4 1 6 5 9
4 5 9 2 8 3
–
_ _ _
_
_ _ _ _
_ _
1 1 8 4 3 6
1 1 7 6 2 4
–1 1 7 6 2 4
—´Šœ´Êœ 341659 – 45983 = – 117624
ª·›¸š¸É 2 3 4 1 6 5 9
4 5 9 2 8 3
_ _ _
–
_
1 1 8 4 3 6
_
_
– (1 1 8 4 3 6 )
–1 1 7 6 2 4
—´Šœ´Êœ 341609 – 45983 = –117624
š«œ·¥¤ š«œ·¥¤ÄœÁªš‡–·˜„´š«œ·¥¤Äœ¦³˜´ªÁ¨…“µœ­·‡¨oµ¥„´œ „¨nµª‡º° ˜´ªÁ¨…®¨´Š‹»—š«œ·¥¤¤¸
‹ÎµœªœÁšnµ„´œ ¨³˜ÎµÂ®œnŠ…°Š‹»—š«œ·¥¤°¥¼nš¸ÉÁ—¸¥ª„´œ Á¤ºÉ°Áž¨¸É¥œ‹Îµœªœš¸É¤‹¸ »—š«œ·¥¤‹µ„¦³˜´ªÁ¨…
“µœ 10 Áž}œ˜´ªÁ¨…ÄœÁªš‡–·˜
˜´ª°¥nµŠš¸É 1.1.44 ‹ŠÂž¨Š 120 . 981 Áž}œ˜´ªÁ¨…Äœœ·…¨· ´¤­¼˜¦
œª‡·—
1 2 0
x
9 8 1
1 2 1
x
0 2 1
_
_
_ _
—´Šœ´Êœ 120 x 981 = 1 2 1 x 0 2 1
45
_
_
˜´ª°¥nµŠš¸É 1.1.45 ‹ŠÂž¨Š 1 x 7 3 2 8 Áž}œ˜´ªÁ¨…Äœ¦³“µœ­·
_
_
1x 7 3 2 8
œª‡·—
1x 6 7 1 2
_
_
—´Šœ´Êœ 1 x 7 3 2 8 = 1 x 6712
_
_
_
˜´ª°¥nµŠš¸É 1.1.46 ‹ŠÂž¨Š 3 x 2 1 4 6 Áž}œ˜´ªÁ¨…Äœ¦³“µœ­·
_
œª‡·— 3
_
_
x
_
2 1 4 6 = – (– ( 3
_
_
_
x
_
2 1 4 6 ))
= – (3 x 2 1 4 6)
= –2 x 8 0 6 6
_
—´Šœ´Êœ 3
_
_
x
2 1 4 6 = –2 x 8 0 6 6
˜´ª°¥nµŠš¸É 1.1.47 ‹Š®µ‡nµ…°Š
4 8 3 1 0
3 7 4 6 1
5 2 1 4 8
1 2 3 4 5
–
+
–
„
œª‡·— ‡·—ª„ – ¨ ‡¨³„´œÄœÂ˜n¨³®¨´„Ÿ¨¨´¡›rš¸Éŗo‹³Áž}œ ‹ÎµœªœÁ‡¦ºÉ°Š®¤µ¥‡¨³ ¨oªÂž¨ŠÁž}œ
‹ÎµœªœÄœ¦³“µœ­·—´Šœ¸Ê
4 8 3 1 0 –
3 7 4 6 1
5 2 1 4 8
+
–
1 2 3 4 5
_ _
5 1 3 5 2„
50 6 5 2
—´Šœ´Êœ Ÿ¨¨ ‡º° 5 0 6 5 2
46
˜´ª°¥nµŠš¸É 1.1.48 ‹Š®µ‡nµ…°Š
3 7 4 6 1 –
8 8 3 9 9
–
1 2 3 4 5
5 1 3 4 6
+
„
œª‡·—
3 7 4 6 1–
8 8 3 9 9
–
1 2 3 4 5
+
5 1 3 4 6
_ _
_ _
1 2 1 3 7
_ _ _ _ _
1 1 9 3 7
–1 1 9 3 7
—´Šœ´Êœ Ÿ¨¨‡º° –1 1 9 3 7
47
 f„®´—š¸É 3
1. ‹Š®µŸ¨¨˜n°Åžœ¸Ê
(1) 100 – 36 = ______________
(2) 100 – 47 = ______________
(3) 100 – 89 = ______________
(4) 1000 – 327 = ____________
(5) 1000 – 638 = ____________
(6) 1000 – 95 = _____________
(7) 10000 – 6759 = __________
(8) 10000 – 328 = ___________
(9) 10000 – 2149 = __________
(10) 100000 – 12345 = _______
(11) 100000 – 39393 = _______
(12) 100000 – 567 = _________
2. ‹Š®µŸ¨¨˜n°Åžœ¸Ê ץčoÁŒ¡µ³‹ÎµœªœšÁ„oµ š­· c ¨³„µ¦ª„
(1) 3 6
2 8
(2) 3 2 4
-
1 1 8
„
-
„
(3) 4 3 7
(4) 5 0 6
2 6 9
1 0 7
Á
„
--
-
(5) 7 2 8
(6) 8 7 4 2
8 9
3 7 9 6
- „„
„
(7) 4 9 0 6 2
-
-
8 6 4 8
„
(9) 9 8 3 5 8
-
7 9 4 6 7
Á
(8) 8 7 2 1 1
-
3 0 8 8
„
-
(10) 8 6 4 2 3 7
2 7 0 0 3 8
d
-
48
3. ‹Š®µ‡nµ˜n°Åžœ¸Ê ×¥ª·›š¸ ¸É˜„˜nµŠ„´œ 3 ª·›¸
(1) 4 7 9
(2) 6 6 6
2 9 8
1 2 3
--
3 4 5
+
5 4 8
Á
+
-
„
(3) 9 9 9
(4) 4 5 1 8
6 7 4
1 2 3 8
1 8
- 8
9 4 2„
3 1 2 6
„
+
-
+
d
(5)
3 2 1
-
4 7 2 9
6 5 8 7„
1 2 3 4
+
-
(6) 1 4 8 7
8 2 6
7 2 8
d
d
(7) 4 1 4
(8) 7 6 8
5 8 2
1 7 9
- -
6 7 9
6 1 6
+
-
5 4 6
5 2 8
„
„
(9) 4 4 8
(10) 2 7 4
-
6 2 3
8 1 2
1 7 4
„
+
-
9„
4
2 3 6 7
5 8 5
4 1 6
7 3 1
„
+
+
-
49
(11) 4 7 1 8 4
-
-
6 5 3 2 5
7 8 6 2 9
3 2 4 4 4
+
1 1 1 2 1
„
(12) 5 9 4 8 9
-
6 6 6 6 6
9 8 7 6 5
+
3 4 5 6 7
-
5 4 3 2 5
„
4. ‹Š®µ‡nµ…°Š 31456 – 47474 + 38641 – 27264 + 38886 = ______________
50
˜°œš¸É 1.2 š´„¬³„µ¦‡Îµœª– 2 („µ¦‡¼–)
œª‡·— 1. “µœÄœ¦³š«œ·¥¤ ®¤µ¥™¹Š ‹ÎµœªœÄœ¦¼ž 10n Á¤ºÉ° n Áž}œ‹ÎµœªœÁ˜È¤ª„ ¨³Á¦¸¥„
‹ÎµœªœÁ®¨nµœ´Êœªnµ“µœ 10n
2. ‹Îµœªœš¸É°¥¼nĄ¨o 10n ¦³¥³®nµŠ¦³®ªnµŠ‹Îµœªœ„´“µœ 10n š¸É‡·—š·«šµŠ Á¦¸¥„ªnµ ‡nµÁ¸É¥Š“µœ
…°Š‹Îµœªœœ´Êœ
3. Ĝ„µ¦®µŸ¨‡¼–…°Š‹Îµœªœš¸É°¥¼nĄ¨o“µœÁ—¸¥ª„´œ Á¤ºÉ°šÎµ„µ¦‡¼–ץčo‡nµÁ¸É¥Š“µœ‹³
­³—ª„„ªnµ
4. ª·›¸„µ¦‡¼–‹Îµœªœ¤¸Å—o®¨µ„®¨µ¥ª·›¸š´ÊŠ„µ¦‡¼–×¥„µ¦‹´—˜ÎµÂ®œnŠŸ¨‡¼– „µ¦‡¼–ץčo
˜µ¦µŠ ®¦º°„µ¦‡¼–Âœª˜´ŠÊ ¨³„µ¦‡¼–Å…ªo Ž¹ÉŠÂ˜n¨³ª·›¸‹³¤¸„¦³ªœ„µ¦š¸É˜„˜nµŠ„´œ
ª´˜™»ž¦³­Š‡r
Á¤ºÉ°«¹„¬µ®œnª¥š¸É 2 ‹Â¨oªœ´„Á¦¸¥œ­µ¤µ¦™
1. ¤¸„¦³ªœ„µ¦‡¼–Å—o®¨µ„®¨µ¥ª·›¸
2. °°„„µ¦‡¼–‹Îµœªœ˜µ¤Âœª‡·—­¦oµŠ­¦¦‡r…°Š˜œÁ°Š
„·‹„¦¦¤¦³®ªnµŠÁ¦¸¥œ
1. °µ‹µ¦¥r°›·µ¥ÁœºÊ°®µ „¦³ªœ„µ¦ª·›¸‡·—‡Îµœª–„µ¦‡¼–ץčo‡nµÁ¸¥É Š“µœ„µ¦‡¼–×¥
„µ¦‹´—˜ÎµÂ®œnŠŸ¨‡¼– „µ¦‡¼–ץčo˜µ¦µŠ„µ¦‡¼–Âœª˜´ÊŠÂ¨³„µ¦‡¼–Å…ªo
2. œ´„Á¦¸¥œšÎµ„·‹„¦¦¤˜µ¤˜´ª°¥nµŠÂ¨³Â f„®´—
3. œ´„Á¦¸¥œž¦³Á¤·œ¡´•œµ„µ¦…°Š˜œÁ°Š
­ºÉ°„µ¦­°œ
1. Á°„­µ¦„µ¦­°œ
2.  f„ž’·´˜·
3. Á‡¦ºÉ°ŠŒµ¥…oµ¤«¸¦¬³
ž¦³Á¤·œŸ¨
ž¦³Á¤·œŸ¨‹µ„ f„®´—¨³„µ¦š—­°
51
„µ¦‡¼–ץčo‡nµÁ¸É¥Š“µœ
Á¦ºÉ°Šš¸É 1 ‡nµÁ¸É¥Š“µœ
˜´ªÁ¨…ÄœÁªš‡–·˜ ÁnœÁ—¸¥ª„´˜´ªÁ¨…¦³“µœ­· „¨nµª‡º°°·Š“µœ…°Š¦³“µœ­· ¨³
…–³Á—¸¥ª„´œ‹³¦³»‡nµÁ¸¥É Š“µœ‡ª‡¼nŞ—oª¥Äœ„¦–¸˜°o Š„µ¦‡¼–®¦º°®µ¦ ×¥˜´ªÁ¨…ÄœÁªš‡–·˜‹³¥¹—
“µœ 10, 100, 1000, ..., 10n ¨³¦³»‡nµÁ¸¥É Š“µœ…°Š‹ÎµœªœÁ®¨nµœ´Êœ Ž¹ÉŠ‡nµÁ¸É¥Š“µœ¤¸š´ÊŠ‡nµª„ ‡nµ¨
¨³«¼œ¥r ‹³°›·µ¥ ‡nµÁ¸É¥Š“µœÃ—¥Äo˜´ª°¥nµŠž¦³„°—´Šœ¸Ê
˜´ª°¥nµŠš¸É 1.2.1
103 ¤¸‡nµ¤µ„„ªnµ 100 °¥¼n 3 „¨nµªªnµ 103 ¤¸‡nµÁ¸É¥Š“µœ ‹µ„ 100 Áž}œ + 03
94 ¤¸‡nµœo°¥„ªnµ 100 °¥¼n 6 „¨nµªªnµ 94 ¤¸‡nµÁ¸É¥Š“µœ ‹µ„ 100 Áž}œ – 06
28 ¤¸‡nµ¤µ„„ªnµ 10 °¥¼n 18 „¨nµªªnµ 28 ¤¸‡nµÁ¸É¥Š“µœ ‹µ„ 10 Áž}œ + 18
22 ¤¸‡nµœo°¥„ªnµ 100 °¥¼n 78 „¨nµªªnµ 22 ¤¸‡nµÁ¸É¥Š“µœ ‹µ„ 100 Áž}œ –78
Ĝ„µ¦Á…¸¥œ˜´ªÁ¨…¡¦o°¤¦³“µœÄœÁªš‡–·˜œ´Êœ ‹Îµœªœ˜´ªÁ¨…×—…°Š‡nµÁ¸É¥Š“µœ‹³˜o°Š
Ášnµ„´‹Îµœªœ 0 š¸Éž¦µ„’Äœ“µœœ´Êœ Ç —´Š˜´ª°¥nµŠ˜n°Åžœ¸Ê
˜´ª°¥nµŠš¸É 1.2.2
103 Á…¸¥œÄœ¦³“µœ 100 ¡¦o°¤‡nµÁ¸É¥Š“µœÁž}œ 103 + 03
94 Á…¸¥œÄœ¦³“µœ 100 ¡¦o°¤‡nµÁ¸É¥Š“µœÁž}œ 94 – 06
115 Á…¸¥œÄœ¦³“µœ 100 ¡¦o°¤‡nµÁ¸É¥Š“µœÁž}œ 115 + 15
1012 Á…¸¥œÄœ¦³“µœ1000 ¡¦o°¤‡nµÁ¸É¥Š“µœÁž}œ 1012 + 012
971 Á…¸¥œÄœ¦³“µœ 1000 ¡¦o°¤‡nµÁ¸É¥Š“µœÁž}œ 971 – 029
®¤µ¥Á®˜» Ĝ„µ¦„ε®œ—“µœÁ¡ºÉ°®µ‡nµÁ¸É¥Š“µœ­Îµ®¦´‹ÎµœªœÄ— Ç œ´Êœ‹³Å¤nÁ¨º°„“µœš¸ÉšÎµÄ®o…œµ—
…°Š‡nµÁ¸É¥Š“µœ¤¸‹Îµœªœ¤µ„Á„·œÅž Ánœ 12 ‹³Á¨º°„“µœÁž}œ 10 ¤¸‡nµÁ¸É¥Š“µœÁž}œ +2 ‹³Å¤nÁ¨º°„“µœ
100 Á¡¦µ³¤¸‡µn Á¸É¥Š“µœÁž}œ –88 Ž¹ÉŠ¤¸…œµ—Áž}œ 88 Ž¹ÉŠ¤µ„Á„·œÅž œ°„‹µ„œ¸Ê­Îµ®¦´µŠ‹ÎµœªœÁnœ 41
™oµÁ¨º°„“µœ 10 ‹³Å—o‡nµÁ¸É¥Š“µœÁž}œ +31 Ĝ…–³š¸ÉÁ¨º°„“µœ 100 ‹³Å—o‡nµÁ¸É¥Š“µœÁž}œ –59 Ž¹ÉŠš´ÊŠ
+31 ¨³ -59 ¤¸…œµ—Áž}œ 31 ¨³ 59 Ž¹ÉŠ¤µ„Á„·œÅž ĜšµŠž’·´˜·°µ‹‹³Á¨º°„“µœ¥n°¥­Îµ®¦´nª¥Äœ
„µ¦‡Îµœª– Ž¹ÉŠ“µœ¥n°¥Å—o„n¡®»‡–
¼ …°Š“µœ 10, 100, 1000, ...Ánœ 20, 30, ..., 200, 300,... ­Îµ®¦´“µœ
¥n°¥‹³„¨nµª™¹ŠÄœš˜n°Åž
52
 f„®´—š¸É 4
1. ‹Š¦³»“µœ…°Š‹Îµœªœ˜n°Åžœ¸Ê ¡¦o°¤š´ÊŠ°„‡nµÁ¸É¥Š“µœ
(1.1) 102 (1.2) 12
(1.3) 97 (1.4) 19 (1.5) 112
(1.6) 975 (1.7) 10008 (1.8) 89 (1.9) 75 (1.10) 29
2. ‹Š˜¦ª‹­°ªnµ„µ¦¦³»“µœÂ¨³„µ¦Á…¸¥œ‹Îµœªœ¡¦o°¤‡nµÁ¸É¥Š“µœ…°Š‹Îµœªœ˜n°Åžœ¸Ê™¼„˜o°Š
®¦º°Å¤n ­Îµ®¦´„µ¦¦³»š¸ÉŤn™¼„˜o°Š‹ŠÂ„oŅ
‹Îµœªœ
111
6
97
107
992
9972
10025
“µœ
100
10
100
100
1000
1000
10000
‹Îµœªœ¡¦o°¤‡nµÁ¸É¥Š“µœ
111 + 11
6–4
100 – 3
100 + 3
992 + 8
1000 – 22
10025 – 25
„oŅ
3. ‹µ„“µœÂ¨³‡nµÁ¸É¥Š“µœš¸„É 宜—¤µÄ®o ‹Š®µ‹ÎµœªœÁ®¨nµœ´ÊœÂ¨³ Á…¸¥œ‹Îµœªœ¡¦o°¤‡nµÁ¸É¥Š“µœ
“µœ
‡nµÁ¸É¥Š“µœ
‹Îµœªœ
‹Îµœªœ¡¦o°¤‡nµÁ¸É¥Š“µœ
(3.1)
10
+3
13
13 + 3
(3.2)
100
– 03
(3.3)
100
– 07
(3.4)
1000
+ 014
(3.5)
1000
– 019
(3.6)
10000
– 0031
(3.7)
1000
+ 374
53
­œ»„„´˜´ªÁ¨… (3)
Ÿ¨‡¼–š¸Éŗo˜´ªÁ¨…Á¦¸¥Š‹µ„œo°¥Åž¤µ„
1u1
=
1
22 u 3
=
12
3 u 41
=
123
2 u 617
=
1234
3 u 5 u 823
=
12345
26 u 3 u 643
=
123456
127 u 9721
=
1234567
2 u 32 u 47 u 14953
=
12345678
32 u 13717421
=
123456789
2 u 32 u 5 u 13717421
=
1234567890
Á¦ºÉ°Šš¸É 2 „µ¦‡¼–ץčo‡µn Á¸É¥Š“µœ
ª·›¸„µ¦‡¼–š´ªÉ Ç Åžœ´ÊœÄoŗo„´š»„„¦–¸…°Š„µ¦‡¼–Ž¹ÉŠÅ¤n¥»nŠ¥µ„Ž´Žo°œ‹œÁ„·œÅž °¥nµŠÅ¦„È
˜µ¤ ª·›¸ÁŒ¡µ³ÄœÁªš‡–·˜œ´œÊ ­³—ª„¨³Šnµ¥„ªnµ „¨nµª‡º° ‹³Äo„µ¦‡¼–ÁŒ¡µ³®¨´„šµŠ—oµœ…ªµ…°Š
­°Š‹Îµœªœš¸‡É ¼–„´œ ­nªœ®¨´„šµŠŽoµ¥™´—¤µ‹³Äoª·›¸„µ¦ª„ œ°„‹µ„œ¸Ê„µ¦‡¼–®¨´„šµŠ…ªµœ´œÊ ‹³°·Š
œ·…·¨´¤­¼˜¦ ª·›¸„µ¦‡¼–‹³‹ÎµÂœ„˜µ¤œ·——´Š˜´ª°¥nµŠ˜n°Åžœ¸Ê
œ·—š¸É 1 Ÿ¨‡¼–‡nµÁ¸É¥Š“µœ¤¸‹ÎµœªœÁ¨…×—ŤnÁ„·œ‹Îµœªœ˜´ªÁ¨… 0 …°Š“µœ
˜´ª°¥nµŠš¸É 1.2.3 ‹Š®µŸ¨‡¼–…°Š 13 u 11
œª‡·— …´Êœš¸É 1 Á…¸¥œ 13 ¨³ 11 ‹Îµœªœ¡¦o°¤‡nµÁ¸É¥Š“µœ 10 ¨³˜´ÊŠ‡¼–
13 + 3 u
11 + 1
®¤µ¥Á®˜» ­Îµ®¦´“µœ 10 ¤¸ 0 Á¡¸¥Š˜´ªÁ—¸¥ª —´Šœ´Êœ‡nµÁ¸É¥Š“µœ‹¹ŠÁ…¸¥œÁ¨…×—˜´ªÁ—¸¥ª
54
…´Êœš¸É 2 nŠŸ¨‡¼–š¸É‹³Å—o°°„Áž}œ­°Š­nªœ ץčo / Áž}œ˜´ªÂnŠ Ž¹ÉŠ‹³Â¥„Ÿ¨‡¼–‹µ„
‡nµÁ¸É¥Š“µœÅªo˜nµŠ®µ„—´Šœ¸Ê
13 + 3
11 + 1
/
…´Êœš¸É 3 ®µŸ¨‡¼–…°Š‡nµÁ¸É¥Š“µœÂ¨oªÄ­nŸ¨¨´¡›rŪošµŠ…ªµ…°Š / š¸É‹³Å—oÁž}œ 3 Á¡¦µ³ 3 u 1 = 3
13 + 3
11 + 1
u
/3
…´Êœš¸É 4 šµŠ—oµœŽoµ¥…°Š / ®µŸ¨¨´¡›r×¥®µŸ¨ª„…°Š˜´ª˜´ÊŠ„´‡nµÁ¸É¥Š“µœ…°Š˜´ª‡¼– ‹³Å—o
13 + 1 = 14 (®¦º°®µŸ¨ª„…°Š˜´ª‡¼–„´‡nµÁ¸É¥Š“µœ…°Š˜´ª˜´ÊŠ ‹³Å—o 11 + 3 = 14 Ž¹ÉŠ‹³Ášnµ„´ 13 + 1)
„µ¦®µŸ¨ª„œ¸ÊÁ¦¸¥„ªnµ „µ¦®µŸ¨ª„šÂ¥Š Á¡¦µ³ 13 + 1 ®¦º° 11 + 3 Áž}œ„µ¦ª„
œªšÂ¥Š
13 + 3
œ´Éœ‡º°
u
11 + 1
14 / 3
…´Êœš¸É 5 Á°µ…¸— / °°„‹³Å—o 143 Áž}œŸ¨¨´¡›r
œ´Éœ‡º° 13 u 11 = 143
˜´ª°¥nµŠš¸É 1.2.4 ˜´ª°¥nµŠ˜n°Åžœ¸ÊÁž}œŸ¨¨´¡›r‹µ„ª·›¸‡·—˜µ¤˜´ª°¥nµŠš¸É 1.2.3
14 + 4
12 + 2
11 + 1
u
16 / 8
‹³Å—o 14 u 12 = 168,
17 + 7
11 + 1
u
18 / 7
‹³Å—o 17 u 11 = 187,
15 + 5
19 + 9
u
11 + 1
u
16 / 5
20 / 9
11 u 15 = 165,
19 u 11 = 209
12 + 2
13 + 3
13 + 3
15 / 6
u
13 + 3
u
16 / 9
12 u 13 = 156 ¨³ 13 u 13 = 169
55
œ·—š¸É 2 ˜n°Åž‹³„¨nµª™¹Š„¦–¸š¸ÉŸ¨‡¼–‡nµÁ¸É¥Š“µœÁž}œÁ¨…­°Š®¨´„
˜´ª°¥nµŠ˜n°Åž‹³Áž}œ„µ¦‡¼–˜´ªÁ¨…š¸É“µœÁž}œ 100
˜´ª°¥nµŠš¸É 1.2.5 ‹Š®µ‡nµ…°Š 106 u 105
106 + 06
œª‡·—
105 + 05
ŗo‹µ„ 106 + 05
®¦º° 105 + 06
‹³Å—o 106 u 105 = 11130
111 / 30
u
ŗo‹µ„ 06 u 05
˜´ª°¥nµŠš¸É 1.2.6 ‹Š®µ‡nµ…°Š
(1) 111 u 112
(2) 113 u 114
(3) 117 u 118
œª‡·—
(1)
111 + 11
112 + 12
u
123 / 132 = 123 + 1/32 = 12432
(ÁœºÉ°Š‹µ„ 11 u 12 = 132 ˜n‡nµÁ¸É¥Š“µœÁž}œÁ¨…­°Š®¨´„‹¹ŠÁ…¸¥œ 132
˜´ªš—
œ´Éœ‡º° 111 u 112 = 12432
®¤µ¥Á®˜» Ĝ„µ¦®µ‡nµ 11 u 12 šÎµÅ—o×¥
11 + 1
12 + 2
13 / 2 = 132
Áž}œ 132 Á¡ºÉ°Â­—Šªnµ 1 Áž}œ
56
Ĝ„µ¦®µ 111 u 112 °µ‹Á…¸¥œ˜n°ÁœºÉ°ŠÁž}œ—´Šœ¸Ê
“µœ 100 “µœ 10
111 + 11 + 1
112 + 12 + 2
u
123 / 13 / 2 = 123/132 = 123 + 1/32
= 12432
(2)
“µœ 100 “µœ 10
113 + 13 + 3
114 + 14 + 4
u
127 / 17 / 12 = 127/17 + 1/2
= 127/18/2
= 127 + 1/82
= 12882
(3)
117 + 17 + 7
118 + 18 + 8
u
135 / 25 / 56 = 135 + 2/5 + 5/6
= 137/10/6
= 137 + 1/0/6
= 13806
®¤µ¥Á®˜» Ĝ„µ¦®µŸ¨‡¼–˜n°ÁœºÉ°Š —´Š˜´ª°¥nµŠš¸É 1.2.6 ˜´ªÁ¨… šµŠ—oµœ…ªµ…°Š/œ´˜´ÊŠÂ˜n…¸—/
¦„‹µ„…ªµ‹³Áž}œ˜´ªÁ¨…×— ™oµÅ—oŸ¨¨´¡›rÁž}œ
Á¨…×—¤µ„„ªnµ®œ¹ÉŠ˜´ª‹³‡Š˜´ªÁ¨…šµŠ…ªµ¤º°Åªo
­nªœ˜´ªÁ¨…šµŠŽoµ¥‹³Áž}œ ˜´ªš— œ°„‹µ„œ¸ÊĜ„µ¦‡Îµœª–®µŸ¨‡¼–‹µ„Ëš¥rŸ¨¨´¡›rĜn°Š…ªµ­»—
Ášnµœ´Êœ‹³Å—o‹µ„Ÿ¨‡¼– ­nªœŸ¨¨´¡›rĜn°Š™´—ÅžšµŠŽoµ¥‹³Å—o‹µ„Ÿ¨ª„˜µ¤ª·›¸š¸Éŗo„¨nµª¤µÂ¨oª
57
œ·—š¸É 3 „¦–¸š¸É˜´ª˜´ÊŠÂ¨³˜´ª‡¼–¤¸‹Îµœªœ®¨´„˜nµŠ„´œ „µ¦Á¨º°„“µœ‹³Á¨º°„“µœ…°Š˜´ªÁ¨…š¸É¤¸‹Îµœªœ
®¨´„œo°¥„ªnµ
˜´ª°¥nµŠš¸É 1.2.7 ‹Š®µ‡nµ…°Š 103 u 2312
œª‡·— čo“µœ 100
103 + 03
‹³Å—o
2312 + 2212
u
2315 / 6636
= 2315 + 66/36
= 238136
œ´Éœ‡º° 103
u
2312 = 238136
®¤µ¥Á®˜» ­Îµ®¦´“µœ 100 ‹³Á®Èœªnµ 2312 ¤¸‡nµÁ¸É¥Š“µœÁž}œ +2212 ‹¹ŠÁ…¸¥œ 2312 + 2212 Ĝ­nªœš¸É
Áž}œ˜´ª‡¼–
ĜŸ¨‡¼–šµŠ…ªµ…°Š / ÁœºÉ°Š‹µ„ 3 u 2212 = 6636 ˜n“µœÁž}œ 100 ‹¹Š˜o°Š„µ¦Á¨…×—Äœ
­nªœœ¸ÊÁ¡¸¥Š 2 ˜´ª‡º° 36 ­nªœ 66 œ´ÊœÁž}œ˜´ªš—Äœ®¨´„ 100 ¨³®¨´„ 1000
­Îµ®¦´Ÿ¨ª„šÂ¥Š ‡º° 103 + 2212 ®¦º° 2312 + 3
˜´ª°¥nµŠš¸É 1.2.8 ‹Š®µ‡nµ…°Š 103 u 13
œª‡·— čo“µœ 10
103 + 93
‹³Å—o
13 + 3
106 / 279
u
= 106 + 27/9
= 1339
œ´Éœ‡º° 103
u
13 = 1339
œ·—š¸É 4 ‹µ„˜´ª°¥nµŠ„µ¦‡¼–š´ÊŠ 3 œ·—œ´Êœ‹³¤¸°¥nµŠœo°¥®œ¹ÉŠ‹Îµœªœ Ž¹ÉŠ°µ‹‹³Áž}œ˜´ª˜´ÊŠ ®¦º°˜´ª‡¼–š¸É
‡nµÁ¸É¥Š“µœ¤¸‡nµÁž}œª„Á¨È„œo°¥Å¤nÁ„·œ‡nµ…°Š“µœ „µ¦‡¼–œ·—š¸É 4 ‹³Áž}œ„µ¦‡¼–Ž¹ÉŠ˜´ª˜´ÊŠÂ¨³˜´ª‡¼–
¤¸‡nµÁ¸É¥Š“µœ¤¸‡nµÁž}œ¨
58
˜´ª°¥nµŠš¸É 1.2.9 ‹Š®µ‡nµ…°Š
(1) 9
œª‡·— (1)
u
6
(2) 6
9 – 1
6 – 4
u
5
u
5 / 4
ŗo‹µ„ (–1) u (–4) = 4
ŗo‹µ„ 9 + (–4) = 9 – 4 = 5
®¦º° 6 + (–1) = 6 – 1 = 5
—´Šœ´Êœ 9
(2)
u
6 = 54
6 – 4
5 – 5
u
1 / 20
= 1 + 2/0 = 30
ŗo‹µ„ (– 4) u (–5) = 20
¨³˜o°Šš— 2
ŗo‹µ„ 6 – 5 = 1 (®¦º° 5 – 4)
—´Šœ´Êœ 6
u
5 = 30
˜´ª°¥nµŠš¸É 1.2.10 ˜n°Åžœ¸ÊÁž}œ„µ¦®µŸ¨‡¼–…°Š‹Îµœªœ­°Š‹Îµœªœš¸É¤¸‡nµÁ¸É¥Š“µœÁž}œ¨
8
u
6
8 – 2
6 – 4
91
x
4 / 8
—´Šœ´Êœ 8 u 6 = 48
u
93
91 – 09
93 – 07
89
x
u
97
89 – 11
97 – 03
x
84 / 63
86 / 33
—´Šœ´Êœ 91 u 94 = 8463
—´Šœ´Êœ 89 u 97 = 8633
59
64
u
98
64 – 36
98 – 02
79
995 – 005
998
988 – 002
x
986 / 024
—´Šœ´Êœ 79 u 97 = 7663
99976
u
—´Šœ´Êœ 988 u 988 = 986024
99998
99976 – 00024
x
u
988 – 012
76 / 63
995
998 – 002
988
x
97 – 03
—´Šœ´Êœ 64 u 96 = 6272
u
97
79 – 21
x
62 / 72
998
u
99998 – 00002
x
99974 / 00048
993 / 010
—´Šœ´Êœ 998 u 995 = 993010
—´Šœ´Êœ 99976 u 99998 = 9997400048
œ·—š¸É 5 „¦–¸­°Š‹Îµœªœš¸É‡¼–„´œ‹Îµœªœ®œ¹ÉŠ¤¸‡nµÁ¸É¥Š“µœÁž}œª„ ¨³°¸„‹Îµœªœ®œ¹ÉŠ¤¸
‡nµÁ¸É¥Š“µœÁž}œ¨
˜´ª°¥nµŠš¸É 1.2.11 ‹Š®µ‡nµ…°Š 13 u 9
œª‡·—
13 + 3
9 – 1
u
_
12 / 3
_
= 12 3 = 117
_
ŗo‹µ„ 3 u (–1) = –3 = 3
ŗo‹µ„ 13 – 1 = 12 (®¦º° 9 + 3)
—´Šœ´Êœ 13
u
9 = 117
60
 f„®´—š¸É 5
1. ‹Š®µŸ¨‡¼–˜n°Åžœ¸Ê
(1.1) 12 u 12
(1.2) 14 u 13
(1.3) 17 u 17
(1.4) 108 u 103
(1.5) 102 u 103
(1.6) 112 u 108
(1.7) 113 u 109
(1.8) 112 u 111
(1.9) 1002 u 1007
(1.10) 1007 u 1012
(1.11) 1110 u 1004
(1.12) 1003 u 1002
(2.1) 89 u 99
(2.2) 87 u 88
(2.3) 91 u 93
(2.4) 989 u 995
(2.5) 999 u 988
(2.6) 998 u 992
(2.7) 9998 u 9995
(2.8) 9989 u 9980
(2.9) 9988 u 9996
(2.10) 98 u 102
(2.11) 108 u 98
(2.12) 1004 u 995
(2.13) 1007 u 988
(2.14) 1110 u 989
(2.15) 1003 u 998
2. ‹Š®µŸ¨‡¼–˜n°Åžœ¸Ê
­œ»„„´˜´ªÁ¨… (4)
Ÿ¨‡¼–š¸Éŗo˜´ªÁ¨…Á¦¸¥Š„¨´
1u
1
=
1
3u
7
=
21
3u
107
=
321
29 u
149
=
4321
3u
19u953
=
54321
3u
218107
=
654321
19 u
402859
=
7654321
32 u
9739369
=
87654321
32 u
109739369
=
987654321
61
„µ¦‡¼–×¥„µ¦‹´—˜ÎµÂ®œnŠŸ¨‡¼–
Á¦ºÉ°Šš¸É 3 „µ¦‡¼–š¸É˜´ª˜´ŠÊ ¨³˜´ª‡¼–ž¦³„°—oª¥Á¨…×—­°Š˜´ª
„µ¦‡¼–×¥„µ¦‹´—˜ÎµÂ®œnŠŸ¨‡¼– ­Îµ®¦´ a ¨³ b Ž¹ÉŠÁž}œÁ¨…×—­°Š˜´ª‹³¤¸‡nµÁž}œ‹Îµœªœ
Á˜È¤ ˜´ÊŠÂ˜n 0 ™¹Š 9 Ÿ¨‡¼– a u b ‹³¤¸‡nµÁšnµ„´ c u 10 + d ­Îµ®¦´µŠÁ¨…×— c ¨³ d
™oµÁ…¸¥œ c u 10 + d Áž}œ˜´ªÁ¨…š¸É¤¸‡nµž¦³‹Îµ˜ÎµÂ®œnŠ ‹³Å—o c u 10 + d = cd œ´Éœ‡º° a u b = cd ‹¹Š
Á®ÈœÅ—oªnµ Ÿ¨‡¼–…°ŠÁ¨…×— a ¨³ b ¤¸‡nµÁž}œ‹Îµœªœš¸žÉ ¦³„°—oª¥˜´ªÁ¨…š¸ÉŤnÁ„·œ­°Š®¨´„
Ánœ 9 u 9 = 81 (‡nµ¤µ„š¸­É »—…°ŠŸ¨‡¼–Á¨…×—­°Š˜´ª)
6 u 5 = 30
1 u 7 = 7 ®¦º° 07 Áž}œ˜oœ
—´Šœ´Êœ ™oµ a ¨³ b Áž}œÁ¨…×—Äœ®¨´„®œnª¥ ‹³¤¸Ÿ´Š„µ¦‡¼– —´Šœ¸Ê
­·
®œnª¥
­°Šn°Š ‡º° ˜ÎµÂ®œnŠš¸ÉčoÁ…¸¥œŸ¨‡¼–
…°Š a u b Ž¹ÉŠ‹³°¥¼nĜ˜ÎµÂ®œnŠ®¨´„®œnª¥
¨³˜ÎµÂ®œnŠ®¨´„­· („¦–¸Ÿ¨‡¼–š¸ÉŗoÁž}œ
Á¨…×—˜´ªÁ—¸¥ª Ĝ®¨´„­· ®¤µ¥™¹Š 0)
a
b
u
™oµ a Áž}œÁ¨…×—Äœ®¨´„®œnª¥ ¨³ b Áž}œÁ¨…×—Äœ®¨´„­· ¨³Á…¸¥œ a ¨³ b Ĝ¦¼ž
‡nµž¦³‹Îµ˜ÎµÂ®œnŠ ‹³Å—o
a ¤¸‡nµÁšnµ„´ a ¨³ b ¤¸‡nµÁšnµ„´ b u 10
—´Šœ´Êœ Ÿ¨‡¼–…°Š‹Îµœªœš´ÊŠ­°Š‹³°¥¼nĜ¦¼ž
a u (b u 10) = cd u 10 (a u b = cd)
= cdo
( cdo Á…¸¥œÄœ¦¼ž‡nµž¦³‹Îµ˜ÎµÂ®œnŠ)
­—Šªnµ ™oµ a Áž}œÁ¨…×—Äœ®¨´„®œnª¥ ¨³ b Áž}œÁ¨…×—Äœ ®¨´„­· ‹³¤¸Ÿ´Š„µ¦‡¼– —´Šœ¸Ê
¦o°¥
­·
®œnª¥
a
b
u
­°Šn°Š ‡º° ˜ÎµÂ®œnŠš¸Éčo
Á…¸¥œŸ¨‡¼–…°Š a u (b u 10) Ž¹ÉŠ
‹³°¥¼Än œ˜ÎµÂ®œnŠ®¨´„­· ¨³
˜ÎµÂ®œnŠ®¨´„¦o°¥
62
šÎµœ°ŠÁ—¸¥ª„´œ ™oµ a Áž}œÁ¨…×—Äœ®¨´„­· ¨³ b Áž}œÁ¨…×—Äœ®¨´„®œnª¥‹³¤¸Ÿ´Š„µ¦‡¼–
—´Šœ¸Ê
¦o°¥
­·
®œnª¥
a
b
u
¨³™oµ a ¨³ b ˜nµŠ„ÈÁž}œÁ¨…×—Äœ®¨´„­· ‹³¤¸Ÿ´Š„µ¦‡¼– —´Šœ¸Ê
¡´œ ¦o°¥ ­· ®œnª¥
a
u
b
­°Šn°Š ‡º° ˜ÎµÂ®œnŠš¸ÉčoÁ…¸¥œ
Ÿ¨‡¼–…°Š (a u 10) u (b u 10) Ž¹ÉŠ‹³
°¥¼nĜ˜ÎµÂ®œnŠ®¨´„¦o°¥ ¨³®¨´„¡´œ
™oµ a, b, c, d ˜nµŠÁž}œÁ¨…×— ¨³ ab „´ cd Áž}œ˜´ªÁ¨…š¸ÉÁ…¸¥œÄœ¦¼žš¸¤É ¸‡nµž¦³‹Îµ˜ÎµÂ®œnŠ
ab u cd ‹³¤¸ŸŠ´ „µ¦‡¼– —´Šœ¸Ê
¡´œ
¦o°¥
­·
®œnª¥
a
b
c
d
u
(1)
(2)
(3)
(4)
63
¦¦š´— (1) n°Š ­°Šn°Š ­—Š˜ÎµÂ®œnŠ…°ŠŸ¨‡¼– b u d
¦¦š´— (2) n°Š ­°Šn°Š ­—Š˜ÎµÂ®œnŠ…°ŠŸ¨‡¼– (a u 10) u d
¦¦š´— (3) n°Š ­°Šn°Š ­—Š˜ÎµÂ®œnŠ…°ŠŸ¨‡¼– c u (b u 10)
¦¦š´— (4) n°Š ­°Šn°Š ­—Š˜ÎµÂ®œnŠ…°ŠŸ¨‡¼– (a u 10) u (c u 10)
™oµ¥oµ¥˜ÎµÂ®œnŠ…°Šn°Š ­°Šn°ŠÄœ¦¦š´— (4) Ş¥´Š¦¦š´— (1) Ä®o°¥¼nĜ®¨´„¡´œÂ¨³
®¨´„¦o°¥ ‹³Å—oŸ´Š„µ¦‡¼– —´Šœ¸Ê
¡´œ
¦o°¥
­·
®œnª¥
a
b
c
d
(2)
(3)
u
(1)
Ÿ¨¨´¡›rš¸Éŗo‹µ„„µ¦ª„‹ÎµœªœÄœ¦¦š´— (1), (2) ¨³ (3) š¸É¤¸®¨´„˜¦Š„´œ˜µ¤ž„˜·
˜´ª°¥nµŠš¸É 1.2.12 ‹Š®µŸ¨‡¼–…°Š 48 u 69
48
ª·›¸‡·—
69
(24 ŗo‹µ„ 4 u 6)
u
2472
( 72 ŗo‹µ„ 8 u 9)
36
( 36 ŗo‹µ„ 4 u 9)
48
(48 ŗo‹µ„ 8 u 6)
3312
—´Šœ´Êœ 48 u 69 = 3312
64
˜´ª°¥nµŠš¸É 1.2.13 ‹Š®µŸ¨‡¼–…°Š 41 u 69
41
ª·›¸‡·—
69
(24 ŗo‹µ„ 4 u 6)
u
2409
( 09 ŗo‹µ„ 1 u 9)
36
( 36 ŗo‹µ„ 4 u 9)
06
(06 ŗo‹µ„ 1 u 6)
2829
—´Šœ´Êœ 41 u 69 = 2829
Á¦ºÉ°Šš¸É 4 „µ¦‡¼–š¸É˜´ª˜´ŠÊ ž¦³„°—oª¥Á¨…×—­µ¤˜´ª ¨³˜´ª‡¼–ž¦³„°—oª¥Á¨…×—­°Š˜´ª
™oµ a, b, c, d ¨³ e Áž}œÁ¨…×—
Ĝ„µ¦®µŸ¨‡¼–…°Š abc u de š¸ÉÁž}œŸ¨‡¼–š¸É˜´ª˜´ÊŠž¦³„°—oª¥Á¨…×—­µ¤˜´ª ¨³˜´ª‡¼–
ž¦³„°—oª¥Á¨…×—­°Š˜´ª čoœª‡·—šÎµœ°ŠÁ—¸¥ª„´œ„´¦¼žÂš¸É 1 ˜nÁ¡·É¤®¨´„¦o°¥š¸É˜´ª˜´ÊŠ…¹œÊ °¸„
®œ¹ÉŠ®¨´„ Ÿ´Š„µ¦‡¼–Áž}œ—´Šœ¸Ê
®¤ºÉœ
¡´œ
¦o°¥
a
­·
b
®œnª¥
c
d
e
(1)
(2)
(3)
(4)
(5)
(6)
e Áž}œ˜´ª‡¼–
d Áž}œ˜´ª‡¼–
¦¦š´— (1) n°Š ­—Š˜ÎµÂ®œnŠ…°ŠŸ¨‡¼– c
u
e
¦¦š´— (2) n°Š ­—Š˜ÎµÂ®œnŠ…°ŠŸ¨‡¼– (b u 10) u e
¦¦š´— (3) n°Š ­—Š˜ÎµÂ®œnŠ…°ŠŸ¨‡¼– (a u 100) u e
65
¦¦š´— (4) n°Š ­—Š˜ÎµÂ®œnŠ…°ŠŸ¨‡¼– c u (d u 10)
¦¦š´— (5) n°Š ­—Š˜ÎµÂ®œnŠ…°ŠŸ¨‡¼– (b u 10) u (d u 10)
¦¦š´— (6) n°Š ­—Š˜ÎµÂ®œnŠ…°ŠŸ¨‡¼– (a u 100) u (d u 10)
œ°„‹µ„œ¸Ê ™oµœÎµ Ĝ¦¦š´— (3) ŞŪoš¸É¦¦š´— (1) Ĝ®¨´„š¸É˜¦Š„´œÂ¨³œÎµ Ĝ¦¦š´— (6) ŞŪoš¸É¦¦š´— (4) Ĝ®¨´„š¸É˜¦Š„´œ ‹³Å—oš·«šµŠ„µ¦ªµŠ˜ÎµÂ®œnŠš¸¤É ¸ e ¨³ d Áž}œ˜´ª
‡¼– —´Šœ¸Ê
®¤ºÉœ ¡´œ
¦o°¥
a
­·
b
d
(3)
®œnª¥
c
e
(1)
u
(1), (2) ¨³ (3) ­—Šš·«šµŠ
„µ¦ªµŠ˜ÎµÂ®œnŠ …°Š
„µ¦‡¼– c, b ¨³ a —oª¥ e
˜µ¤¨Îµ—´
(2)
(6)
(4), (5) ¨³ (6) ­—Šš·«šµŠ
„µ¦ªµŠ˜ÎµÂ®œnŠ …°Š
„µ¦‡¼– c, b ¨³ a —oª¥ d
˜µ¤¨Îµ—´
n°ŠŸ¨¨´¡›r
(4)
(5)
Ÿ´Š„µ¦‡¼– Áž}œ—´Šœ¸Ê
®¤ºÉœ
¡´œ
¦o°¥
a
­·
b
d
®œnª¥
c
e
u
66
˜´ª°¥nµŠš¸É 1.2.14 ‹Š®µŸ¨‡¼–…°Š 987 u 36
ª·›¸‡·—
987
36
(54 ŗo‹µ„ 9 u 6)
5442
48
2721
(27 ŗo‹µ„ 9 u 3)
(24 ŗo‹µ„ 8 u 3)
24
35532
—´Šœ´Êœ 987 u 36 = 35532
(42 ŗo‹µ„ 7 u 6)
(48 ŗo‹µ„ 8 u 6)
(21 ŗo‹µ„ 7 u 3)
™oµÁž}œŸ¨‡¼–…°ŠÁ¨…­µ¤®¨´„„´Á¨…­µ¤®¨´„„ȚεŗošµÎ œ°ŠÁ—¸¥ª„´œÃ—¥Á¡·É¤Â™ª…°ŠŸ¨¨´¡›r
„µ¦‡¼–—oª¥˜´ªÁ¨…Äœ®¨´„¦o°¥…°Š˜´ª‡¼–˜n°°¸„­°ŠÂ™ª ¨oª‹¹Š®µŸ¨ª„…°ŠŸ¨‡¼–š´ÊŠ®¤—
—´Š˜´ª°¥nµŠ˜n°Åžœ¸Ê
˜´ª°¥nµŠš¸É 1.2.15 ‹Š®µŸ¨‡¼–…°Š 987 u 436
987
ª·›¸‡·—
436
u
5442
48
2721
24
3628
32
„
43 0 3 3 2
—´Šœ´Êœ 987 u 436 = 430332
67
 f„®´—š¸É 6
‹Š®µ‡nµ…°Š
1. 47
u
68
2. 63
u
327
3. 472
u
384
4. 615
u
738
5. 1234
u
389
­œ»„„´˜´ªÁ¨… (8)
®œ¹ÉŠ‡¼–„´®œ¹ÉŠÅ—o ®œ¹ÉŠ ­°Š ­µ¤
11 u
111 u
1111 u
11111 u
111111 u
1111111 u
11111111 u
111
111
111
111
111
111
111
=
=
=
=
=
=
=
1221
12321
123321
1233321
12333321
123333321
1233333321
68
„µ¦‡¼–ץčo˜µ¦µŠ
Á¦ºÉ°Šš¸É 5 „µ¦‡¼–›¦¦¤—µ
„µ¦‡¼–ץčo˜µ¦µŠ˜µ¤ª·›¸š¸É‹³„¨nµª˜n°Åžœ¸Ê¤¸‡ªµ¤­³—ª„ ×¥…–³š¸É‡¼–Ťn˜o°ŠœÎµ˜´ªš—
ޝª„Äœ®¨´„™´—Åž Á¤ºÉ°‹´‡¼n‡¼–Á¨…×—š´ÊŠ®¤—¨oª®µŸ¨ª„¦ª¤¥°—Áž}œŸ¨‡¼–—´Š˜´ª°¥nµŠ˜n°Åžœ¸Ê
˜´ª°¥nµŠš¸É 1.2.16 ‹Š®µ‡nµ…°Š 109 u 12
œª‡·— …´œÊ š¸É 1 Á…¸¥œ˜´ª˜´ÊŠÅªo—oµœœ…°Š˜µ¦µŠ ¨³Á…¸¥œ˜´ª‡»–Ūo—µo œ…ªµ…°Š˜µ¦µŠ ×¥˜µ¦µŠšÎµ
Áž}œn°Š—´Šœ¸Ê
1 0 9
1
2
…´Êœš¸É 2 Ä­nŸ¨‡¼–¨ŠÄœn°Š
ץĭn°n Š¨nµŠ„n°œ ( )
­Îµ®¦´˜´ªš—™oµ¤¸Ä­nĜn°Šœ ( )
™ªœÄœ˜µ¦µŠ‹³Áž}œŸ¨‡¼–…°Š 109 u 1 ¨³Â™ª¨nµŠÄœ˜µ¦µŠ
‹³Áž}œŸ¨‡¼–…°Š 109 u 2 —´Šœ¸Ê
1
0
9
™ªœ
1
0
9 1
1
™ª¨nµŠ
2
0
8 2
…´Êœš¸É 3 ®µŸ¨ª„˜µ¤ÂœªšÂ¥Š // Ÿ¨ª„š¸Éŗo‹³Áž}œŸ¨‡¼–˜µ¤˜o°Š„µ¦
1
1
0
1
0
2
2
9
9
1
0
10
8
8
œ´Éœ‡º° 109 u 12 = 1308
˜´ª°¥nµŠš¸É 1.2.17 ‹Š®µ‡nµ…°Š 387 u 24
œª‡·—
3 8
1
8
12
œ´Éœ‡º° 387 u 24 = 9288
2
7
6
3
2
8
= 12108 = 1308
1
6
1
1
2
4
2
8
8
2
4
= 81288 = 9288
69
˜´ª°¥nµŠš¸É 1.2.18 ‹Š®µ‡nµ…°Š 89 u 9988
œª‡·—
7
1
7
1
1
8
7
2
7
2
6
7
6
8
1
8
4
7
4
7
3
9
8
1
9
1
9
2
8
2
8
= 7171718132 = 888932
2
œ´Éœ‡º° 89 u 9988 = 888932
Á¦ºÉ°Šš¸É 6 „µ¦‡¼–×¥‹´—Áž}œ‹ÎµœªœÁ‡¦ºÉ°Š®¤µ¥‡¨³
Ĝ„¦–¸š¸É˜´ª˜´ÊŠ®¦º°˜´ª‡¼–¤¸Á¨…×—š¸ÉÁ„·œ 5 °¥¼n®¨µ¥˜´ª °µ‹Âž¨ŠÁž}œ˜´ªÁ¨…Äœœ·…·¨´¤­¼˜¦
„n°œÂ¨oª‡¼–ץčo˜µ¦µŠ„È‹³šÎµÄ®očo„µ¦‡¼– ˜´ªÁ¨…š¸ÉŤnÁ„·œ 5 —´Š˜´ª°¥nµŠ˜n°Åžœ¸Ê
˜´ª°¥nµŠš¸É 1.2.19 ‹ŠÂž¨Š 89 u 9988 Áž}œ˜´ªÁ¨…Äœœ·…·¨´¤­¼˜¦Â¨oª®µŸ¨‡¼–ץčo˜µ¦µŠ
œª‡·—
__
89 u 9988 = 11 1 u 10 1 2
_ _
1
1
1
1
1
_
1
_
1
1
1
1
0
0
0
0
0
0
1
1
1
2
2
2
1
3
2
_
1
0
0
1
2
____
= 1 1 1 1 1 32
= 888932
œ´Éœ‡º° 89 u 9988 = 888932
70
˜´ª°¥nµŠš¸É 1.2.20 ‹Š®µ‡nµ…°Š 10012 u 9997
œª‡·— ª·›¸š¸É 1
1
0
0
1
2
9
0
0
9
9
9
0
0
9
9
9
0
0
9
9
7
0
0
7
7
1
1
8
1
8
1
1
8 9
1
8 9
1
8 9
1
4 7
6
4
= 100089964
œ´Éœ‡º° 10012 u 9997 = 100089964
ª·›¸š¸É 2 10012 u 9997 = 10012 u 10003
1
0
1
0
0
1
2
0
1
2
1
1
0
0
0
0
0
1
3
1
0
0
0
0
œ´Éœ‡º° 10012 u 9997 = 100089964
3
3
6
6
3
=100110036
=100089964
…o°­¦»ž „µ¦‡¼–ץčo˜µ¦µŠœ¸ÊÁž}œ„µ¦‡¼–Á¨…×—„´Á¨…×—Å—oŸ¨¨´¡›rÁž}œÁšnµÄ— Á…¸¥œŸ¨¨´¡›r
Á®¨nµœ´Êœ¨ŠÄœn°Š˜µ¦µŠÃ—¥Å¤n˜o°Šš— ®¨´Š‹µ„œ´Êœ‹¹Šª„šÂ¥Š ‹³Å—oŸ¨¨´¡›r…°Š„µ¦‡¼– Ž¹ÉŠšÎµÄ®o
ŗoŸ¨‡¼–¦ª—Á¦ÈªÂ¨³Ÿ·—¡¨µ—œo°¥ œ°„‹µ„œ¸Ê„¦–¸š¸ÉÁ¨…×—Äœ˜´ª˜´ÊŠ®¦º°˜´ª‡¼–¤¸‡nµÁ„·œ 5 ­µ¤µ¦™Äo
œ·…·¨´¤­¼˜¦Âž¨ŠÄ®očo˜´ªÁ¨…š¸ÉŤnÁ„·œ 5 ¨oªšÎµ„µ¦‡¼–˜µ¦µŠ„È‹³­³—ª„¨³¦ª—Á¦Èª…¹Êœ
71
 f„®´—š¸É 7
‹Š®µ‡nµ…°ŠŸ¨‡¼–˜n°Åžœ¸Ê
1. 287 u 315
2. 38644 u 9998
3. 27413 u 5615
4. 374 u 89 u 463
5. 27135 u 246 u 4138
6. 99998 u 819 u 19210
„µ¦‡¼–Âœª˜´ÊŠÂ¨³„µ¦‡¼–Å…ªo
Á¦ºÉ°Šš¸É 7 „µ¦‡¼–›¦¦¤—µ
„µ¦®µŸ¨‡¼–ץčo„µ¦‡¼–Âœª˜´ÊŠÂ¨³„µ¦‡¼–Å…ªoœ¸ÊÁž}œª·›¸„µ¦‡¼–ª·›¸®œ¹ÉŠ Ž¹ÉŠ¦¼žÂ„µ¦‡¼–
‹³Áž}œ¦¼žÂš¸É­¤¤µ˜¦ —´Š˜´ª°¥nµŠ„µ¦‡¼–˜n°Åžœ¸Ê
˜´ª°¥nµŠš¸É 1.2.21 42 u 31 ¤¸Ÿ´Š„µ¦‡¼–Âœª˜´ÊŠÂ¨³„µ¦‡¼–Å…ªo—´Šœ¸Ê
4
2
3
1
Á¤ºÉ° x šœ˜ÎµÂ®œnŠ…°ŠÁ¨…×—…°Š˜´ª˜´ÊŠÂ¨³˜´ª‡¼–‹³¤¸Ÿ´Š„µ¦‡¼–Áž}œ¦¼ž­¤¤µ˜¦ ¨³¤¸ 3
…´Êœ˜°œ„µ¦‡¼–‹µ„…ªµÅžŽoµ¥—´Šœ¸Ê
4
4
2
2
x x
x
x
x x
x
x
3
…´Êœš¸É 3
3
1
1
…´Êœš¸É 2
…´Êœš¸É 1
4 u 3 = 12 (4 u 1) + (3 u 2) = 10
2u1=2
Ÿ¨¨´¡›rš¸Éŗo‹µ„„µ¦‡¼–Ĝ˜n¨³…´Êœ‹³Å—o˜ÎµÂ®œnŠ…°ŠÁ¨…×—ÄœŸ¨¨´¡›r ‡º° ˜ÎµÂ®œnŠ
®¨´„®œnª¥ ®¨´„­· ®¨´„¦o°¥ ˜µ¤¨Îµ—´‹µ„…´Êœš¸É 1, 2 ¨³ 3 ­Îµ®¦´Äœ…´ÊœÄ—š¸ÉŸ¨‡¼–Áž}œÁ¨…×—­°Š
˜´ª Á¨…×—˜´ª®œoµ‹³š—ŞĜ®¨´„š¸É­¼Š…¹Êœ®œ¹ÉŠ®¨´„ —´Šœ¸Ê
72
2
1
4
2
3
1
0
2
1
= 1302
˜´ª°¥nµŠš¸É 1.2.22 ‹Š®µŸ¨‡¼–…°Š 89 u 23
œª‡·— ÁœºÉ°Š‹µ„˜´ª˜´ÊŠÂ¨³˜´ª‡¼–Áž}œÁ¨…­°Š®¨´„ ‹¹Š¤¸Ÿ´Š„µ¦‡¼–—´Šœ¸Ê
x
x
x
…´Êœš¸É 3
x
…´Êœš¸É 1
…´Êœš¸É 2
…´Êœ˜°œ„µ¦‡¼–
1) 6 u 3 = 18
2) (8 u 3) + (2 u 6) = 36
3) 8 u 2 = 16
„µ¦‡Îµœª–
8 6u
x
2 3
16 36 18 = 1978
˜´ª°¥nµŠš¸É 1.2.23 ‹Š®µŸ¨‡¼–…°Š 302 u 514
œª‡·— Ÿ´Š„µ¦‡¼–nŠÁž}œ 5 …´Êœ˜°œ —´Šœ¸Ê
xxx
xxx
xxx
xxx
…´Êœš¸É 5
xxx
…´Êœš¸É 4
xxx xxx
xxx
…´Êœš¸É 3 …´Êœš¸É 2 …´Êœš¸É 1
(‡¼– 1 ‡¼n)
(‡¼– 2 ‡¼n)
(‡¼– 3 ‡¼n) (‡¼– 2 ‡¼n) (‡¼– 1 ‡¼n)
„µ¦‡Îµœª–
3 0 2
u
5 1 4
15 3 22 2 8 = 155228
xxx
xxx
…´Êœ˜°œ„µ¦‡¼–
1) 2 u 4 = 8
2) (0 u 4) + (1 u 2) = 2
3) (3 u 4) + (0 u 1) + (5 u 2) = 22
4) (3 u 1) + (5 u 0) = 3
5) 3 u 5 = 15
73
˜´ª°¥nµŠš¸É 1.2.24 ‹Š®µŸ¨‡¼–…°Š 321 u 43
œª‡·— ÁœºÉ°Š‹µ„‹Îµœªœ®¨´„…°Š˜´ª˜´ÊŠ¤µ„„ªnµ˜´ª‡¼– ‹¹ŠÄ­nÁ¨… 0 ®œoµ˜´ª‡¼–Ä®o‹µÎ œªœÁ¨…×—…°Š
˜´ª˜´ÊŠÂ¨³˜´ª‡»–Ášnµ„´œÁ­¸¥„n°œ ¨oªšÎµ„µ¦‡Îµœª–
3 2 1
0 4 3
u
2 17 10 3 = 13803
1
˜´ª°¥nµŠš¸É 1.2.25 ‹Š®µŸ¨‡¼– 3251 u 7604
œª‡·— Ÿ´Š„µ¦‡»–nŠÁž}œ 7 …´Êœ˜°œ—´Šœ¸Ê
xxxx
xxxx xxxx xxxx
xxxx
xxxx xxxx
…´Êœš¸É 7
…´Êœš¸É 6
…´Êœš¸É 5
xxxx xxxx
xxxx
xxxx xxxx xxxx
xxxx
…´Êœš¸É 4
…´Êœš¸É 3
…´Êœš¸É 2
„µ¦‡Îµœª–
3 2 5 1
7 6 0 4
1 32 47 49 14 20 4 = 24720604
2
…´Êœ˜°œ„µ¦‡¼–
1)
2)
3)
4)
5)
6)
7)
1u4=4
(5 u 4) + (0 u 1) = 20
(2 u 4) + (5 u 0) + (6 u 1) = 14
(3 u 4) + (2 u 0) + (6 u 5) + (7 u 1) = 49
(3 u 0) + (2 u 6) + (7 u 5) = 47
(3 u 6) + (7 u 2) = 32
3 u 7 = 21
…´Êœš¸É 1
74
…o°­´ŠÁ„˜ Ĝ„µ¦‡¼–Âœª˜´ŠÊ ¨³„µ¦‡¼–Å…ªo¤¸¨´„¬–³—´Šœ¸Ê
1. ‹ÎµœªœÁ¨…×—…°Š˜´ª˜´ÊŠÂ¨³˜´ª‡¼–˜o°ŠÁšnµ„´œ Ĝ„¦–¸Å¤nÁšnµ Ä®oÁ˜·¤ 0 ®œoµ˜´ª˜´ÊŠ®¦º°
˜´ª‡¼–š¸œÉ o°¥„ªnµÄ®o¤˜¸ ´ªÁ¨…×—Ášnµ„´˜´ª˜´ÊŠ ®¦º°˜´ª‡¼–š¸É¤¸‹ÎµœªœÁ¨…×—¤µ„„ªnµ
2. Ÿ¨‹µ„ 1. ‹³šÎµÄ®o‹ÎµœªœÁ¨…×—Äœ„µ¦‡¼–Ášnµ„´ 2n Á¤ºÉ° n = 1,2,3... ‹³¤¸…´Êœ˜°œ„µ¦‡¼–
Ášnµ„´ 2n - 1 …´Êœ˜°œ
3. ˜´ÊŠÂ˜n…Êœ´ š¸É 1 ™¹Š…´Êœš¸É n ‹Îµœªœ„µ¦‹´‡¼nÁ¡ºÉ°‡¼–„´œš´ÊŠÂœª˜´ÊŠÂ¨³‡¼–Å…ªo‹³Ášnµ„´˜´ªÁ¨…
­—Š°´œ—´š¸…É °Š…´Êœœ´Êœ Ç ­Îµ®¦´…´Êœš¸É n + 1 ‹œ™¹Š…´Êœš¸É 2n - 1 ‹Îµœªœ„µ¦‹´‡¼n‡¼–„´œÄœÂ˜n¨³…´Êœš¸É
Á¡·É¤…¹Êœ‹³¨—¨Šš¸¨³ 1 ˜µ¤¨Îµ—´
˜´ª°¥nµŠš¸É 1.2.26 ‹Š®µŸ¨‡¼–…°Š 12131 u 20412
œª‡·— ÁœºÉ°Š‹µ„š´ÊŠ˜´ª˜´ÊŠÂ¨³˜´ª‡¼–˜nµŠ„ÈÁž}œ˜´ªÁ¨… 5 ®¨´„ —´Šœ´œÊ …´Êœ˜°œ„µ¦‡¼–¤¸š´ÊŠ®¤—
2(5) - 1 = 9 …´Êœ˜°œ —´Šœ¸Ê
„µ¦‡Îµœª–
1 2131
2 0412
u
2 4 6 15 10 17 9 7 2 = 247617972
…´Êœ˜°œ„µ¦‡¼–
1) 1 u 2 = 2
2) (3 u 2) + (1 u 1) = 7
3) (1 u 2) = (3 u 1) + (4 u 1) = 9
4) (2 u 2) + (1 u 1) + (4 u 3) + (0 u 1) = 17
5) (1 u 2) + (2 u 1) + (1 u 4) + (0 u 3) + (2 u 1) = 10
6) (1 u 1) + (2 u 4) + (0 u 1) + (2 u 3) = 15
7) (1 u 4) + (2 u 0) + (2 u 1) = 6
8) (1 u 0) + (2 u 2) = 4
9) 1 u 2 = 2
75
Ĝ„¦–¸š¸É˜´ª˜´ŠÊ ¨³˜´ª‡¼–¤¸Á¨…×—š¸É¤µ„„ªnµÁ¨… 5 Á¦µ­µ¤µ¦™Âž¨ŠÁž}œÁ¨…×—š¸É¤¸‡nµœo°¥
„ªnµÁ¨… 5 ŗoץčoœ·…·¨´¤­¼˜¦‹³šÎµÄ®o„µ¦‡¼–Šnµ¥…¹Êœ—´Š˜´ª°¥nµŠ˜n°Åžœ¸Ê
Á¦ºÉ°Šš¸É 8 „µ¦‡¼–×¥‹´—Áž}œ‹ÎµœªœÁ‡¦ºÉ°Š®¤µ¥‡¨³
˜´ª°¥nµŠš¸É 1.2.27 ‹Š®µŸ¨‡¼–…°Š 39 u 49
_
_
39 = 4 1 ¨³ 49 = 5 1
œª‡·—
_
4 1
_ u
5 1
_
2 0 9 1 = 1911
˜´ª°¥nµŠš¸É 1.2.28 ‹Š®µŸ¨‡¼–…°Š 291 u 388
_
_ _
291 = 3 1 1 ¨³ 388 = 4 1 2
œª‡·—
…´Êœ˜°œ„µ¦‡¼–
„µ¦‡Îµœª–
_
_
_
3 1 1
1) 1 u 2 = 2
4 1 2
2) ( 1 u 2 ) + ( 1 u 1) = 1
_ _u
_ _
_ _
_
1 2 7 1 1 2 = 112908
_
_
_ _
_
_
_
3) (3 u 2 ) + ( 1 u 1 ) + (4 u 1) = 1
_
4) (3 u 1 ) + (4 u 1 ) = 7
5) 3 u 4 = 12
˜´ª°¥nµŠš¸É 1.2.29 ‹Š®µŸ¨‡¼–…°Š 818 u 39
_
_
_
8 1 8 = 1 2 2 2 ¨³ 3 9 = 4 1
œª‡·—
„µ¦‡Îµœª–
_
_
12 22
_ u
_
004 1
_ _
4 9 10 1 0 2 = 4 8 1 0 2 = 3 1 9 0 2
76
˜´ª°¥nµŠš¸É 1.2.30 ‹Š®µŸ¨‡¼–…°Š 37898 u 19989
_ _
_
__
37898 = 4 2 1 0 2 ¨³ 19989 = 200 1 1
œª‡·—
„µ¦‡Îµœª–
_ _
_
42 102
_ _u
2001 1
_ _ _ _
8 4 2 4 6 3 1 2 2 = 757543122
…´Êœ˜°œ„µ¦‡¼–
_ _
1) 2 u 1 = 2
_
_ _
_ _
_
_ _
_ _
_
_ _
_
_
_
_
2) (0 u 1 ) + ( 1 u 2 ) = 2
_
3) ( 1 u 1 ) + (0 u 1 ) + (0 u 2 ) = 1
_
4) ( 2 u 1 ) + ( 1 u 1 ) + (0 u 0) + (0 u 2 ) = 3
_
_
5) (4 u 1 ) + ( 2 u 1 ) + ( 1 u 0) + (0 u 0) + (2 u 2 ) = 6
_
6) (4 u 1 ) + ( 2 u 0) + ( 1 u 0) + (2 u 0) = 4
_
_
_
7) (4 u 0) + ( 2 u 0) + (2 u 1 ) = 2
_
_
8) (4 u 0) + (2 u 2 ) = 4
9) 4 u 2 = 8
Ĝ„µ¦‡¼–‹Îµœªœš¸É¤¸‹—» š«œ·¥¤­µ¤µ¦™šÎµ„µ¦‡¼–Å—o×¥ž¦„˜· ×¥˜ÎµÂ®œnŠ…°Š‹»—š«œ·¥¤
Ÿ¨‡¼–Ášnµ„´Ÿ¨ª„…°Š˜ÎµÂ®œnŠš«œ·¥¤…°Š‹Îµœªœš´ÊŠ­°Šš¸É‡¼–„´œ
˜´ª°¥nµŠš¸É 1.2.31 ‹Š®µŸ¨‡¼–…°Š 34.1 u 4.54
œª‡·— ÁœºÉ°Š‹µ„˜´ª˜´ÊŠ¤¸š«œ·¥¤ 1 ˜ÎµÂ®œnŠ ¨³˜´ª‡¼–¤¸š«œ·¥¤ 2 ˜ÎµÂ®œnŠ —´Šœ´ÊœŸ¨¨´¡›r¤¸š«œ·¥¤
3 ˜ÎµÂ®œnŠ „µ¦‡¼–Áž}œ—´Šœ¸Ê
3 4.1
4.5 4
u
12 31 3 . 6 21 4 = 154.814
77
˜´ª°¥nµŠš¸É 1.2.32 ‹Š®µŸ¨‡¼–…°Š 8.18 u 3.9
œª‡·— ÁœºÉ°Š‹µ„˜´ª˜´ÊŠ¤¸š«œ·¥¤ 2 ˜ÎµÂ®œnŠ˜´ª‡¼–¤¸š«œ·¥¤ 1 ˜ÎµÂ®œnŠ —´Šœ´œÊ Ÿ¨¡´š›r¤¸š«œ·¥¤ 3
˜ÎµÂ®œnŠ œ°„‹µ„œ¸Ê
_
_
_
8 . 1 8 = 1 2 . 2 2 ¨³ 3 . 9 = 4 . 1 „µ¦‡¼–Áž}œ—´Šœ¸Ê
_
_
1 2 .2 2
_ u
_
0 0 4. 1
4 9 . 10
_ _
10 2 = 4 8 . 1 0 2 = 3 1 . 9 0 2
…o°­¦»ž
„µ¦‡¼–ץčoŸ´Š„µ¦‡¼–œ¸ÊŸ´Š„µ¦‡¼–Áž}œŸ´Šš¸ÉÁž}œ¦¼ž­¤¤µ˜¦ Ĝ„¦–¸š¸É˜´ª˜´ÊŠ ¨³˜´ª‡¼–¤¸
‹Îµœªœ®¨´„ŤnÁšnµ„´œ˜o°ŠÁ˜·¤«¼œ¥r®œoµ‹Îµœªœš¸É¤®¸ ¨´„œo°¥„ªnµÃ—¥Äo 0 šœ®¨´„š¸É…µ—Åž ¨oª‹¹ŠšÎµ
„µ¦‡¼–Á¨…×—š¸¨³ 1,2,...n,...,2, 1 ‡¼n˜µ¤Ÿ´Š„µ¦‡¼–
78
 f„®´—š¸É 8
‹Š®µŸ¨‡¼–˜n°Åžœ¸Ê
1. 45 u 58
2. 314 u 67
3. 8121 u 374
4. 31.28 u 4.35
5. 3561 u 62.5
­œ»„„´˜´ªÁ¨… (7)
„µ¦‡¼–š¸É¤¸Á‹È—„´Á„oµÅ—o…°ŠÂ™¤Áž}œ®œ¹ÉŠ„´­°Š
779 u
99
=
77121
7779 u
999
=
7771221
77779 u
9999
=
777712221
777779 u
99999
=
77777122221
7777779 u
999999
=
7777771222221
77777779 u
9999999
=
777777712222221
777777779 u
99999999
=
77777777122222221
79
˜°œš¸É 1.3 š´„¬³„µ¦‡Îµœª– 3 („µ¦®µ¦)
œª‡·— 1. „µ¦®µ¦š¸É˜´ª®µ¦œo°¥„ªnµ 10 ®¦º° 100 ®¦º° 1000 Á¨È„œo°¥œ´Êœ čoª·›¸„µ¦®µ¦­´ŠÁ‡¦µ³®r‹³
­³—ª„„ªnµ
2. Ĝ„µ¦®µŸ¨®µ¦Áž}œš«œ·¥¤ ‹³˜o°ŠÁ˜·¤ 0 ˜n°šoµ¥˜´ª˜´ÊŠÁšnµ„´‹Îµœªœ˜ÎµÂ®œnŠ…°Šš«œ·¥¤
š¸É˜o°Š„µ¦
ª´˜™»ž¦³­Š‡r
Á¤ºÉ°«¹„¬µ®œnª¥š¸É 3 ‹Â¨oª œ´„Á¦¸¥œ­µ¤µ¦™
1. ®µŸ¨®µ¦Ã—¥ª·›¸„µ¦®µ¦­´ŠÁ‡¦µ³®rŗo
2. ®µŸ¨®µ¦š¸ÉÁž}œš«œ·¥¤Ã—¥ª·›¸„µ¦®µ¦­´ŠÁ‡¦µ³®rŗo
„·‹„¦¦¤¦³®ªnµŠÁ¦¸¥œ
1. °µ‹µ¦¥r°›·µ¥Âœª‡·—¨³Â­—Š˜´ª°¥nµŠ„µ¦®µ¦š¸É˜´ª®µ¦œo°¥„ªnµ 10 ®¦º° 100 ®¦º° 1000
Á¨È„œo°¥ ץčoª·›¸„µ¦®µ¦­´ŠÁ‡¦µ³®r
2. œ´„Á¦¸¥œšÎµ„·‹„¦¦¤˜µ¤˜´ª°¥nµŠÂ¨³Â f„®´—
3. œ´„Á¦¸¥œž¦³Á¤·œ¡´•œµ„µ¦…°Š˜œÁ°Š
­ºÉ°„µ¦­°œ
1. Á°„­µ¦„µ¦­°œ
2.  f„ž’·´˜·
3. Á‡¦ºÉ°ŠŒµ¥…oµ¤«¸¦¬³
ž¦³Á¤·œŸ¨
ž¦³Á¤·œŸ¨‹µ„ f„®´—¨³„µ¦š—­°
80
„µ¦®µ¦­´ŠÁ‡¦µ³®r
Á¦ºÉ°Šš¸É 1 „µ¦®µ¦š¸É˜´ª®µ¦Áž}œÁ¨…×—š¸É¤µ„„ªnµ 5
Ĝ„µ¦‡¼–×¥ÁŒ¡µ³Á¨…×—š¸ÉœÎµ¤µ‡¼–„´œœ´Êœ¤¸‡nµ¤µ„„ªnµ 5 ™oµÄo‹ÎµœªœÁ‡¦ºÉ°Š®¤µ¥‡¨³
šÎµÄ®o‡¼–Šnµ¥…¹ÊœÄœšÎµœ°ŠÁ—¸¥ª„´œ­Îµ®¦´„µ¦®µ¦ ™oµ˜´ª®µ¦¤¸Á¨…×—š¸É¤¸‡nµ¤µ„„ªnµ 5 Á¤ºÉ°Äo‹Îµœªœ
Á‡¦ºÉ°Š®¤µ¥‡¨³Ÿ­¤„´„µ¦®µ¦­´ŠÁ‡¦µ³®r„È‹³šÎµÄ®o„µ¦®µ¦œ´ÊœŠnµ¥…¹Êœ —´Š˜´ª°¥nµŠ˜n°Åžœ¸Ê×¥‹³
­—Šª·›¸‡·—Áž}œ…´Êœ˜°œ
œ·—š¸É 1 ˜´ª®µ¦Áž}œÁ¨…×—š¸É¤µ„„ªnµ 5
˜´ª°¥nµŠš¸É 1.3.1 ‹Š®µ¦ 34 —oª¥ 9
œª‡·—
…´Êœš¸É 1 ®µ‹Îµœªœš­·…°Š˜´ª®µ¦‡º°‹Îµœªœ­·…°Š 9 Ž¹ÉŠ‡º° 1 Á…¸¥œ 1 ŪoĘo 9 ¨oªÁ…¸¥œ
„µ¦˜´ÊŠ®µ¦—´Šœ¸Ê
9)34
1
…´Êœš¸É 2 čo 1 Ž¹ÉŠÁž}œ‹Îµœªœš­·…°Š˜´ª®µ¦ šÎµ„µ¦®µ¦Âšœ˜´ª®µ¦Á—·¤ ¨oªÁ…¸¥œ…¸— |
nŠ˜´ª˜´ÊŠ ×¥Á…¸¥œ | ®¨´ŠÁ¨…×— Ĝ˜ÎµÂ®œnŠš¸ÉÁšnµ„´‹ÎµœªœÁ¨…×—…°Š˜´ª®µ¦ Ĝš¸Éœ¸Ê˜´ª®µ¦Áž}œ
Á¨…×— 1 ˜´ª —´Šœ´Êœ ‹µ„˜´ª˜´ÊŠ‡º° 34 œ´Á¨…×—‹µ„šµŠ…ªµÅžšµŠŽoµ¥ 1 ˜´ªÂ¨oªÁ…¸¥œ…¸— | ‡´Éœ ‹³
ŗo 3 | 4 Ä®o¥µª¡°­¤‡ª¦—´Šœ¸Ê
9) 3 4
1
®¤µ¥Á®˜» ‹³Äo MD šœ˜´ª®µ¦š¸Éž¦´ž¦»ŠÄ®¤n Ž¹ÉŠÄœš¸Éœ¸Ê‡º°‹Îµœªœš­·…°Š˜´ª®µ¦Á—·¤ ¨³‹³Äo
D šœ˜´ª˜´ÊŠ
…´Êœš¸É 3 …¸—Á­oœÄ˜o¦¦š´—š¸É 2 ¨oª´„Á¨…×—šµŠŽoµ¥­»—…°Š D (Ĝš¸Éœ¸Ê‡º° 3 ¨³°¥¼nšµŠŽoµ¥
…°Š | Ĝ˜´ª˜´ÊŠ) ¨Š¤µ˜¦ŠÇ ¨³Á…¸¥œÅªoĜ¦¦š´—š¸É 3
9) 3 4
MD
1
3
81
…´Êœš¸É 4 ‡¼–Á¨…×—š¸É´„¨Š¤µÄœ¦¦š´—š¸É 3 —oª¥Á¨…×—˜´ªŽoµ¥­»—…°Š MD (Ĝš¸Éœ¸Ê MD ¤¸
Á¨…×—˜´ªÁ—¸¥ª‡º° 1) ‹³Å—o 3 u 1 = 3 ¨oªÁ…¸¥œŸ¨‡¼–Ūo¦¦š´—š¸É 2 Ęo˜´ª˜´ÊŠÄœ®¨´„™´—ÅžšµŠ…ªµ
(Ĝš¸Éœ¸Ê‡º°®¨´„®œnª¥)
9)3 4
1
3
3
…´Êœš¸É 5 ª„Á¨…Äœ®¨´„®œnª¥…°ŠÂ™ªš¸É 1 ¨³ 2 ‹³Å—o 4+3 = 7 Á…¸¥œ 7 Ūo¦¦š´—š¸É 3 Ä®o˜¦Š
„´˜´ªÁ¨…š¸Éª„„´œ‹³Á®Èœªnµ | °¥¼n®œoµ 7
9)3 4
1
3
3
7
Ÿ¨¨´¡›r‡º° 3
Á«¬‡º°
7
—´Šœ´Êœ 34 y 9 = 3 Á«¬ 7
˜´ª°¥nµŠš¸É 1.3.2 ‹Š®µ¦ 60 —oª¥ 9
œª‡·—
…´Êœš¸É 1-2 ÁœºÉ°Š‹µ„˜´ª®µ¦‡º° 9 —´Šœ´Êœ MD = 1 Á…¸¥œ£µ¡˜´ÊŠ®µ¦Å—o—´Šœ¸Ê
9) 6 0
1
…´Êœš¸É 3 ´„ 6 Ž¹ÉŠÁž}œÁ¨…×—Žoµ¥­»—…°Š˜´ª˜´ÊŠ¨Š¤µ¦¦š´—š¸É 3
9) 6 0
1
6
…´Êœš¸É 4 ‡¼– 6 š¸É´„¨Š¤µ—oª¥ MD ‡º° 1 ‹³Å—o 6 u 1 = 6 ¨oªÁ…¸¥œÅªo¦¦š´—š¸É 2 Ĝ®¨´„™´—Åž
šµŠ…ªµ
82
9) 6 0
1
6
6
…´Êœš¸É 5 ª„Á¨…Äœ®¨´„®œnª¥…°Š¦¦š´—š¸É 1 ¨³ 2 ŗo 0 + 6 = 6 ¨oªÁ…¸¥œÅªo¦¦š´—š¸É 3 ‹³
Á®Èœªnµ | °¥¼n®œoµ 6 š¸ÉÁž}œŸ¨ª„
9) 6 0
1
6
6
6
Ÿ¨¨´¡›r‡º° 6
Á«¬‡º° 6
—´Šœ´Êœ 60 y 9 = 6 Á«¬ 6
˜´ª°¥nµŠš¸É 1.3.3 ‹Š®µ¦ 213 —oª¥ 9
œª‡·—
…´Êœš¸É 1-2
9) 2 1 3
1
…´Êœš¸É 3 ´„ 2 š¸ÉÁž}œÁ¨…×—Žoµ¥­»—…°Š˜´ª˜´ÊŠ¨Š¤µ¦¦š´—š¸É 3
9) 2 1 3
1
2
…´Êœš¸É 4 ‡¼– 2 š¸É´„¨Š¤µ—oª¥ 1 š¸ÉÁž}œÁ¨…×—Á¡¸¥Š˜´ªÁ—¸¥ªÄœ MD ‹³Å—o 2 u 1 = 2 Á…¸¥œ 2 Ūo
¦¦š´—š¸É 2 Ĝ®¨´„™´—ÅžšµŠ…ªµ (Ĝš¸Éœ¸Ê‡º°®¨´„­·) ¨oª®µŸ¨ª„Á¨…×—Äœ®¨´„­·…°Š¦¦š´—š¸É
1 ¨³ 2 ‹³Å—o 1 + 2 = 3 Á…¸¥œÅªo¦¦š´—š¸É 3 Ä®o˜¦Š„´Á¨…×—š¸Éª„„´œ
9) 2 1 3
1
2
23
83
…´Êœš¸É 4 („) œÎµ 1 Ĝ MD އ¼– 3 Ĝ¦¦š´—š¸É 3 š¸ÉÁž}œŸ¨ª„Äœ®¨´„­·‹³Å—o 3 u 1 = 3
Á…¸¥œ 3 Ūoœ®¨´„®œnª¥Äœ¦¦š´—š¸É 2
9) 2 1 3
1
2 3
23
…´Êœš¸É 5 ®µŸ¨ª„…°ŠÁ¨…×— Ĝ®¨´„®œnª¥…°Š¦¦š´—š¸É 1 ¨³ 2 ‹³Å—o 3 + 3 = 6 Á…¸¥œ 6
¨ŠÄœ¦¦š´—š¸É 3 Ä®o˜¦Š„´Á¨…×—š¸Éª„„´œ
9) 2 1 3
1
2 3
23 6
Ÿ¨¨´¡›r‡º° 23
6
Á«¬‡º°
—´Šœ´Êœ 213 y 9 = 23 Á«¬ 6
˜´ª°¥nµŠš¸É 1.3.4 ‹Š®µ¦ 323 —oª¥ 9
9)3 2 3
œª‡·—
1
3 5
35 8
Ÿ¨¨´¡›r‡º° 35
8
Á«¬‡º°
—´Šœ´Êœ 323 y 9 = 35 Á«¬ 8
˜´ª°¥nµŠš¸É 1.3.5 ‹Š®µ¦ 12121 —oª¥ 9
œª‡·—
9)1 2 1 2 1
1
1 3 4 6
1 3 4 6 7
84
Ÿ¨¨´¡›r‡º° 1346
7
Á«¬‡º°
—´Šœ´Êœ 12121 y 9 = 1346 Á«¬ 7
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Ÿ¨¨´¡›r‡º° 7
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—´Šœ´Êœ 167 y 9 = 18 Á«¬ 5
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˜´ª°¥nµŠš¸É 1.3.8 ‹Š®µ¦ 1011638 —oª¥ 9
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1 1 1 2 3 9 12
1 1 2 3 9 12 20
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Ÿ¨¨´¡›r‡º° 112404
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Ÿ¨¨´¡›r‡º° 126
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—´Šœ´Êœ 1012 y 8 = 126 Á«¬ 4
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(Ĝš¸Éœ¸Ê‡º° 2) ¨Š¤µ˜¦Š Ç Â¨³Á…¸¥œÅªo¦¦š´—š¸É 3
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Ĝ­°Š®¨´„™´—ÅžšµŠ…ªµ
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Á®Èœªnµ¤¸ | ®œoµ 38 Ž¹ÉŠ 38 ‡º°Á«¬ ­nªœ 2 š¸É´„¨Š¤µ˜´ÊŠÂ˜n˜oœ‡º°Ÿ¨¨´¡›r
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—´Šœ´Êœ 216 y 89 = 2 Á«¬ 38
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—´Šœ´Êœ 112 y 87 = 1 Á«¬ 25
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˜´ª°¥nµŠš¸É 1.3.13 ‹Š®µ¦ 34567 —oª¥ 89
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33
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…´Êœš¸É 4 („) ª„ 4 „´ 3 Ž¹ÉŠÁž}œÁ¨…×—Äœ¦¦š´—š¸É 1 ¨³ 2 °¥¼nĜ®¨´„¡´œÁ®¤º°œ„´œ‹³Å—o
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33
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…´Êœš¸É 4 (…) œÎµ MD = 11 ‡¼– 7 Ĝ¦¦š´—­»—šoµ¥ ‹³Å—o
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3 3
7 7
3 7
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…´Êœš¸É 4 (Š) œÎµ MD = 11 ‡¼– 15 Ĝ¦¦š´—­»—šoµ¥ ‹³Å—o 15 u 11 = 15 15 Ūo¦¦š´—š¸É 4 Ĝ
­°Š®¨´„™´—ÅžšµŠ…ªµ
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3 3
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1 1
3 7 15
…´Êœš¸É 5 ®µŸ¨ª„…°Š 6 „´ 7 „´ 15 ¨³ 7 „´ 15 ‹³Å—o
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ªnµ¤¸ | ®œoµ 28
89)3 4 5 67
11
3 3
7 7
55
1 1
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…´Êœš¸É 6 Á«¬š¸Éŗo‡º° 28 22 = 302 Ž¹ÉŠ¤¸‡nµÁ„·œ 89 ‹¹ŠšÎµ„µ¦®µ¦Á«¬˜n°Åž —´Šœ¸Ê
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1 1
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91
‹³Å—oŸ¨¨´¡›r‡º° 3
35
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Ÿ¨¨´¡›r­»—šoµ¥Â¨³Á«¬­»—šoµ¥‡º° 35 Ž¹ÉŠÅ—o‹µ„…´Êœš¸É 6
‹³Å—oŸ¨¨´¡›r‡º° 388
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—´Šœ´Êœ 34567 y 89 = 388 Á«¬ 35
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3 3
7 7
15 15
3 7 15 28 22
3 7 15 3 0 2
33
3 35
3 7 15 3 3 5
= 385 + 3 | 35
= 388 | 35
Ÿ¨¨´¡›r‡º° 388
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—´Šœ´Êœ 34567 y 89 = 388 Á«¬ 35
3715 + 3 = 385 + 3 = 388 ༡
92
˜´ª°¥nµŠš¸É 1.3.14 ‹Š®µ¦ 1011638 —oª¥ 987
œª‡·— MD …°Š 987 ‡º° 013 (˜o°ŠÁ…¸¥œÁž}œÁ¨… 3 ®¨´„Á®¤º°œ˜´ª®µ¦ ‹¹ŠÁ˜·¤ 0 ®œoµ 13)
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1 0 2 4 81 41 0 = 1024 | 950
Ÿ¨¨´¡›r‡º° 1024
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—´Šœ´Êœ 1011638 y 987 = 1024 Á«¬ 950
˜´ª°¥nµŠš¸É 1.3.15 ‹Š®µ¦ 300000 —oª¥ 9888
9888)30 0000
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Ÿ¨¨´¡›r‡º° 30
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—´Šœ´Êœ 300000 y 9888 = 30 Á«¬ 3360
Ĝ„¦–¸š¸É˜´ª˜´ÊŠ¤¸Á¨…×—š¸É¤¸‡nµ¤µ„„ªnµ 5 °¥¼n®¨µ¥˜´ª Á¦µ°µ‹‹³Âž¨ŠÁ¨…×—Á®¨nµœ´ÊœÃ—¥Äo
…¸—œ Á¡ºÉ°‡·—‡Îµœª–ץčoÁ¨…×—š¸Éœo°¥„ªnµ 5 ‹³­³—ª„„ªnµ—´Šœ¸Ê
93
˜´ª°¥nµŠš¸É 1.3.16 ‹Š®µ¦ 98564318 —oª¥ 9886
_ _ _
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œª‡·— 98564318 = 1 0 1 4 4 4 3 2 2
9886)98564318
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0114)10 1 4 4 4 3 2 2
01 14
0 00 0
0 0 00
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0 3 31 2
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00 0 0
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1 0 0 3 0 1 01 0 2 = 1 0 0 3 0 | 1 2 0 2
= 9970 | 0898
Ÿ¨¨´¡›r‡º° 9970
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—´Šœ´Êœ 98564318 y 9886 = 9970 Á«¬ 898
­œ»„„´˜´ªÁ¨… (9)
Á„oµ‡¼–„´Á„oµÅ—o Á„oµ ž— «¼œ¥r ®œ¹ÉŠ
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99u99
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94
˜´ª°¥nµŠš¸É 1.3.17 ‹Š®µ¦ 9876534201 —oª¥ 8876
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œª‡·— 9876534201 = 1 0 1 2 4 5 3 4 2 0 1
8876)9876534201
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1124)101 2 453 4 2 0 1
11 2 4
1124
1124
224 8
5 5 10 20
18 18 36 72
34 34 68 136
1 1 1 2 5 18 34 74 92 140 137
1 1 1 2 6 11 4 83 2 153 7
1 1 1 2 7 1 4 84 7 3 7
8 4737
8 8 16 32
8 12 15 19 39
1 1 1 2 7 2 2 13 7 2 9
1 3729
1124
1 4 8 4 13
1112723 4 853
Ÿ¨¨´¡›r‡º° 1112723
Á«¬‡º°
4853
—´Šœ´Êœ 9876534201 y 8876 = 1112723 Á«¬ 4853
95
˜´ª°¥nµŠš¸É 1.3.18 ‹Š®µ¦ 157689 —oª¥ 887
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887)157 6 8 9
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113)24 2 3 1 1
22 6
_ _
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2
26
_ _ _
2 2 6
_ _
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22 2 1 9 7
1 7 8 1_ 8 0 3
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178 180 3
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178 169 0
177
69 0
Ÿ¨¨´¡›r‡º° 177
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—´Šœ´Êœ 157689 y 887 = Á«¬ 690
­œ»„„´˜´ªÁ¨… (10)
«¼œ¥r ¦³®ªnµŠ®œ¹ÉŠ „´ ž—
12u9
112u99
1112u999
11112u9999
111112u99999
1111112u999999
11111112u9999999
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1111111112u999999999
=
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108
11088
1110888
111108888
11111088888
1111110888888
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96
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‹Š®µŸ¨®µ¦˜n°Åžœ¸Ê (˜°Äœ¦¼žŸ¨¨´¡›r ¨³Á«¬)
1. 68 y 9
2. 221 y 9
3. 3128 y 8
4. 6153 y 89
5. 212132 y 989
­œ»„„´˜´ªÁ¨… (11)
ŽÊε¨oªŽÊε°¸„
1/ 89 = 0. 01234567 9
1/ 891 = 0. 011223344 55667789
1/ 8991 = 0. 0 001112223 334445556667778 8 9
1/ 89991= 0. 0 0001111222 233 33444 45555666677 77888 9
1/ 899991 =
0. 0 0000111112 2222333 33444 445 5555666667 77778888 9
x
x
x
x
x
x
x
x
x
1/ 27 = 0. 0 3 7
1/ 297 = 0. 0 0336 7
1/ 2997 = 0. 0 0033366 7
1/ 29997 = 0. 0 00033336667
1/ 299997 = 0. 0 0000333336 666 7
x
x
x
x
x
x
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x
x
x
97
š«œ·¥¤
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Ášnµ„´‹Îµœªœ˜ÎµÂ®œnŠ…°Šš«œ·¥¤š¸É˜o°Š„µ¦ —´Šœ¸Ê
˜´ª°¥nµŠš¸É 1.3.19 ‹Š®µ‡nµ…°Š 345675 y 9 ˜o°Š„µ¦š«œ·¥¤ 10 ˜ÎµÂ®œnŠ 1
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1 3 7 12 18 25
3 7 12 18 25
38405 30
3
38405 3 3
3 8 4 0 5 +3 3
38408
3 0 0 0 0 0 0 0 0 0 0 (®µ¦ 3 š¸ÉÁž}œÁ«¬
3 3 3 3 3 3 3 3 3 3 ×¥Á˜·¤ 0 ®¨´Š 3
3 3 3 3 3 3 3 3 3 3 š¸ÉÁž}œÁ«¬ 10 ˜´ª
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0 ˜´ª…ªµ­»—)
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«¼œ¥r ¦³®ªnµŠÂž— „´ ®œ¹ÉŠ
89u9
889u99
8889u999
88889u9999
888889u99999
8888889u999999
88888889u9999999
888888889u99999999
=
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=
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801
88011
8880111
888801111
88888011111
8888880111111
888888801111111
88888888011111111
98
˜´ª°¥nµŠš¸É 1.3.20 ‹Š®µ‡nµ…°Š 2341 y 89 ˜o°Š„µ¦š«œ·¥¤ 3 ˜ÎµÂ®œnŠ
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89)23 41
11 2 2
5 5
2 5 11 6
25 1 16
11
1 27
2 5+1 2 7
2 6 2 7 0 0 0 (®µ¦Á«¬ 27 Á¤ºÉ°˜o°Š„µ¦š«œ·¥¤ 3 ˜ÎµÂ®œnŠ
22
Á˜·¤ 0 ‹Îµœªœ 3 ˜´ª®¨´Š 27 ¨oªšÎµ„µ¦®µ¦)
99
11 11
2 9 112011
3 0 3 11
26 303
26.303
—´Šœ´Êœ 2341 y 89 = 26.303
­œ»„„´˜´ªÁ¨… (13)
šÎµ°¥nµŠÅ¦„È®œ¸Å¤n¡oœÂž—
(
(
(
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0u9)+8
9u9)+7
98u9)+6
987u9)+5
9876u9)+4
98765u9)+3
987654u9)+2
9876543 u9)+1
98765432 u9)+0
9 8 7 6 5 4 3 2 1 u 9 ) + -1
=
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8
88
888
8888
88888
888888
8888888
88888888
888888888
88888888888
99
˜´ª°¥nµŠš¸É 1.3.21 ‹Š®µ‡nµ…°Š 2467 y 8 ˜o°Š„µ¦š«œ·¥¤ 2 ˜ÎµÂ®œnŠ
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8)246 7
2 4 16 44
2 8 22 51
302 5 1
10
5 11
3 0 7 11
307 1 1
2
1 3
308 300
6 12
3 6 12
372
308 372
308 .37
—´Šœ´Êœ 2467 y 8 = 308.37
­œ»„„´˜´ªÁ¨… (14)
„µ¦ª„¡µ¨·œÃ—¦¤Â¨³„ε¨´Š­°Š
1
1+2+1
1+2+3+2+1
1+2+3+4+3+2+1
1+2+3+4+5+4+3+2+1
1+2+3+4+5+6+5+4+3+2+1
1+2+3+4+5+6+7+6+5+4+3+2+1
1+2+3+4+5+6+7+8+7+6+5+4+3+2+1
1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1
=
=
=
=
=
=
=
=
=
1
4
9
16
25
36
49
64
81
=
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=
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=
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12
22
32
42
52
62
72
82
92
100
˜´ª°¥nµŠš¸É 1.3.22 ‹Š®µ‡nµ…°Š 64532 y 98 ˜o°Š„µ¦š«œ·¥¤ 3 ˜ÎµÂ®œnŠ
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98)645 32
0 2 0 12
0 8
0 34
6 4 17 11 36
6 5 7 14 6
657 1 46
02
1 48
658 4800
08
0 16
4 8 8 16
4896
658 4896
658 .489
—´Šœ´Êœ 64532 y 98 = 658.489
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²¨² …°Š˜´ª®µ¦Á®¨nµœ´Êœ¤¸˜´ªÁ¨…×—˜n¨³˜´ª¤¸‡nµœo°¥ œ°„‹µ„œ¸Ê„µ¦®µ¦­´ŠÁ‡¦µ³®r‹³Áž}œ„µ¦
ž’·´˜·„µ¦„¨´„´®µ¦›¦¦¤—µ „¨nµª‡º° čo„µ¦‡¼– ¨³„µ¦ª„ Ĝ…–³š¸É„µ¦®µ¦›¦¦¤—µ˜o°ŠÄo„µ¦
®µ¦ ¨³„µ¦¨ „µ¦®µ¦Ã—¥ª·›¸œ¸Ê‹¹Š¦ª—Á¦Èª„ªnµ
 f„®´—š¸É 10
‹Š®µŸ¨®µ¦˜n°Åžœ¸Ê (˜°Äœ¦¼žš«œ·¥¤®oµ˜ÎµÂ®œnŠ)
1. 68 y 9
2. 221 y 9
3. 3128 y 8
4. 6153 y 89
5. 212132 y 989
101
®œnª¥š¸É 2
¨Îµ—´…°Š‹ÎµœªœÁ·Š¦¼žÁ¦…µ‡–·˜
˜°œš¸É 2.1 ¨Îµ—´…°Š‹ÎµœªœÁ·Š ¦¼ž­µ¤Á®¨¸É¥¤ ¨³‹ÎµœªœÁ·Š¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­
˜°œš¸É 2.2 ¨Îµ—´…°Š‹ÎµœªœÁ·Š¦¼ž¡¸¦µ¤·— “µœ¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ ¨³“µœ¦¼ž
­¸ÉÁ®¨¸É¥¤‹´˜»¦´­
˜°œš¸É 2.3 ‡ªµ¤­´¤¡´œ›r¦³®ªnµŠ‹ÎµœªœÁ·Š¦¼žÁ¦…µ‡–·˜
œª‡·— 1. ¦¼žšµŠ‡–·˜«µ­˜¦r Áž}œ¦¼ž›¦¦¤š¸É­ºÉ°Ä®oÁ„·—„¦³ªœ„µ¦šµŠ‡ªµ¤‡·—Å—o°¥nµŠ
­¦oµŠ­¦¦‡r ¨³¤¸Á­¦¸£µ¡œÎµÅž­¼nž’·´˜·„µ¦šµŠ‡–·˜«µ­˜¦rÁ¡ºÉ°­¦oµŠ°Š‡r‡ªµ¤¦¼oÄ®¤n
¡´•œµ‡ªµ¤¦¼oÄ®o„ªoµŠÂ¨³¨¹„Ž¹ÊŠ…¹Êœ šÎµÄ®o¤„¸ µ¦¡´•œµ„µ¦Äœ„¦³ªœ„µ¦Â„ož{®µš¸É—¸
2. ¨Îµ—´…°Š‹ÎµœªœÁ·Š¦¼žÁ¦…µ‡–·˜Áž}œÂ¦¼ž®œ¹ÉŠšµŠ‡–·˜«µ­˜¦rž¦³„°—oª¥ ¨Îµ—´
…°Š‹ÎµœªœÁ·Š¦¼ž­µ¤Á®¨¸É¥¤ ¦¼ž­¸ÉÁ®¨¸É¥¤ ¨³¡¸¦µ¤·— Áž}œ˜oœ
3. ‹ÎµœªœÁ·Š¦¼žÁ¦…µ‡–·˜­µ¤µ¦™Á…¸¥œÁž}œ Ÿ¨ª„…°Š‹ÎµœªœÂ¨³¦¼žš´ÉªÅžÅ—o
œ°„‹µ„œ¸Ê ‹ÎµœªœÁ®¨nµœ¸¥Ê ´Š¤¸‡ªµ¤­´¤¡´œ›r„´œÅ—o ®¨µ„®¨µ¥¦¼žÂ
ª´˜™»ž¦³­Š‡r
Á¤ºÉ°«¹„¬µ®œnª¥š¸É 9 ‹Â¨oª œ´„Á¦¸¥œ­µ¤µ¦™
1. ­¦oµŠ¦¼žÁ¦…µ‡–·˜š¸ÉÁ„¸¥É ª…o°Š„´ ‹ÎµœªœÁ·Š¦¼žÁ¦…µ‡–·˜Å—o
2. ®µ‡nµ…°Š‹ÎµœªœÁ·Š¦¼žÁ¦…µ‡–·˜Å—o
3. ®µ‡ªµ¤­´¤¡´œ›r¦³®ªnµŠ‹ÎµœªœÁ·Š¦¼žÁ¦…µ‡–·˜Å—o
4. čo­´¨´„¬–r ¦ š¸ÉÁ„¸É¥ª…o°Š„´‹ÎµœªœÁ·Š¦¼žÁ¦…µ‡–·˜Å—o
„·‹„¦¦¤¦³®ªnµŠÁ¦¸¥œ
1. °µ‹µ¦¥r°›·µ¥‡ªµ¤®¤µ¥…°Š¦¼žÂšµŠ‡–·˜«µ­˜¦r ¨Îµ—´…°Š‹ÎµœªœÁ·Š¦¼ž
Á¦…µ‡–·˜ ¡¦o°¤š´ÊŠÄ®o˜ª´ °¥nµŠ¨Îµ—´…°Š‹ÎµœªœÁ·Š¦¼ž­µ¤Á®¨¸É¥¤ ‹ÎµœªœÁ·Š¦¼ž
­¸ÉÁ®¨¸É¥¤‹´˜»¦´­ ‹ÎµœªœÁ·Š¦¼ž¡¸¦µ¤·—“µœ¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ ¨³“µœ¦¼ž­¸ÉÁ®¨¸É¥¤
‹´˜»¦´­
2. œ´„Á¦¸¥œ°£·ž¦µ¥‡ªµ¤­´¤¡´œ›r¦³®ªnµŠ‹ÎµœªœÁ·Š¦¼žÁ¦…µ‡–·˜ š´ÊŠ‡ªµ¤­´¤¡´œ›r‹µ„
Ÿ¨ª„ ¨³‡ªµ¤­´¤¡´œ›r‹µ„¦¼žš´ÉªÅž
3. œ´„Á¦¸¥œšÎµ„·‹„¦¦¤˜µ¤˜´ª°¥nµŠ ¨³Â f„®´—Á¡ºÉ°®µŸ¨ª„…°Š‹Îµœªœ ¨³¦¼ž
š´ÉªÅž…°Š‹ÎµœªœÁ·Š¦¼žÁ¦…µ‡–·˜
102
­ºÉ°„µ¦­°œ
1. Á°„­µ¦„µ¦­°œ
2.  f„ž’·´˜·
3. Á‡¦ºÉ°ŠŒµ¥…oµ¤«¸¦¬³
ž¦³Á¤·œŸ¨
ž¦³Á¤·œŸ¨‹µ„ f„®´—¨³„µ¦š—­°
˜°œš¸É 2.1 ¨Îµ—´…°Š‹ÎµœªœÁ·Š¦¼ž­µ¤Á®¨¸É¥¤ ¨³‹ÎµœªœÁ·Š¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­
Á¦ºÉ°Šš¸É 1 ¨Îµ—´…°Š‹ÎµœªœÁ·Š¦¼ž­µ¤Á®¨¸É¥¤
1. ¨Îµ—´…°Š‹ÎµœªœÁ·Š¦¼ž­µ¤Á®¨¸É¥¤
Á¤ºÉ°Á…¸¥œ‹»— x Ä®oÁ¦¸¥Š˜´ª„´œÁž}œ¦¼ž­µ¤Á®¨¸É¥¤‹³Å—o ‹Îµœªœ‹»—š¸ÂÉ šœ‹ÎµœªœÁ¦¸¥Š˜´ª
Áž}œ¨Îµ—´—´Šœ¸Ê
•
•
••
•
••
•••
•
••
•••
••••
•
••
•••
••••
•••••
‹Îµœªœ 1
3
6
10
5
¨Îµ—´š¸É 1
2
3
4
5
... n
ŗo¨Îµ—´…°Š‹ÎµœªœÁ·Š¦¼ž­µ¤Á®¨¸É¥¤ ‡º° 1, 3, 6, 10, 15,…, 12 n ( n + 1 )
šÎµÅ¤‹ÎµœªœÁ·Š¦¼ž­µ¤Á®¨¸É¥¤¨Îµ—´š¸É n ¤¸¦¼žš´ÉªÅž ‡º° 12 n ( n + 1 )
¦¼žš´ÉªÅžœ¸Ê°µ‹Â­—ŠÁ®˜»Ÿ¨ž¦³„°Å—o—Š´ œ¸Ê
™oµÁ…¸¥œ‹»—Á¦¸¥Š„´œÁž}œ¦¼ž­¸ÉÁ®¨¸É¥¤ŸºœŸoµ…œµ— 4 u 5 ‹»— ¨oªÂnŠÁž}œ‹ÎµœªœÁ·Š¦¼ž­µ¤Á®¨¸É¥¤
¨Îµ—´š¸É 4 ‹³Å—o­°Š¦¼ž—´Šœ¸Ê
‹³¡ªnµ‹ÎµœªœÁ·Š¦¼ž­µ¤Á®¨¸É¥¤ ¨Îµ—´š¸É 4 ‡º° 10 ¨³ 10 = 4 x 5
2
šÎµœ°ŠÁ—¸¥ª„´œ ™oµÁ…¸¥œ‹»—Á¦¸¥Š„´œÁž}œ¦¼ž­¸ÉÁ®¨¸É¥¤ŸºœŸoµ…œµ— n u (n+1) ‹»— ‹³ÂnŠ‹ÎµœªœÁ·Š¦¼ž
­µ¤Á®¨¸É¥¤¨Îµ—´š¸É n ŗo ­°Š¦¼ž
103
‹¹ŠÅ—o ‹ÎµœªœÁ·Š¦¼ž­µ¤Á®¨¸É¥¤¨Îµ—´š¸É n ‡º° 12 n(n+1)
™oµÂšœ‹ÎµœªœÁ·Š¦¼ž­µ¤Á®¨¸É¥¤¨Îµ—´š¸É n —oª¥ ' n‹³Å—o
'1 = 1 '2 = 3 '3 = 6 '4 = 10 '5 = 15 'n = 12 n(n+1)
2. Ÿ¨ª„‹Îµœªœœ´ n ‹ÎµœªœÂ¦„ 1 + 2 + 3 + ... + n (Á…¸¥œÂšœ—oª¥
Á¤ºÉ°Å°¤¸‡nµ˜´ÊŠÂ˜n®œ¹ÉŠ™¹ŠÁ°Èœ)
1
Ÿ¨ª„
'n
¦i
i 1
i 1
°nµœªnµ Ž·„¤µÅ°
1+2 = 3 1+2+3 = 6 1+2+3+4 = 1+2+3+4+5 =
10
15
1
‹Îµœªœ
n
¦i
2
¦i
1 i 1
'1
3
3
¦i
i 1
'2
6
4
¦i
i 1
'3
10
'4
5
¦i
i 1
15
'5
 f„®´— 11
1. ‹Š®µ‡nµ…°Š 'n ¨³‡nµ…°Š‹Îµœªœš¸É„ε®œ——´Š˜n°Åžœ¸Ê
(1) '10 , '11 , '99 , '100 ¨³ '101
(2) '9+10 , '10+11 , '98+99 , '99+100 ¨³ '100+101
(3) '1+'2 , '2 +'3 , '3 +'4 , '99+ '100
2. ‹Š®µ‡ªµ¤­´¤¡´œ›r¦³®ªnµŠ 'n-1 , 'n , n ¨³ n2 Á¤ºÉ° n Áž}œ‹ÎµœªœÁ˜È¤ª„š¸É¤µ„„ªnµ 1
104
Á¦ºÉ°Šš¸É 2 ¨Îµ—´…°Š‹ÎµœªœÁ·Š¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­
1. ¨Îµ—´…°Š‹ÎµœªœÁ·Š¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­
Á¤ºÉ°Á…¸¥œ‹»—Ä®oÁ¦¸¥Š˜´ª„´œÁž}œ¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­‹³Å—o‹µÎ œªœ‹»—š¸ÂÉ šœ‹Îµœªœ‹³Á¦¸¥Š
˜´ªÁž}œ¨Îµ—´—´Šœ¸Ê
‹Îµœªœ 1
4
9
16
25
¨Îµ—´š¸É 1
2
3
4
5
ŗo¨Îµ—´…°Š‹ÎµœªœÁ·Š¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­ ‡º° 1, 4, 9, 16, 25, …, n2
ŗo¨Îµ—´…°Š‹ÎµœªœÁ˜È¤ª„š¸É¥„„ε¨´Š­°Š®¦º°¨Îµ—´…°Š n2 ‡º°
... n
12, 22, 32, 42, 52,…, n2 ™oµÂšœ‹ÎµœªœÁ·Š¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦­´ š¸É n —oª¥ F n ‹³Å—o
F 1 = 12 F 2 = 22 F 3 = 32 F 4 = 42 F 5 = 52 ¨³ F n = n2
2. Ÿ¨ª„…°Š‹Îµœªœ‡¸Éª„ n ‹ÎµœªœÂ¦„ 1 + 3 + 5 + 7 + ... + (2n – 1)
‹µ„‹ÎµœªœÁ·Š¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜¦» ´­Á¤ºÉ°¡·‹µ¦–µÂœª®´„Áž}œ¤»¤Œµ„( )‹³¡„µ¦Á¦¸¥Š˜´ª…°Š‹»—
Á¡·É¤…¹ÊœÁž}œ¨Îµ—´…°Š‹Îµœªœ‡¸É —´Š¦¼ž
1 1+3 = 4 1+3+5=9 1+3+5+7=16 1+3+5+7+9=25
1=12 1+3 = 22 1+3+5=32 1+3+5+7=42 1+3+5+7+9=52
­°—‡¨o°Š„´œ„´¦¼žš´ÉªÅžš¸ªÉ nµŸ¨ª„…°Š‹Îµœªœª„‡¸É n ‹ÎµœªœÂ¦„Ášnµ„´ n2
œ´Éœ‡º° 1+3+5+...+(2n-1)= n2
Ž¹ÉŠ 1+3+5+...+(2n-1) Á…¸¥œÄœ¦¼žŽ·„¤µÅ—oÁž}œ
Å°¤¸‡nµ˜´ÊŠÂ˜n®œ¹ÉŠ™¹ŠÁ°Èœ
‹¹ŠÅ—o F n = n2 =
n
¦ (2i
i 1
1)
n
¦ (2i
i 1
1) °nµœªnµ Ž·„¤µ…°Š­°ŠÅ°¨®œ¹ÉŠÁ¤ºÉ°
105
 f„®´— 12
1. ‹ŠÁ…¸¥œ F n Ĝ¦¼žŸ¨ª„…°Š‹Îµœªœ‡¸Éª„ n ‹ÎµœªœÂ¦„‹µ„ F n ˜n°Åžœ¸Ê
F 3 , F 5 , F 7 , F 10 , F 21 ¨³ F 25
2. ‹Š«¹„¬µÂ¦¼ž…°Š n2 ¨³ n3 —´Šœ¸Ê
1
= 12
1
= 13
1+3
= 22
3+5
= 23
7 + 9 + 11
= 33
1+3+5
= 32
1+3+5+7
= 42
13 + 15 + 17 + 19 = 43
1 + 3 + 5 + 7 + 9 = 52 21 + 23 + 25 + 27 + 29 = 53
¡ªnµ
13 = 12 = F 1
13 + 23 = 1 + 3 + 5 = F 3
13 + 23 + 33 = 1 + 3 + 5 + 7 + 9 + 11 = 62 = F 6
‹ŠÁ…¸¥œ F 10 Ĝ¦¼ž…°ŠŸ¨ª„„ε¨´Š­µ¤…°Š‹Îµœªœœ´š¸ÉÁ¦¸¥Š„´œ Á¦·É¤‹µ„ 13
3. ‹Š®µ‡ªµ¤­´¤¡´œ›r¦³®ªnµŠ F n-1, F n , n ¨³ n2 Á¤ºÉ° n Áž}œ‹ÎµœªœÁ˜È¤ª„š¸É¤µ„„ªnµ 1
4. ¡·‹µ¦–µÂ¦¼ž¨Îµ—´…°Š„µ¦ª„˜n°Åžœ¸Ê
1
=1
= 12
1+2+1
=4
= 22
1+2+3+2+1
=9
= 32
1+2+3+4+3+2+1
= 16
= 42
1 + 2 + 3 + 4 +5 + 4 + 3 + 2 + 1
= 25
= 52
x
x
x
x
x
x
x
x
x
‹³¡ªnµ 1 +2+3+…+ ( n –1 ) + n + ( n –1 ) + … +3 + 2 +1 = n2
™oµÁ¦¸¥„„µ¦ª„¨´„¬–³œ¸ªÊ nµ “„µ¦ª„…´Êœ´œÅ—” ¨³Á¦¸¥„Ÿ¨ª„š¸Éŗoªnµ‹Îµœªœ…´ÊœÁ·Š
´œÅ— Á…¸¥œÂšœ—oª¥ n ‹³¡ªnµ
n = Fn
‹Š®µ‡ªµ¤­´¤¡´œ›r¦³®ªnµŠ n-1 , n , n ¨³ n2 Á¤ºÉ° n Áž}œ ‹ÎµœªœÁ˜È¤ª„š¸É¤µ„„ªnµ 1
106
˜°œš¸É 2.2 ¨Îµ—´…°Š‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ ¨³
“µœ¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­
Á¦ºÉ°Šš¸É 1 ¨Îµ—´…°Š‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ
1. ¨Îµ—´…°Š‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ
™oµªµŠ¨¼„„¨¤Ä®oÁž}œ¡¸¦µ¤·—“µœ¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ ‹³Å—o‹Îµœªœ¨¼„„¨¤Âšœ‹Îµœªœš¸ÉÁ¦¸¥Š˜´ª
„´œÁž}œ¨Îµ—´—´Šœ¸Ê
‹Îµœªœ 1
4
10
20
35
¨Îµ—´š¸É 1
2
3
4
5
... n
ŗo¨Îµ—´…°Š‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ ‡º° 1, 4, 10, 20, 35, …, 16 n ( n+ 1 )( n +2 )
šÎµÅ¤‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ ¨Îµ—´š¸É n ¤¸¦¼žš´ÉªÅž‡º° 16 n ( n+ 1 )( n +2 )
Ĝ„µ¦­¦oµŠ‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ¨Îµ—´š¸É n ‹³¡ªnµ Á¦·É¤Â¦„„µ¦­¦oµŠ“µœ
‹³­¦oµŠÁž}œ¦¼ž 'n „n°œ ¨oª­¦oµŠ´Êœ˜n°ÅžÁž}œ¦¼ž 'n-1 ´Êœ­¼Š…¹ÊœÁž}œ 'n-2 ‹œ´Êœ­»—šoµ¥‡º° '1 =1
¦¼žš´ÉªÅž°µ‹Â­—ŠÁ®˜»Ÿ¨ž¦³„°Å—o—´Šœ¸Ê
™oµÂšœ‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ ¨Îµ—´š¸É n —oª¥ 'n ‹³Å—o
1 = '1 = 1
2 = '1 + '2 = 1 + 3 = 4
3 = '1 + '2 + '3 = 1 + 3 + 6 = 10
4 = '1 + '2 + '3 + '4 = 1 + 3 + 6 + 10 = 20
5 = '1 + '2 + '3 + '4 + '5 = 1 + 3 + 6 +10 + 15 = 35
¡·‹µ¦–µ
4 = 1 + 3 + 6 + 10
‹³Å—o
4 = 1 + (1+2) + (1+2+3) + (1+2+3+4)
®¦º°
4 = 1 + (2+1) + (3+2+1) + (4+3+2+1)
®¦º°
4 = 4 + (3+3) + (2+2+2) + (1+1+1+1)
Á¤ºÉ°ª„˜µ¤Âœª˜´ÊŠ‹³Å—o 3 4 = (1+1+4) + (1+2+3) + (2+1+3) + (1+3+2) + (2+2+2) + (3+1+2)
+ (1+4+1) + (2+3+1) + (3+2+1) + (4+1+1)
= 6 +6 +6 +6 +6 +6 +6 +6 +6 +6
107
= 10 u 6
= (1+2+3+4 ) u (4+2)
3 4 = '4 ( 4+2 )
™oµÁ…¸¥œ
4
Ĝ¦¼ž„µ¦ª„‹³Á…¸¥œ—´Šœ¸Ê
4
=
‹³Á®ÈœÅ—oªnµ 3
4
=
+
=
¨³
3
3
šÎµœ°ŠÁ—¸¥ª„´œ‹³Å—o
—´Šœ´Êœ
+
=
u 6 = (1+2+3+4) u (4+2)
= '5 (5 + 2)
n = 'n (n + 2)
1
n = 3 'n (n + 2)
= 16 n (n + 1)(n + 2)
5
2. Ÿ¨ª„…°Š‹ÎµœªœÁ·Š¦¼ž­µ¤Á®¨¸É¥¤ (Á…¸¥œÂšœ—oª¥
n
¦
i 1
ǻ i °nµœªnµ Ž·„¤µ¦¼ž­µ¤Á®¨¸É¥¤Å°
Á¤ºÉ°Å°¤¸‡nµ˜´ÊŠÂ˜n®œ¹ÉŠ™¹ŠÁ°Èœ)
‹µ„‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ ‹³¡ªnµ„µ¦Á¦¸¥Š˜´ª¨¼„„¨¤‹µ„“µœš¸ÁÉ ž}œ¦¼ž
­µ¤Á®¨¸É¥¤—oµœÁšnµ‹³Žo°œ„´œÁž}œ¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ…¹Êœ¤µÁž}œ´ÊœÇ ×¥¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ´Êœ
œ‹³¤¸‡ªµ¤¥µª—oµœœo°¥„ªnµ ‡ªµ¤¥µª—oµœ…°Š¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ´Êœ¨nµŠš¸É˜·—„´œ°¥¼n 1 ®œnª¥
108
™oµ¡¸¦µ¤·—­¼Š 5 ´Êœ‹³¡ªnµ ´Êœš¸É 1 ™¹Š´Êœš¸É 5 ‹³¤¸¨¼„„¨¤°¥¼nÁž}œ‹Îµœªœ 5˜6 , 4 ˜5 , 3˜4 , 2˜3
2
¨³ 1˜2 Á¤ºÉ°Á…¸¥œÁ¦¸¥ŠÄ®¤n ‹³Å—o 1˜2 , 2˜3 , 3˜4 , 4 ˜5 ¨³ 5˜6
2
2
2 2 2 2
2
™oµœÎµ¤µª„„´œ‹³Å—o 1˜2 + 2˜3 + 3˜4 + 4 ˜5 + 5˜6 = '1 + '2 + '3 + '4 + '5
2
2
2
2
2
5
œ´Éœ‡º°
¦
'i = '1 + '2 + '3 + '4 + '5
¦
'i = '1 + '2 + '3 + ... + 'n
i1
n
‹³Å—o
i 1
= 1+ 2 + 3 + ... + i
ÁœºÉ°Š‹µ„ 'i
=
i
¦
j
j 1
‹µ„ (1) ¨³ (2) ‹¹ŠÅ—o
n
¦
i 1
˜n
—´Šœ´Êœ
n
'i =
=
§ i ·
¨¨ ¦ j ¸¸
i 1 ©j 1¹
n
¦
n
¦
'i
n
=
i 1
n § i ·
¨¨ ¦ j ¸¸
i 1 ©j 1¹
¦
…(1)
…(2)
2
2
109
n
§ i ·
¨¨ ¦ j ¸¸ „´ ¦ 'i
i 1 ©j 1¹
i 1
Ÿ¨ª„Äœ¦¼žŽ·„¤µ…°ŠŽ·„¤µ Ÿ¨ª„Äœ¦¼žŽ·„¤µ…°Š 'i
n
˜µ¦µŠ˜n°Åžœ¸ÁÊ ž}œ„µ¦Áž¦¸¥Áš¸¥„µ¦®µ‡nµ
¦
1 § i ·
1=
= ¦ ¨¨ ¦ j ¸¸
i 1 ©j 1¹
1
=¦
1
j
=
1
2 § i ·
¦ ¨¨ ¦ j ¸¸
i 1 ©j 1 ¹
1
¦
j+
2
¦
=
2=
j 1 j 1
1
¦
j+
2
¦
j 1 j 1
j+
3=
3
¦
j
j 1
= 1 + (1+2) + (1+2+3)
= 10
œ°„‹µ„œ¸Ê Áœº°É Š‹µ„
n
§i · n
n
¨³ n = ¦ ¨¨¦j¸¸ = ¦ ( 12 i(i +1)) = ¦ ( 12 (i2 + i))
i = 1 © J = 1¹ i = 1
i=1
§i ·
n
‹¹ŠÅ—o ¦ ¨¨¦j¸¸ = 1 n (n + 1)(n + 2)
i = 1© J =¹1 6
¨³ ¦( 1 (i2 + i)) = 1 n (n + 1)(n + 2)
n
i=1
2
6
i 1
'i
3
¦
i 1
'i
= '1+ '2+ '3
= 1 + (1+2) + (1+2+3)
= 10
= 16 n (n + 1)(n + 2)
n
2
¦
= '1+ '2
= 1 + (1+2)
=4
j
= 1 + (1+2)
=4
3 § i ·
3= ¦ ¨
¨ ¦ j ¸¸
i 1 ©j 1¹
i 1
'i
= '1
=1
=1
2=
1
¦
110
 f„®´— 13
1. ‹Š®µ‡nµ…°Š n —´Š˜n°Åžœ¸Ê
6,
7,
10 ,
50 ¨³
100
2. ‹ŠÁ…¸¥œ n Ĝ¦¼ž n-1 ¨³ n Á¤ºÉ° n Áž}œ‹ÎµœªœÁ˜È¤ª„¤µ„„ªnµ 1
3. ‹ŠÁ…¸¥œ 4 ¨³ 5 ¨³ 6 Ĝ¦¼žŽ·„¤µ…°Š 'i ¨³Ž·„¤µ…°ŠŽ·„¤µ
Á¦ºÉ°Šš¸É 2 ¨Îµ—´…°Š‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­
1.¨Îµ—´…°Š‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­¸ÉÁ®¨¸¥É ¤‹´˜»¦´­
™oµªµŠ¨¼„„¨¤Ä®oÁž}œ¡¸¦µ¤·—“µœ¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­‹³Å—o‹µÎ œªœ¨¼„„¨¤Âšœ‹Îµœªœš¸É Á¦¸¥Š˜´ª
„´œÁž}œ¨Îµ—´—´Šœ¸Ê
‹Îµœªœ 1
¨Îµ—´š¸É 1
5
2
14
3
30
4
55
5
ŗo¨Îµ—´…°Š‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­‡º° 1, 5, 14, 30, 55, .., 16 n (n + 1)(n + 2)
šÎµÅ¤‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜¦» ´­ ¨Îµ—´š¸É n ¤¸¦¼žš´ÉªÅž‡º° 16 n (n + 1)(n + 2)
Ĝ„µ¦­¦oµŠ‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­¸ÉÁ®¨¸¥É ¤‹´˜»¦´­ ¨Îµ—´š¸É n ‹³¡ªnµÁ¦·É¤Â¦„Áž}œ¦¼ž n= n2
‹œ™¹Š´Êœ­»—šoµ¥‡º° 1 = 1 „µ¦­¦oµŠ“µœ‹³­¦oµŠÁž}œ¦¼ž n „n°œÂ¨oª­¦oµŠ´Êœ˜n°ÅžÄœ¦¼ž F n-1
´Êœ­¼Š…¹Êœ ™oµÂšœ‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­—oª¥ n ‹³Å—o
1=
1,
2=
1+
2,
n=
1+
2+
3 + ... +
n
¦¼žš´ÉªÅžœ¸Ê °µ‹Â­—ŠÃ—¥Â¦¼ž…°Š‹Îµœªœš¸Éª„„´œ—´Šœ¸Ê
111
121
222
323
424
1
1 2 1
1 2 3 2 1
1 2 3 4 3 2 1
1 2 3 4
4 3 2 1
2 3 4
4 3 2
3 4
4 3
4
4
cdefghijk
12
22
32
42
122
222
3232
422
‹µ„¦¼ž‹³¡ªnµ 3 ( 12 + 22 + 32 + 42 ) = 9 ( 1+2+3+4 )
= 9 '4
3 4
= (( 2 u 4 ) + 1 ) '4
šÎµœ°ŠÁ—¸¥ª„´œ­Îµ®¦´ 5 ‹³Å—o
= (( 2 u 5 ) + 1 ) '5
3 5
¨³
3 n
= ( 2n + 1) 'n
= ( 2n + 1) 12 n(n+1)
= 16 n (n + 1)(2n + 1)
n
2. Ÿ¨ª„…°Š„ε¨´Š­°Š…°Š‹Îµœªœœ´ n ‹ÎµœªœÂ¦„ (Á…¸¥œÂšœ—oª¥
n 2
¦i
i 1
°nµœªnµ Ž·„¤µÅ°
„ε¨´Š­°Š Á¤ºÉ°Å°¤¸‡nµ˜´ÊŠÂ˜n®œ¹ÉŠ™¹ŠÁ°Èœ)
‹µ„‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­‹³¡ªnµ„µ¦Á¦¸¥Š˜´ª¨¼„„¨¤‹µ„“µœš¸ÉÁž}œ¦¼ž
­¸ÉÁ®¨¸É¥¤‹´˜»¦´­ ‹³Žo°œ„´œÁž}œ¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­…¹Êœ¤µÁž}œ´ÊœÇ ×¥¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­´Êœœ‹³¤¸
‡ªµ¤¥µª—oµœœo°¥„ªnµ‡ªµ¤¥µª—oµœ…°Š¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦­´ ´Êœ¨nµŠš¸É˜·—„´œ°¥¼n 1 ®œnª¥
™oµ¡¸¦µ¤·—­¼Š 5 ´Êœ‹³¡ªnµ ´Êœš¸É 1 ™¹Š´Êœš¸É 5 ‹³¤¸¨¼„„¨¤Áž}œ‹Îµœªœ 52 , 42 , 32 , 22 ¨³ 12
Á¤ºÉ°Á…¸¥œÁ¦¸¥ŠÄ®¤n‹³Å—o 12 , 22 , 32 , 42 , 52
™oµœÎµ¤µª„„´œ‹³Å—o 12 + 22 + 32 + 42 + 52 = 55
‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­‡·—˜µ¤‡ªµ¤¥µª—oµœ“µœÅ—o—Š´ œ¸Ê
“µœ 1 ®œnª¥ Ášnµ„´ 1
“µœ 2 ®œnª¥ Ášnµ„´ 12 + 22 = 5
“µœ 3 ®œnª¥ Ášnµ„´ 12 + 22 + 32 = 14
112
“µœ 4 ®œnª¥ Ášnµ„´ 12 + 22 + 32 + 42 = 30
“µœ 5 ®œnª¥ Ášnµ„´ 12 + 22 + 32 + 42 + 52 = 55
1=
‹³Å—o
2=
¨³
3
=
4
=
5
=
2 2
1 2
¦i
i 1
= 12 = 1
¦i
= 12 + 22 = 5
¦i
= 12 + 22 + 32 = 14
¦i
= 12 + 22 + 32 + 42 = 30
¦i
= 12 + 22 + 32 + 42 + 52 = 55
i 1
3 2
i 1
4 2
i 1
5 2
i 1
n=
n 2
¦i
i 1
= 16 n (n + 1)(2n + 1)
 f„®´— 14
1. ‹Š®µ‡nµ n —´Š˜n°Åžœ¸Ê
6,
7,
10 ,
50 ¨³
100
2. ‹ŠÁ…¸¥œ n Ĝ¦¼ž n-1 ¨³ n Á¤ºÉ° n Áž}œ‹ÎµœªœÁ˜È¤ª„¤µ„„ªnµ 1
˜°œš¸É 2.3 ‡ªµ¤­´¤¡´œ›r¦³®ªnµŠ‹ÎµœªœÁ·Š¦¼žÁ¦…µ‡–·˜
Á¦ºÉ°Šš¸É 1 ‡ªµ¤­´¤¡´œ›r¦³®ªnµŠ‹ÎµœªœÁ·Š¦¼ž­µ¤Á®¨¸É¥¤ ‹ÎµœªœÁ·Š¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­Â¨³‹Îµœªœ
Á·Š…´Êœ´œÅ—
1. ‡ªµ¤­´¤¡´œ›r‹µ„Ÿ¨ª„
™oµÄ®o 'n Áž}œ‹ÎµœªœÁ·Š¦¼ž­µ¤Á®¨¸É¥¤¨Îµ—´š¸É n („ε®œ— '0 = 0 )
n Áž}œ‹ÎµœªœÁ·Š¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­¨Îµ—´š¸É n („ε®œ—
0=0 )
n Áž}œ‹ÎµœªœÁ·Š…´Êœ´œÅ— („ε®œ—
0 = 0)
‹³Å—o
'n = 1+2+3+ … + n
113
n
—´Šœ´Êœ
n
=
= n2 = 1+3+5+ … + (2n – 1)
2
n= n = 1+2+3+ … + (n – 1) + n + (n – 1) + … + 3+ 2 + 1
= (1+2+3+ … + (n – 1) + n) + (1+2+3+ … + (n – 1))
= 'n + 'n – 1
n = 'n + 'n-1
2. ‡ªµ¤­´¤¡´œ›r‹µ„¦¼žš´ÉªÅž
‹µ„¦¼žš´ÉªÅž…°Š 'n = 12 n(n+1)
¨³ 'n-1 = 12 (n – 1) n
‹³Å—o 'n + 'n-1 = 1 n(n+1) + 1 (n – 1) n
2
2
= 1 (n(n+1) + (n – 1) n)
2
= 1 (n2 + n + n2 – n)
2
—´Šœ´Êœ
n
=
= n2
= n= n
n = 'n + 'n-1
Á¦ºÉ°Šš¸É 2 ‡ªµ¤­´¤¡´œ›r¦³®ªnµŠ‹Îµœªœ¡¸¦µ¤·—“µœ¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ ¨³‹Îµœªœ¡¸¦µ¤·—“µœ¦¼ž
­¸ÉÁ®¨¸É¥¤‹´˜»¦´­
1. ‡ªµ¤­´¤¡´œ›r‹µ„Ÿ¨ª„
™oµÄ®o n Áž}œ ‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­µ¤Á®¨¸É¥¤—oµœÁšnµ ¨Îµ—´š¸É n („ε®œ— 0 = 0 )
n Áž}œ ‹ÎµœªœÁ·Š¡¸¦µ¤·—“µœ¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­ ¨Îµ—´š¸É n („ε®œ—
0=0)
‹³Å—o
n = '1 + '2 + '3 + ... + ' n
n=
1+
2+
3 + ... +
n
¡·‹µ¦–µ n = 1 + 2 + 3 + ... + n
= ('0 + '1) + ('1 + '2) + ('2 + '3) + … + ('n-2 + ' n-1) + ('n-1 + 'n)
= ('0 + '1 + '2 + … + 'n-1 ) + ('1 + '2 + '3 + … + 'n)
= n-1 + n
—´Šœ´Êœ n
= n-1 + n
œ°„‹µ„œ¸Ê n = n-1 + 'n ¨³ n-1 = n – 'n
= n-1 + n-1 + 'n
‹³Å—o
n
114
‹µ„
‹³Å—o
n
n
=2
=2
=2
=2
+ 'n
n-1 + 'n ¨³
n – 2'n + 'n
n – 'n
n-1
2. ‡ªµ¤­´¤¡´œ›r‹µ„¦¼žš´ÉªÅž
‹µ„¦¼žš´ÉªÅž …°Š n =
¨³
n-1 =
‹³Å—o n + n-1
=
=
n-1
=
– 'n
n (n + 1)(n + 2)
1
6 (n – 1) n(n + 1)
1
1
6 n (n + 1)(n + 2) + 6 (n – 1) n(n + 1)
1
6 n (n + 1) ((n + 2) + (n – 1))
1
6
= 1 n (n + 1) (2n + 1)
6
=
n
n
115
 f„®´— 15
Įo
¨³
n
šœ‹ÎµœªœÁ·Š¨¼„µ«„r¨Îµ—´š¸É n „ε®œ—×¥
n
¦
i 1
i
=
1
+
2
+
3
+…+
n
= n3
n
= 13 + 23 + 33 + … + n3
=
n 3
¦i
i 1
¡·‹µ¦–µ ¦¼ž˜n°Åžœ¸Ê
1
= 13
3+5
= 23
7 + 9 + 11
= 33
13 + 15 + 17 + 19
= 43
21 + 23 + 25 + 27 + 29
= 53
‹³Á®Èœªnµ 13 + 23 + 33 + 43 + 53 = 1 + 3 + 5 +…+ 29
Ÿ¨ª„‹Îµœªœ‡¸Éª„ 15 ‹ÎµœªœÂ¦„ Ž¹ÉŠŸ¨ª„‹Îµœªœ‡¸Éª„ n
‹ÎµœªœÂ¦„Ášnµ„´ n2
—´Šœ´Êœ 13 + 23 + 33 + 43 + 53 = 152
¨³
15 = '5
‹¹ŠÅ—o
13 + 23 + 33 + 43 + 53 = ('5)2
ĜšÎµœ°ŠÁ—¸¥ª„´œ‹³Å—o
13 + 23 + 33 + 43 + 53 + 63 = ('6)2
œ°„‹µ„œ¸Ê 13 + 23 + 33 + ... + n3 = ('n)2
n 3 § n ·2
4
3
2
¦ i = ¨ ¦ i¸ = 1 n + 1 n + 1 n
1. ‹ŠÂ­—Šªnµ
2
4
i 1 ©i 1 ¹ 4
2. ‹ŠÂ­—Šªnµ
n = n ('n-1 + 'n)
3. ‹ŠÁ…¸¥œ n – n Ä®o¤¸Ÿ¨¨´¡›rĜ¦¼žš¸É¤¸ n , 'n-1 ¨³ 'n ž¦µ„’
4. ‹ŠÁ…¸¥œ n – n Ä®o¤¸Ÿ¨¨´¡›rĜ¦¼žš¸É¤¸ n ¨³ 'i ž¦µ„’Á¤ºÉ° i = 1, 2,…, n
116
®œnª¥š¸É 3
‡–·˜«µ­˜¦r„´ ICT
˜°œš¸É 3.1
˜°œš¸É 3.2
œ´„‡–·˜«µ­˜¦r (Mathematicians)
¨Îµ—´¢eݜ´„¸ (Fibonacci Sequences)
˜°œš¸É 3.1
œ´„‡–·˜«µ­˜¦r (Mathematicians)
„µ¦­º‡oœž¦³ª´˜· ¨³Ÿ¨Šµœ…°Šœ´„‡–·˜«µ­˜¦r ­º‡oœÅ—o‹µ„ Web sites Ánœ
http://www.yahoo.com/Science/Mathematics/History/Mathematics
http://www-groups.des.st-andrew.ac.uk/~history/
http://forum.swarthmore.edu/~steve/steve/mathhistory.html
http://www-groups.dcs.st-and.ac.uk/~history/Day-files/Year.html
http://dimacs.rutgers.edu/~judyann/calendar/Calendar.html
Á¤ºÉ°­º‡oœÂ¨oª«¹„¬µÁ¦ºÉ°Š˜n°Åžœ¸Ê
1. œ´„‡–·˜«µ­˜¦rš¸É¤¸ºÉ°Á­¸¥Š 6 ‡œ µ¥ 3 ‡œ ®·Š 3 ‡œ
2. œ´„‡–·˜«µ­˜¦rš¸Éœ´„Á¦¸¥œ°œ°„Á®œº°‹µ„ 6 ‡œÂ¦„ °¸„ 4 ‡œ
3. Ÿ¨Šµœš¸Éœnµ­œÄ‹…°Šœ´„‡–·˜«µ­˜¦r
4. ž¦³ª´˜·Ÿ¨Šµœ ¨³ÂŸœš¸É oµœÁ„·—…°Šœ´„‡–·˜«µ­˜¦rš¸É¤¸ª´œÁ„·—˜¦Š„´ª´œÁ„·—…°Šœ´„Á¦¸¥œ
Á…¸¥œ®´ª…o°š´ÊŠ 4 Áž}œ¦µ¥Šµœ
˜°œš¸É 3.2 ¨Îµ—´¢eݜ´„¸ (Fibonacci Sequences)
šœÎµ
¨Îµ—´¢eݜ´„¸Å—ož¦µ„’‡¦´ÊŠÂ¦„Äœ¦¼ž…°Š‡Îµ˜°…°ŠÃ‹š¥rž{®µÄœ®œ´Š­º° The Liber
Abaci š¸ÉÁ…¸¥œ…¹ÊœÄœže ‡.«. 1202 ×¥ Leonardo Fibonacci ®nŠÁ¤º°Š Pisa Á¡ºÉ°š¸É‹³Âœ³œÎµ˜´ªÁ¨… HinduArabic š¸ÉčoĜ¸ª·˜ž¦³‹Îµª´œ˜n°µª¥»Ã¦žš¸É¥´Š‡ŠÄo˜ª´ Á¨…椴œŽ¹ÉŠÅ¤n­³—ª„Äœ„µ¦Äoœ´„ ˚¥r
—´Š„¨nµª¤¸°¥¼nªµn ‹³¤¸„¦³˜nµ¥Áž}œ‹Îµœªœ„¸É‡¼nš¸ÉÁ„·—‹µ„„¦³˜nµ¥Á¡¸¥Š‡¼nÁ—¸¥ª ™oµÄœÂ˜n¨³Á—º°œ„¦³˜nµ¥‡¼šn ¸É°¥¼n
Ĝª´¥Á‹¦·¡´œ›»r‹³Ä®o„εÁœ·—„¦³˜nµ¥‡¼Än ®¤n ¨³Äœ‹³Á…oµ­¼nª´¥Á‹¦·¡´œ›»rĜÁ—º°œš¸É­°Š®¨´Š‹µ„Á„·—
¨Îµ—´…°ŠÁ¨…¢eݜ´„¸š¸Éŗo‡º° 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … ¨³Áž}œ®´ª…o°š¸Éŗo«¹„¬µ„´œ¤µ
Á¦ºÉ°¥ Ç ™¹ŠÂ¤oªnµ¨Îµ—´¢eݜ´„¸‹³—¼ªnµÅ—o¤µŠnµ¥ Ç Â˜n„µ¦ž¦³¥»„˜r¤¸¤µ„¤µ¥Â¨³¤¸‡ªµ¤­ª¥Šµ¤Áž}œš¸É
117
®¨ŠÄ®¨Â¨³œnµŒŠœ˜n°œ´„‡–·˜«µ­˜¦r¤µÁž}œ¦³¥³Áª¨µ„ªnµ 800 že …o°¤¼¨Á„¸É¥ª„´¨Îµ—´¢eݜ´„¸Å—o
Á¡·É¤…¹ÊœÁ¦ºÉ°¥ Ç Â¨³‡–·˜«µ­˜¦r¢eݜ´„¸Å—o…¥µ¥°¥nµŠ˜n°ÁœºÉ°ŠÃ—¥ÁŒ¡µ³°¥nµŠ¥·ÉŠÄœ­µ…µš§¬‘¸‹Îµœªœ
(Number Theory)
™oµ Ä®o Fn Áž}œÁ¨…¢eݜ´„¸¡‹œrš¸É n ¨oªÁ¦µ­µ¤µ¦™®µÁ¨…¡‹œrœ¸Êŗo×¥„µ¦¦ª¤‡nµÁ¨…
¢eݜ´„¸­°Š¡‹œrš¸ÉŸnµœ¤µ —´Šœ¸Ê Fn1 Fn Fn1 Á¤ºÉ° n > 1 ×¥š¸É F1 1 ¨³ F2 1
¡ªnµÁ¨…¢eݜ´„¸¤¸‡ªµ¤­´¤¡´œ›r¤µ„¤µ¥„´­´—­nªœš°Š (Golden Ration)
F
1 5
2
Á¦¸¥„ªnµ‡nµ‡Š
˜´ª…°Š›¦¦¤µ˜· (a constant of nature) ¨³œnµ¤®´«‹¦¦¥r¥·ÉŠ‡nµš¸É‡nµœ¸Êž¦µ„’Äœ«·¨ž³Â¨³­™µž{˜¥„¦¦¤
ݦµ–…°Š„¦¸„ ¨³Á¦µ­µ¤µ¦™¡·­¼‹œrŗoªnµ°´˜¦µ­nªœ¦³®ªnµŠ Fn1 ¨³ Fn ¤¸‡nµž¦³¤µ–
1.6180339… Ž¹ÉŠÁž}œ‡nµÁ—¸¥ª„´‡nµ­´—­nªœš°ŠÄœ¦¼ž…°Šš«œ·¥¤
œ°„‹µ„‡ªµ¤­´¤¡´œ›ršµŠ‡–·˜«µ­˜¦r°¥nµŠœnµ¤®´«‹¦¦¥r¨oª ¥´Š¤¸„µ¦ž¦³¥»„˜r…°ŠÁ¨…¢eݜ´„¸
°¸„¤µ„¤µ¥Äœ®¨µ„®¨µ¥­µ…µ Ánœ šµŠ—oµœ¡§„¬«µ­˜¦r ¸ªª·š¥µ ¢d­·„­r —œ˜¦¸ ¨³«·¨ž³ Áž}œ˜oœ
Leonardo Fibonacci (‡.«. 1175 - 1240)
Leonardo Fibonacci Á„·—¦µªže ‡.«. 1175 š¸ÉÁ¤º°Š Pisa ž¦³Áš«°·˜µ¨¸ ¤¸ºÉ°Á¦¸¥„®¨µ¥º°É
ÁœºÉ°Š‹µ„°¥¼nÁ¤º°Š Pisa Á…µÁ¨¥™¼„Á¦¸¥„ªnµ Leonardo ®nŠ Pisa ®¦º°Äœ£µ¬µ°·˜µÁ¨¸¥œÁ¦¸¥„ªnµ Leonardo
Pisano Á…µ¤¸ºÉ°‹¦·Šªnµ Leonardo Pisano Bigollo œ´„ž¦³ª´˜·«µ­˜¦rŤnœnċªnµ Bigollo ¤¸‡ªµ¤®¤µ¥ªnµ
°¥nµŠÅ¦ °µ‹®¤µ¥™¹Š traveller ®¦º° good-for-nothing ¡n°…°ŠÁ…µºÉ° Guglielmo Bonaccio Ĝže 1828
®¨´Š‹µ„¥»‡…°Š Fibonacci œµ¥ Guillaume Libri ŗo‡oœ¡ºÉ° Fibonacci ‹µ„ filius Bonacci Ĝ£µ¬µ
¨³˜·œ®¤µ¥™¹Š The son of Bonacci ¨³ºÉ° Fibonacci š¸ÉÁ¦¸¥„„´œ°¥¼nž‹{ ‹»´œ¤µ‹µ„„µ¦Á¦¸¥„­´Êœ Ç …°Š‡Îµ
ªnµ Filius Bonacci œ´ÉœÁ°Š
Guglielmo Bonacci ¡n°…°ŠÁ…µÁž}œÁ‹oµ®œoµš¸É—nµœ«»¨„µ„¦­Îµ®¦´ Pisa š¸ÉÁ¤º°ŠšnµÂ°¢¦·„µÁ®œº°
…°Š¼Á„¸¥ (The North African port town of Bugia) Ž¹ÉŠÄœž{‹‹»´œ ‡º° Bejala, Algeria ĜnªŠª´¥¦»œn
Fibonacci ŗo°¥¼n„´¡n°Â¨³Å—o¦´„µ¦«¹„¬µ‹µ„ The Moors Ž¹ÉŠÁž}œµª°µ®¦´ —oª¥ž¦³­„µ¦–rĜ
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118
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L
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Hindu - Arabic
1999
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Abbaci Ž¹ÉŠ®¤µ¥™¹Š ®œ´Š­º°Â®nŠ„µ¦‡Îµœª– (Book of Calculating) Ž¹ÉŠÁ„¸É¥ª„´¦³Á¸¥ª·›¸Á¨…‡–·˜Äœ
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Ĝže ‡.«. 1202 ¢eݜ´„¸­œÄ‹„µ¦…¥µ¥¡´œ›»r…°Š„¦³˜nµ¥ Á…µÅ—o­¦oµŠÁŽ˜…°ŠÁŠºÉ°œÅ…Äœ°»—¤‡˜·Ã—¥š¸É
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Áž}œ¦n¤ÁŠµ„ªnµ—oµœš¸É™¼„­Š°µš·˜¥r×¥˜¦Š ¨°Š¤µ¡·‹µ¦–µ—°„šµœ˜³ª´œÄœ°¸„Šn¤»¤®œ¹ÉŠ Á¦µÁ‡¥
­´ŠÁ„˜»„µ¦Á¦¸¥Š˜´ª…°ŠÁ¤¨È—šµœ˜³ª´œ˜¦Š­nªœ„¨µŠ…°Š—°„®¦º°Å¤n ‹¦·Š Ç Â¨oªÁ¤¨È—šµœ˜³ª´œ¤¸„µ¦
Á¦¸¥Š˜´ª„´œÁž}œÁ­oœ…—¨³¤¸‹ÎµœªœÂ™ªÁž}œÁ¨…¢eݜ´„¸
˜n°Åž¨°Š¡·‹µ¦–µ—°„Ťoš¤¸É ¸‹Îµœªœ„¨¸—°„ŤnÁž}œÁ¨…¢eݜ´„¸ ‹³Á®ÈœªnµÅ¤n­ª¥Šµ¤Ášnµ„´
—°„Ťoš¸É‹Îµœªœ„¨¸—°„Áž}œÁ¨…¢eݜ´„¸ Ánœ
132
133
134
¢eݜ´„¸Â¨³—µª·œ¸ (Leonardo Fibonacci and Leonardo da Vinci)
­¸ÉÁ®¨¸É¥¤š°ŠÅ—o™¼„°oµŠ°·Š™¹ŠÄœÂŠn…°Š‡ªµ¤ž¦³š´Ä‹Äœ‡ªµ¤­ª¥Šµ¤…°Šš»„Ç ­¸ÉÁ®¨¸É¥¤ —oª¥
Á®˜»œ¸Ê‹¹Š™¼„œÎµ¤µÄoĜŠµœ«·¨ž³Â¨³­™µž{˜¥„¦¦¤°¥nµŠÂ¡¦n®¨µ¥Áž}œÁª¨µœµœ¤µÂ¨oª š¸ÉÁ—nœ´—„ȇº°
ŗo¤¸„µ¦œÎµ­¸ÉÁ®¨¸É¥¤š°Š¤µÄoĜŸ¨Šµœ…°Š Leonardo da Vinci «·¨ždœÁ°„ œ´„ž¦³—·¬“r ¨³ œ´„
‡–·˜«µ­˜¦rµª°·˜µÁ¨¸É¥œ
¦¼žÃ¤œµ¨·Žµ
£µ¡ªµ—䜵¨·Žµ (Mona Lisa) Ž¹ÉŠÁž}œŸ¨Šµœš¸É¤¸ºÉ°Á­¸¥ŠÃ—¥Å¤n¤¸…o°…´—Â¥oŠ„Ȥ¸­¸ÉÁ®¨¸¥É ¤š°Š
Á…oµ¤µÁ„¸É¥ª…o°Š°¥nµŠ¤µ„ ™oµÁ¦µªµ—¦¼ž­¸ÉÁ®¨¸É¥¤Ã—¥Á¦·É¤š¸“É µœ‹µ„…o°¤º°…oµŠ…ªµÅž¥´Š…o°«°„Žoµ¥ ¨³
ªµ——oµœ„ªoµŠ…°Š¦¼žÃ—¥¨µ„Á­oœ…¹ÊœÅž‹œ™¹Šœ­»—…°Š«¸¦¬³ ­¸ÉÁ®¨¸É¥¤š¸É¤¸—oµœ¥µªÂ¨³—oµœ„ªoµŠ
—´Š„¨nµªÁž}œ­¸ÉÁ®¨¸É¥¤š°Š ¨³™oµÁ¦µªµ—¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­š¸É¦¦‹»°¥¼n£µ¥Äœ¦¼ž­¸ÉÁ®¨¸É¥¤š°Š —´Š¦¼ž ‹³
Á®Èœªnµ—oµœ˜nµŠÇ …°Š¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­Â˜n¨³¦¼ž‹³Ÿnµœ˜ÎµÂ®œnŠ­Îµ‡´Ç …°Š¦¼ž Ánœ ‡µŠ ˜µ ‹¤¼„ ¨³
¤»¤žµ„…ªµ Áº°É „´œªnµ Leonardo davinci Áž}œœ´„‡–·˜«µ­˜¦r ¨³˜´ÊŠÄ‹ªµ—¦¼žœ¸Êץčo­¸ÉÁ®¨¸É¥¤š°Š¤µ
Á¦¸¥Š„´œ Ĝš¸œÉ ¸ÊÁ¡ºÉ°š¸É‹³ÁºÉ°¤Ã¥Š‡–·˜«µ­˜¦r¨³«·¨ž³Á…oµ—oª¥„´œ
œ°„‹µ„£µ¡ªµ—䜵¨·ŽµÂ¨oª „µ¦«¹„¬µ­´—­nªœ…°Š¤œ»¬¥r„ÈÁž}œŸ¨Šµœš¸É­Îµ‡´°¸„·Êœ®œ¹ÉŠ
…°Š Leonardo da Vinci „ȇº°£µ¡ “The Vetruvian Man” ®¦º° “The Man in Action” Ž¹ÉŠ¤¸¦¼ž­¸ÉÁ®¨¸É¥¤
š°ŠÂ Š°¥¼n¤µ„¤µ¥Ánœ„´œ ŤnÁ®¤º°œ„´¦¼žÃ¤œµ¨·Žµš¸É¤¸„µ¦Á¦·É¤˜oœÃ—¥ªµ—¦¼ž­¸ÉÁ®¨¸É¥¤š°Š„n°œ
Á­¤º°œªnµÁž}œœ´„‡–·˜«µ­˜¦r Ĝ¦¼ž The Vetruvian Man œ¸Ê¤¸„µ¦ªµ—­¸ÉÁ®¨¸É¥¤®¨µ¥¦¼ž ×¥ž¦³„°
—oª¥ÁŽ˜…°Š­¸ÉÁ®¨¸É¥¤š°ŠÄœÂ˜n¨³­nªœš¸ªÉ µ—˜„˜nµŠ„´œ 3 ­nªœ ‡º° ­nªœ®´ª ­nªœ¨Îµ˜´ª ¨³­nªœ…µ
135
¦¼ž The Man in Action
­nªœ®´ª : ªµ—¦¼ž­¸ÉÁ®¨¸É¥¤ŸºœŸoµ¦¼žÂ¦„×¥ªµ—“µœ‹µ„Å®¨n…oµŠ®œ¹ÉŠÅž¥´ŠÅ®¨n°„¸ …oµŠ®œ¹ÉŠÃ—¥
Ÿnµœ¨Îµ‡° ¨³‡ªµ¤­¼Šš¸Éŗo‹³Áž}œ¦³¥³‹µ„“µœÅ®¨n‹œ™¹Š­nªœœ­»—…°Š®´ª ˜n°¤µªµ—¦¼ž­¸ÉÁ®¨¸É¥¤
‹´˜»¦´­Ä®o¦¦‹»°¥¼nšµŠ—oµœŽoµ¥…°Š­¸ÉÁ®¨¸É¥¤ŸºœŸoµ¦¼žÂ¦„ šÎµÄ®oŗo¦¼ž­¸ÉÁ®¨¸É¥¤š°Š¦¼žÁ¨È„¦¦‹»°¥¼nœ®´ª
‡œ—oµœ…ªµ šÎµÁnœÁ—¸¥ª„´œ„´š¸ÉŸnµœ¤µÂ˜n‡œ¨³—oµœ ‹³Å—o¦¼ž­¸ÉÁ®¨¸É¥¤š°Š¦¼žÁ¨È„…œµ—Á—¸¥ª„´œž¦µ„’
š¸É­nªœ®´ª—oµœ…ªµ —oª¥Á®˜»œš¸Ê εĮoŗo¦¼ž­¸ÉÁ®¨¸É¥¤ŸºœŸoµš¸¥É µªÄœÂœª˜´ÊŠÂ¨³Â‡ÄœÂœªœ°œ˜¦Š
­nªœ„¨µŠ…°Š®´ª…°Š¦¼ž
®¤µ¥Á®˜» ¦·Áª–š¸É­¸ÉÁ®¨¸É¥¤‹´˜»¦´­ 2 ¦¼ž š´Žo°œ„´œ‡º°¦·Áª–˜µ…°Š¦¼ž‡œœ´ÉœÁ°Š
­nªœ¨Îµ˜´ª : ªµ—¦¼ž­¸ÉÁ®¨¸É¥¤Ã—¥¤¸‡ªµ¤¥µª‹µ„…o°«°„…oµŠ®œ¹ÉŠÅž¥´Š…o°«°„°¸„…oµŠ ¨³¤¸
‡ªµ¤„ªoµŠ‹µ„¨Îµ‡°‹œ™¹Š­³Ã¡„ ¦¼ž­¸ÉÁ®¨¸É¥¤š¸ÉŗoÁž}œ¦¼ž­¸ÉÁ®¨¸É¥¤š°Š ˜n°¤µšÎµÁnœÁ—¸¥ª„´…´Êœ˜°œÄœ
­nªœ®´ª ×¥ªµ—¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­¦¦‹»…oµŠÄœ¦¼ž­¸ÉÁ®¨¸É¥¤š°Šš´ÊŠ­°Š…oµŠ ‹³Å—o¦¼ž­¸ÉÁ®¨¸É¥¤š°ŠÁ¡·É¤…¹Êœ
°¸„ 2 ¦¼ž
®¤µ¥Á®˜» Ĝ˜°œœ¸Ê¦·Áª–š¸É­¸ÉÁ®¨¸É¥¤‹´˜»¦­´ 2 ¦¼ž š´Žo°œ„´œ‡º°¦·Áª–¨Îµ˜´ªš¸É‡°—š¸É­»—…°Š¦¼ž‡œ
œ´ÉœÁ°Š
­nªœ…µ : ªµ—¦¼ž­¸ÉÁ®¨¸É¥¤Ã—¥¤¸‡ªµ¤¥µª‹µ„œ·Êª®´ªÂ¤nÁšoµ…oµŠ®œ¹ÉŠÅž¥´Š°¸„…oµŠ …°Š¦¼žš¸ÉÁšoµ‡œ
­´¤Ÿ´­ªŠ„¨¤ ¨³¤¸ª—´ ‡ªµ¤„ªoµŠ…¹ÊœÅž‹œ™¹ŠÁ°ª ¦¼ž­¸ÉÁ®¨¸É¥¤š¸ÉŗoÁž}œ¦¼ž­¸ÉÁ®¨¸É¥¤š°Š ˜n°¤µšÎµ
ÁnœÁ—¸¥ª„´…´œÊ ˜°œÄœ­°Š­nªœš¸ÉŸnµœ¤µ ×¥ªµ—¦¼ž­¸ÉÁ®¨¸É¥¤‹´˜»¦´­¦¦‹»…oµŠÄœ¦¼ž­¸ÉÁ®¨¸É¥¤š°Šš´ÊŠ­°Š
…oµŠ ‹³Å—o¦¼ž­¸ÉÁ®¨¸É¥¤š°ŠÁ¡·É¤…¹Êœ°¸„ 2 ¦¼ž
®¤µ¥Á®˜» Ĝ˜°œœ¸Ê¦·Áª–š¸É­¸ÉÁ®¨¸É¥¤‹´˜»¦­´ 2 ¦¼ž š´Žo°œ„´œœnµ‹³Áž}œÂœª…°Š…µš¸ÉÁ®¥¸¥—˜¦Š (­¤¤˜·
ªnµ‡œÅ¤nŗo®´œ…µÅž—´Š¦¼ž)
136
 f„®´—š¸É 16
1. ‹Š¡·­¼‹œrªnµ
­Îµ®¦´ n t 1
1.1 Fn2 Fn21 F2 n1
1.2 Fn1Fn1 Fn2 1n ­Îµ®¦´ n t 2
1.3 Fn31 Fn3 Fn31 F3n ­Îµ®¦´ n t 2
1.4 F12 F22 L Fn2 Fn Fn1
1.5 ™oµ m Áž}œ˜´ªž¦³„°…°Š n ¨oª‹³Å—oªnµ Fm Áž}œ˜´ªž¦³„°…°Š Fn
1.6 „ε®œ—Ä®o Gn
F2n
Fn
¨oª‹³Å—oªnµ Gn
Gn 1 Gn 2
2. čoÁ‡¦ºÉ°Š‡·—Á¨…‡Îµœª–‡nµ 10000/9899 Ä®oŗoÁ¨…š«œ·¥¤°¥nµŠœo°¥™¹Š˜ÎµÂ®œnŠš¸É 8
2.1 ‹Š®µ¦¼žÂ„µ¦Å—o¤µ…°Šš«œ·¥¤Â˜n¨³˜ÎµÂ®œnŠ
2.2 ‹Š®µÁ¨…š«œ·¥¤Äœ˜ÎµÂ®œnŠ™´—Åž
3. ‹Š®µÁ¨…¢eݜ´„¸¨Îµ—´š¸É 10,12 ץčo
3.1 ¦¼žÂšª·œµ¤
3.2 ­¼˜¦…°ŠÅÁœ˜r
4. ‹Š®µ¡‹œr™´—Åž…°ŠÁ¨…¢eݜ´„¸ ¨Îµ—´š¸É 10,12 š¸Éŗo‹µ„…o° 3.
5. ‹Š¡·­¼‹œrªnµ sin18o
M 1
6. ‹Š¡·­¼‹œrªnµ cos 36o
M
2
Á¤ºÉ° M ‡º° °´˜¦µ­nªœš°Š ( M
1 5
2
)
2
7. čo„‘…°ŠÃ‡ÅŽœr‡Îµœª–‡nµ cos 108q
8. čo„‘…°ŠÅŽœr‡Îµœª–‡nµ
sin 72o
cos 36o
9. ‹Š®µ˜´ª°¥nµŠ—°„Ťoš¸É¤¸‹Îµœªœ„¨¸—°„Áž}œÁ¨…¢eݜ´„¸
10. ‹Š®µ˜´ª°¥nµŠ„µ¦ž¦³¥»„˜r°œºÉ Ç š¸É¤¸Á„¸É¥ª„´¨Îµ—´¢eݜ´„¸
web site ¨³Á°„­µ¦°oµŠ°·Š
www.cs.rit.edu/~pga/Fibo/fact_sheet.html
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibInArt.html
Ravi Vakil, A Mathematical Mosaic: Patterns & Problem Solving, Brendan Kelly Publishing Inc.,
1996.
®œnª¥š¸É 4
ª·›¸„µ¦¡·­¼‹œr¨³°»žœ´¥Á·Š‡–·˜«µ­˜¦r
˜°œš¸É 4.1 ª·›¸„µ¦¡·­¼‹œr (Method of Proofs)
˜°œš¸É 4.2 °»žœ´¥Á·Š‡–·˜«µ­˜¦r ( Mathematical Induction )
˜°œš¸É 4.1 ª·›¸„µ¦¡·­¼‹œr (Method of Proofs)
Ĝ„µ¦°oµŠÁ®˜»Ÿ¨š¸É­¤Á®˜»­¤Ÿ¨œ´Êœ Á¤ºÉ°Á¦µ¥°¤¦´ªnµÁ®˜»®¦º°…o°„ε®œ—Áž}œ‹¦·ŠÂ¨oª ‹³˜o°Š
¥°¤¦´Ÿ¨­¦»žš¸Éŗo˜o°ŠÁž}œ‹¦·Š—oª¥
­Îµ®¦´„µ¦¡·­¼‹œrĜª·µ‡–·˜«µ­˜¦r Á¤ºÉ°Á¦µ¥°¤¦´ªnµ šœ·¥µ¤ ­´‹¡‹œr ¨³š§¬‘¸š š¸É¤¸¤µ
„n°œÂ¨oªÄœ¦³œ´ÊœÁž}œ‹¦·Š Á¤ºÉ°Á¦µœÎµ šœ·¥µ¤ ­´‹¡‹œr ¨³š§¬‘¸š —´Š„¨nµª¤µ°oµŠÁž}œÁ®˜»Ÿ¨
Á¡ºÉ°­œ´­œ»œ…o°‡ªµ¤Ä®¤n(Ÿ¨­¦»ž) Á¦µ‹³Å—oªnµ…o°‡ªµ¤Ä®¤nœ´Êœ˜o°ŠÁž}œ‹¦·Š—oª¥
š§¬‘¸˜nµŠ Ç š¸Éŗo¤µ˜o°ŠŸnµœ„µ¦¡·­¼‹œr Á¡ºÉ°¥ºœ¥´œÄ®oœnċªnµÁž}œ„µ¦­¦»žš¸É­¤Á®˜»­¤Ÿ¨
1) „µ¦¡·­¼‹œrªnµ p o q ‹¦·ŠÃ—¥˜¦Š
™oµ S1, S2, S3, ..., Sn Áž}œšœ·¥µ¤ ­´‹¡‹œr ®¦º°š§¬‘¸šš¸É¤¸¤µ„n°œÂ¨oªÄœ„µ¦¡·­¼‹œr p o q
®¦º°¡·­¼‹œrªnµ p o q ¤¸‡nµ‡ªµ¤‹¦·ŠÁž}œ‹¦·Š „¦³šÎµÃ—¥­¤¤»˜·ªnµ p ¨oªÂ­—ŠÄ®oŗoªnµ q
„µ¦¡·­¼‹œr p o q °¥¼nĜ¦¼ž—´Šœ¸Ê
¡·­¼‹œr ­¤¤»˜·ªnµ p
----------
½
---------- °
¾Äo p ¨³ S1, S2, S3, ..., Sn
---------- °
¿
Á¡¦µ³Œ³œ´Êœ q
œ´Éœ‡º° p o q
138
˜´ª°¥nµŠš¸É 4.1.1 „ε®œ—Ä®o x Áž}œ‹ÎµœªœÁ˜È¤
‹Š¡·­¼‹œrªnµ ™oµ x Áž}œ‹Îµœªœ‡¼n¨oª y = x + 3 ‹³Áž}œ‹Îµœªœ‡¸É
¡·­¼‹œr
1) ­¤¤»˜·Ä®o x Áž}œ‹Îµœªœ‡¼n
2) x = 2n ×¥š¸É n Áž}œ‹ÎµœªœÁ˜È¤ (šœ·¥µ¤…°Š‹Îµœªœ‡¼n)
3) y = x + 3 = 2n + 3
(‹µ„ 2) ª„Á…oµš´ÊŠ­°Š…oµŠ—oª¥‹Îµœªœš¸ÉÁšnµ„´œ)
4) y = 2(n+1) + 1
(„‘„µ¦Áž¨¸É¥œ„¨»n¤Å—o¨³„‘„µ¦Â‹„‹Š)
5) Įo n + 1 = r
(„ε®œ—ºÉ°Ä®¤n)
6) ‹¹ŠÅ—o r Áž}œ‹ÎµœªœÁ˜È¤
(‹ÎµœªœÁ˜È¤ª„‹ÎµœªœÁ˜È¤Ÿ¨¨´¡›ršÉŸ —o¥´Š‡ŠÁž}œ
‹ÎµœªœÁ˜È¤)
7) —´Šœ´Êœ y = 2r + 1 ×¥š¸É r Áž}œ‹ÎµœªœÁ˜È¤
8) œ´Éœ‡º° y Áž}œ‹Îµœªœ‡¸É
(šœ·¥µ¤…°Š‹Îµœªœ‡¸É)
—´Šœ´Êœ ™oµ x Áž}œ‹Îµœªœ‡¼n¨oª y = x + 3 ‹³Áž}œ‹Îµœªœ‡¸É
˜´ª°¥nµŠš¸É 4.1.2 ‹Š¡·­¼‹œrªnµ ™oµ 1 = 3 ¨oª 4 = 4
¡·­¼‹œr
1) ­¤¤»˜·Ä®o 1 = 3
2) Á¡¦µ³Œ³œ´Êœ 3 = 1
3) 1 + 3 = 3 + 1
(­·ÉŠš¸ÉÁšnµ„´œª„Á…oµ—oª¥­·ÉŠš¸ÉÁšnµ„´œŸ¨¨´¡›r¥n°¤Ášnµ„´œ)
4) 4 = 4
—´Šœ´Êœ ™oµ 1 = 3 ¨oª 4 = 4
…o°­´ŠÁ„˜ Ĝ„µ¦¡·­¼‹œr p o q œ´Êœ p °µ‹‹³Áž}œÁšÈ‹Å—o —´Š˜´ª°¥nµŠš¸É 4.1.2
˜´ª°¥nµŠš¸É 4.1.3 „ε®œ—Ä®o a Áž}œ‹ÎµœªœÁ˜È¤
‹Š¡·­¼‹œrªnµ ™oµ a Áž}œ‹Îµœªœ‡¼n¨oª a2 ‹³Áž}œ‹Îµœªœ‡¼n—oª¥
¡·­¼‹œr
­¤¤»˜·Ä®o a Áž}œ‹Îµœªœ‡¼n
‹³Å—o a = 2n ×¥š¸É n Áž}œ‹ÎµœªœÁ˜È¤ (šœ·¥µ¤…°Š‹Îµœªœ‡¼n)
ÁœºÉ°Š‹µ„ a2 = a.a
—´Šœ´Êœ a2 = (2n)(2n)
= 2(2n2)
(„‘„µ¦Áž¨¸É¥œ®¤¼nŗo¨³„‘„µ¦­¨´š¸É)
139
Įo 2n2 = r
(˜´ÊŠºÉ°Ä®¤n)
‹³Å—o r Áž}œ‹ÎµœªœÁ˜È¤
—´Šœ´Êœ a2 = 2r ×¥š¸É r Áž}œ‹ÎµœªœÁ˜È¤
œ´Éœ‡º° a2 Áž}œ‹Îµœªœ‡¼n (šœ·¥µ¤…°Š‹Îµœªœ‡¼n)
—´Šœ´Êœ ™oµ a Áž}œ‹Îµœªœ‡¼n¨oª a2 ‹³Áž}œ‹Îµœªœ‡¼n—oª¥
˜´ª°¥nµŠš¸É 4.1.4 „ε®œ—Ä®o a ¨³ b Áž}œ‹ÎµœªœÁ˜È¤
‹Š¡·­¼‹œrªnµ ™oµ a ¨³ b ˜nµŠ„ÈÁž}œ‹Îµœªœ‡¸É¨oª a + b ‹³Áž}œ‹Îµœªœ‡¼n
¡·­¼‹œr
­¤¤»˜·Ä®o a ¨³ b Áž}œ‹Îµœªœ‡¸É
‹³Å—o a = 2n + 1 ×¥š¸É n Áž}œ‹ÎµœªœÁ˜È¤ (šœ·¥µ¤…°Š‹Îµœªœ‡¸É)
¨³ b = 2m + 1 ×¥š¸É m Áž}œ‹ÎµœªœÁ˜È¤
a + b = (2n+1) + (2m+1)
= 2(m+n+1) („‘„µ¦Áž¨¸É¥œ®¤¼nŗo , „‘„µ¦­¨´š¸É¨³„‘„µ¦Â‹„‹Š)
Ä®o m + n + 1 = r ‹³Å—oªnµ r Áž}œ‹ÎµœªœÁ˜È¤
a + b = 2r
×¥š¸É r Áž}œ‹ÎµœªœÁ˜È¤
œ´Éœ‡º° a + b Áž}œ‹Îµœªœ‡¼n (šœ·¥µ¤…°Š‹Îµœªœ‡¼n)
—´Šœ´Êœ ™oµ a ¨³ b ˜nµŠÁž}œ‹Îµœªœ‡¸É¨oª a + b ‹³Áž}œ‹Îµœªœ‡¼n
140
˜´ª°¥nµŠš¸É 4.1.5 „ε®œ—Ä®o a ¨³ b Áž}œ‹ÎµœªœÁ˜È¤Ä— Ç
‹Š¡·­¼‹œrªnµ ™oµ a Áž}œ‹Îµœªœ‡¸É¨oª a + 1 Áž}œ‹Îµœªœ‡¼n
¡·­¼‹œr
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
141
2) „µ¦¡·­¼‹œr p o q ץčo…o°‡ªµ¤Â¥oŠ­¨´š¸É (Proof by Using Contrapositive)
šœš¸ÉÁ¦µ‹³¡·­¼‹œr p o q ×¥˜¦Š Á¦µ°µ‹‹³¡·­¼‹œr p o q ×¥¡·­¼‹œr aq o ap šœ š´ÊŠœ¸Ê
Á¡¦µ³ (p o q) { aq o ap
­nªœ„µ¦¡·­¼‹œr aq o ap „ȗεÁœ·œ„µ¦¡·­¼‹œrÁnœÁ—¸¥ª„´®´ª…o° 4.1 —´Šœ´Êœ Á¦µ°µ‹¡·­¼‹œrªnµ
p o q Áž}œ‹¦·Š ×¥­¤¤»˜·ªnµ aq ¨oª ­—ŠÅ—oªnµ ap „µ¦¡·­¼‹œr°¥¼nĜ¦¼ž—´Šœ¸Ê
¡·­¼‹œr ­¤¤»˜·ªnµ aq
---------½
---------- ° čo aq, šœ·¥µ¤, ­´‹¡‹œr ®¦º°
¾
---------- ° š§¬‘¸šš¸É¤¸¤µ„n°œÂ¨oª
¿
Á¡¦µ³Œ³œ´Êœ ap
œ´Éœ‡º° aq o ap
—´Šœ´Êœ p o q
˜´ª°¥nµŠš¸É 4.1.6 „ε®œ—Ä®o x Áž}œ‹ÎµœªœÁ˜È¤
‹Š¡·­¼‹œrªnµ ™oµ x Áž}œ‹Îµœªœ‡¼n¨oª y = x + 3 ‹³Áž}œ‹Îµœªœ‡¸É
¡·­¼‹œr
­¤¤»˜·Ä®o y = x + 3 Ťnčn‹Îµœªœ‡¸É
Á¡¦µ³Œ³œ´Êœ y = x + 3 Áž}œ‹Îµœªœ‡¼n
‹³Å—o y = 2n ×¥š¸É n Áž}œ‹ÎµœªœÁ˜È¤
Á¡¦µ³Œ³œ´Êœ x = 2n 3
= 2(n2) + 1
Ä®o n 2 = r ‹³Å—o r Áž}œ‹ÎµœªœÁ˜È¤
—´Šœ´Êœ x = 2r + 1 ×¥š¸É r Áž}œ‹ÎµœªœÁ˜È¤
‹³Å—oªnµ x Áž}œ‹Îµœªœ‡¸É
Á¡¦µ³Œ³œ´Êœ x Ťnčn‹Îµœªœ‡¼n
œ´Éœ‡º° ™oµ y = x + 3 Ťnčn‹Îµœªœ‡¸É ¨oª x ‹³Å¤nčn‹Îµœªœ‡¼n
—´Šœ´Êœ ™oµ x Áž}œ‹Îµœªœ‡¼n ¨oª y = x + 3 ‹³Áž}œ‹Îµœªœ‡¸É
142
˜´ª°¥nµŠš¸É 4.1.7 „ε®œ—Ä®o a Áž}œ‹ÎµœªœÁ˜È¤
‹Š¡·­¼‹œrªnµ ™oµ a2 Áž}œ‹Îµœªœ‡¼n ¨oª a ‹³Áž}œ‹Îµœªœ‡¼n—oª¥
¡·­¼‹œr
­¤¤»˜·Ä®o a Ťnčn‹Îµœªœ‡¼n
Á¡¦µ³Œ³œ´Êœ a Áž}œ‹Îµœªœ‡¸É
‹³Å—oªnµ a = 2n + 1 ×¥š¸É n Áž}œ‹ÎµœªœÁ˜È¤ (šœ·¥µ¤…°Š‹Îµœªœ‡¸É)
Á¡¦µ³ a2 = a.a
—´Šœ´Êœ a2 = (2n+1)(2n+1)
= 4n2 + 4n + 1
= 2(2n2+2n) + 1
Ä®o 2n2 + 2n = r ‹³Å—o r Áž}œ‹ÎµœªœÁ˜È¤
—´Šœ´Êœ a2 = 2r + 1 ×¥š¸É r Áž}œ‹ÎµœªœÁ˜È¤
‹³Å—oªnµ a2 Áž}œ‹Îµœªœ‡¸É (šœ·¥µ¤…°Š‹Îµœªœ‡¸É)
Á¡¦µ³Œ³œ´Êœ a2 Ťnčn‹Îµœªœ‡¼n
œ´Éœ‡º° ™oµ a Ťnčn‹Îµœªœ‡¼n¨oª a2 ‹³Å¤nčn‹Îµœªœ‡¼n
—´Šœ´Êœ ™oµ a2 Áž}œ‹Îµœªœ‡¼n¨oª a ‹³Áž}œ‹Îµœªœ‡¼n
˜´ª°¥nµŠš¸É 4.1.8 ‹Š¡·­¼‹œrªnµ ­Îµ®¦´ÁŽ˜ A ¨³ B Ä—Ç ™oµ A  B ¨oª A ˆ Bc I
¡·­¼‹œr
Ä®o A ¨³ B Áž}œÁŽ˜Ä—Ç
­¤¤»˜·Ä®o A ˆ Bc z I
—´Šœ´Êœ¤¸ x  A ˆ Bc
‹³Å—o
(šœ·¥µ¤…°Š “ ˆ ”)
x  A ¨³ x  B c
­—Šªnµ x  A ¨³ x  B
(šœ·¥µ¤…°Š‡°¤¡¨¸Á¤œ˜r)
Á¡¦µ³Œ³œ´Êœ
(œ·Á­›…°Š “  ”)
AΠB
œ´Éœ‡º°
A ˆ Bc z I o A Œ B
—´Šœ´Êœ
A  B o A ˆ Bc I
143
˜´ª°¥nµŠš¸É 4.1.9 „ε®œ—Ä®o a ¨³ b Áž}œ‹ÎµœªœÁ˜È¤
‹Š¡·­¼‹œrªnµ ™oµ ab Áž}œ‹Îµœªœ‡¼n ¨oª a Áž}œ‹Îµœªœ‡¼n®¦º° b Áž}œ‹Îµœªœ‡¼n
¡·­¼‹œr
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
144
 f„®´—š¸É 17
…o°š¸É 1 – 6 „ε®œ—Ä®o a ¨³ b Áž}œ‹ÎµœªœÁ˜È¤
‹Š¡·­¼‹œrªnµ
1. ™oµ a Áž}œ‹Îµœªœ‡¼n¨oª a4 ‹³Áž}œ‹Îµœªœ‡¼n
2. ™oµ a ¨³ b ˜nµŠ„ÈÁž}œ‹Îµœªœ‡¼n¨oª a + b ‹³Áž}œ‹Îµœªœ‡¼n
3. ™oµ (a + b)2 Áž}œ‹Îµœªœ‡¼n¨oª a Áž}œ‹Îµœªœ‡¼n ®¦º° b Áž}œ‹Îµœªœ‡¸É
4. ™oµ a Áž}œ‹Îµœªœ‡¸É¨oª a2 ‹³Áž}œ‹Îµœªœ‡¸É
5. ™oµ (a + b) Áž}œ‹Îµœªœ‡¼n¨oª a Áž}œ‹Îµœªœ‡¼n ®¦º° b Áž}œ‹Îµœªœ‡¸É
6. ™oµ a2 Áž}œ‹Îµœªœ‡¸É¨oª a ‹³Áž}œ‹Îµœªœ‡¸É
7. „ε®œ—Ä®o a, b ¨³ c Áž}œ‹Îµœªœ‹¦·ŠÄ— Ç
‹Š¡·­¼‹œrªnµ ™oµ a > b ¨³ b > c ¨oª a > c
(…o°Âœ³ : x > y „Șn°Á¤ºÉ° ¤¸‹Îµœªœ‹¦·Šª„ t Ž¹ÉŠ x = y + t)
8. „ε®œ—šœ·¥µ¤˜n°Åžœ¸Ê
Ä®o a ¨³ b Áž}œ‹ÎµœªœÁ˜È¤Ã—¥š¸É a z 0
a ®µ¦ b ¨Š˜´ª „Șn°Á¤ºÉ°¤¸‹ÎµœªœÁ˜È¤ c š¸ÉšÎµÄ®o b = ac
čo­´¨´„¬–r a~b šœ a ®µ¦ b ¨Š˜´ª
a~b šœ a ®µ¦ b Ťn¨Š˜´ª
8.1 „ε®œ— a, b ¨³ c Áž}œ‹ÎµœªœÁ˜È¤ ¨³ a z 0
‹Š¡·­¼‹œrªnµ
™oµ a~b ¨³ a~(b+c) ¨oª a~c
8.2 „ε®œ— a, b, c, d Áž}œ‹ÎµœªœÁ˜È¤ ¨³ a z 0 , c z 0
‹Š¡·­¼‹œrªnµ
™oµ a~b ¨³ c~d ¨oª ac~bd
8.3 „ε®œ— a Áž}œ‹Îµœªœ‡¼n ¨³ b Áž}œ‹ÎµœªœÁ˜È¤Ä— Ç
™oµ a~3b ¨oª 10~5b
145
3) „µ¦¡·­¼‹œr×¥„µ¦®µ…o°…´—Â¥oŠ (Proof by Contradiction)
Á¦µ‹³¡·­¼‹œr…o°‡ªµ¤ P ®¦º°‹³¡·­¼‹œrªnµ P Áž}œ‹¦·Š ×¥­¤¤»˜·Ä®o aP Áž}œ‹¦·ŠÂ¨oª
ŗoŸ¨­¦»ž Q š aQ Á¦µÄoª·›¸¡·­¼‹œrÁnœœ¸ÊŗoÁ¡¦µ³ [aP o (Q š aQ)] o P Áž}œ­´‹œ·¦´œ—¦r
Á¦µ°µ‹„¨nµªÅ—oªnµ „µ¦¡·­¼‹œr×¥„µ¦®µ…o°…´—Â¥oŠœ¸ÊÁž}œ„µ¦¡·­¼‹œr×¥­¤¤»˜·Ä®o…o°‡ªµ¤š¸É‹³
¡·­¼‹œrÁž}œÁšÈ‹
‹µ„­¤¤»˜·“µœœ¸Ê¡·­¼‹œr˜n°Åž‹œ¡Ÿ¨­¦»žš¸É…´—Â¥oŠ„´­·ÉŠš¸Éš¦µ¤µ„n°œ®¦º°­·ÉŠš¸É
„ε®œ—Ä®o ®¦º°š§¬‘¸š ®¦º°­´‹¡‹œr ²¨² ¨oª‹¹Š­¦»žªnµ­¤¤»˜·“µœš¸É˜´ÊŠÅªoÁž}œÅžÅ¤nŗo œ´Éœ‡º°š¸É
­¤¤˜·ªnµ…o°‡ªµ¤š¸É˜o°Š„µ¦¡·­¼‹œrÁž}œÁšÈ‹œ´ÊœÁž}œÅžÅ¤nŗo —´Šœ´Êœ…o°‡ªµ¤œ´Êœ‹³˜o°ŠÁž}œ‹¦·Š
„µ¦¡·­¼‹œrªnµ P Áž}œ‹¦·ŠÃ—¥„µ¦®µ…o°…´—Â¥oŠ°¥¼nĜ¦¼ž—´Šœ¸Ê
¡·­¼‹œr ­¤¤»˜·Ä®o aP Áž}œ‹¦·Š
---------- ½
---------- °
¾ čo aP, šœ·¥µ¤, ­´‹¡‹œr ®¦º°
---------- °
¿ š§¬‘¸šš¸É¤¸¤µ„n°œÂ¨oª
Á¡¦µ³Œ³œ´Êœ Q š aQ
œ´Éœ‡º° aP o Q š aQ
—´Šœ´Êœ P Áž}œ‹¦·Š
˜´ª°¥nµŠš¸É 4.1.10 „ε®œ—Ä®o x Áž}œ‹ÎµœªœÁ˜È¤
‹Š¡·­¼‹œrªnµ ™oµ x Áž}œ‹Îµœªœ‡¼n ¨oª y = x + 3 ‹³Áž}œ‹Îµœªœ‡¸É
¡·­¼‹œr
1) Ä®o P šœ…o°‡ªµ¤ ™oµ x Áž}œ‹Îµœªœ‡¼n ¨oª y = x + 3 ‹³Áž}œ‹Îµœªœ‡¸É
2) ­¤¤»˜·Ä®o ~P Áž}œ‹¦·Š
3) —´Šœ´Êœ ‹³Å—oªnµ x Áž}œ‹Îµœªœ‡¼n ¨³ y = x + 3 Ťnčn‹Îµœªœ‡¸É
4) y = x + 3 Ťnčn‹Îµœªœ‡¸É
(‹µ„ 3) čo„‘„µ¦‡´—)
5) x Áž}œ‹Îµœªœ‡¼n
(‹µ„ 3) čo„‘„µ¦‡´—)
6) x =2n ×¥š¸É n Áž}œ‹ÎµœªœÁ˜È¤
(šœ·¥µ¤…°Š‹Îµœªœ‡¼n)
7) ‹³Å—o y = 2n + 3
(‹µ„ y = x + 3)
8) y = 2(n+1) + 1
(„‘„µ¦Áž¨¸É¥œ®¤¼nŗo, „‘„µ¦­¨´š¸É
¨³„‘„µ¦Â‹„‹Š)
146
9) Ä®o n + 1 = r ‹³Å—o r Áž}œ‹ÎµœªœÁ˜È¤
(˜´ÊŠºÉ°Ä®¤n)
10) —´Šœ´Êœ y = 2r + 1 ×¥š¸É r Áž}œ‹ÎµœªœÁ˜È¤
11) œ´Éœ‡º° y Áž}œ‹Îµœªœ‡¸É
(‹µ„ 10) ¨³šœ·¥µ¤…°Š‹Îµœªœ‡¸É)
Ž¹ÉŠ…´—Â¥oŠ„´ 4) š¸Éªnµ y Ťnčn‹Îµœªœ‡¸É
12) ‹³Á®Èœªnµ ™oµÄ®o Q : y Ťnčn‹Îµœªœ‡¸É
aQ : y Áž}œ‹Îµœªœ‡¸É
‹µ„ 4), 11) ¨³ 12) Á¦µÅ—o Q š aQ („‘„µ¦¦ª¤)
13) œ´Éœ‡º° aP o Q š ~Q
14) —´Šœ´Êœ ™oµ x Áž}œ‹Îµœªœ‡¼n ¨oª y = x + 3 ‹³Áž}œ‹Îµœªœ‡¸É
…o°­´ŠÁ„˜ ‹³Á®Èœªnµ˜´ª°¥nµŠš¸É 4.1.1, 4.1.6 ¨³ 4.1.10 Ĝ®´ª…o° 1), 2) ¨³ 3) Áž}œÃ‹š¥r…o°Á—¸¥ª„´œ
—´Šœ´Êœ‹³Å—oªnµÃ‹š¥r…o°Á—¸¥ª„´œ­µ¤µ¦™¡·­¼‹œrŗo®¨µ¥ª·›¸
˜´ª°¥nµŠš¸É 4.1.11 ‹Š¡·­¼‹œrªnµ 2 Áž}œ‹Îµœªœ°˜¦¦„¥³
¡·­¼‹œr
1) ­¤¤»˜·Ä®o 2 Áž}œ‹Îµœªœ˜¦¦„¥³
a
2) Á¦µ­µ¤µ¦™Á…¸¥œÅ—oªnµ 2 = b ×¥š¸É a ¨³ b Áž}œ‹ÎµœªœÁ˜È¤ª„ ¨³
®.¦.¤. (greatest common divisor) …°Š a ¨³ b Ášnµ„´ 1
a2
3) 2 = 2
b
4) a2 = 2b2 ¨³ b2 Áž}œ‹ÎµœªœÁ˜È¤
5) a2 Áž}œ‹Îµœªœ‡¼n (‹µ„ 4) ¨³šœ·¥µ¤…°Š‹Îµœªœ‡¼n)
6)
7)
8)
—´Šœ´Êœ a Áž}œ‹Îµœªœ‡¼n (˜´ª°¥nµŠš¸É 4.1.7 Ĝ®´ª…o° 2))
a = 2n ×¥š¸É n Áž}œ‹ÎµœªœÁ˜È¤ (šœ·¥µ¤…°Š‹Îµœªœ‡¼n)
a2 = (2n)2
9) 2b2 = (2n)2
(‹µ„ 4) ¨³ 8))
10) b2 = 2n2 ×¥š¸É n2 Áž}œ‹ÎµœªœÁ˜È¤
147
11) b2 Áž}œ‹Îµœªœ‡¼n
(‹µ„ 10) ¨³šœ·¥µ¤…°Š‹Îµœªœ‡¼n)
12) b Áž}œ‹Îµœªœ‡¼n
(˜´ª°¥nµŠš¸É 4.1.7 Ĝ…o°®´ª 2))
13) b = 2m ×¥š¸É m Áž}œ‹ÎµœªœÁ˜È¤
14) 2 Áž}œ˜´ª®µ¦¦nª¤…°Š a ¨³ b (‹µ„ 7) ¨³ 13))
15) œ´Éœ‡º° ®.¦.¤. …°Š a ¨³ b ŤnÁšnµ„´ 1
Ž¹ÉŠ…´—Â¥oŠ„´ 2) š¸Éªnµ ®.¦.¤. …°Š a ¨³ b Ášnµ„´ 1
16) —´Šœ´Êœ 2 Áž}œ‹Îµœªœ°˜¦¦„¥³
˜´ª°¥nµŠš¸É 4.1.12 ‹Š¡·­¼‹œrªnµ ‹Îµœªœ˜¦¦„¥³¦ª¤„´‹Îµœªœ°˜¦¦„¥³ ‹³Å—o‹Îµœªœ°˜¦¦„¥³
¡·­¼‹œr
‹³Á®Èœªnµ­·ÉŠš¸É˚¥rÄ®o¡·­¼‹œrœ´Êœ‡º°…o°‡ªµ¤Á—¸¥ª„´…o°‡ªµ¤˜n°Åžœ¸Ê
™oµ x Áž}œ‹Îµœªœ˜¦¦„¥³ ¨³ y Áž}œ‹Îµœªœ°˜¦¦„¥³ ¨oª x + y Áž}œ‹Îµœªœ
°˜¦¦„¥³
Á…¸¥œÄ®o°¥¼nĜ¦¼ž­´¨´„¬–rŗo—´Šœ¸Ê
Ä®o p1 šœ x Áž}œ‹Îµœªœ˜¦¦„¥³
p2 šœ y Áž}œ‹Îµœªœ°˜¦¦„¥³
p3 šœ x + y Áž}œ‹Îµœªœ°˜¦¦„¥³
—´Šœ´Êœ p1 š p2 o p3 ‹³Âšœ ™oµ x Áž}œ‹Îµœªœ˜¦¦„¥³ ¨³ y Áž}œ‹Îµœªœ°˜¦¦„¥³
¨oª x + y Áž}œ‹Îµœªœ°˜¦¦„¥³
Ä®o P šœ p1 š p2 o p3
1) ­¤¤»˜·Ä®o aP Áž}œ‹¦·Š
2) ‹³Å—oªnµ a(p1 š p2 o p3) Áž}œ‹¦·Š
3) (p1 š p2) š ap3
Á¡¦µ³ a{(p1 š p2) o p3} { (p1 š p2) š ap3)
4) œ´Éœ‡º° Á¦µÅ—oªnµ x Áž}œ‹Îµœªœ˜¦¦„¥³ ¨³ y Áž}œ‹Îµœªœ°˜¦¦„¥³ ¨³
x + y Áž}œ‹Îµœªœ˜¦¦„¥³
148
5) Á¦µÅ—o x Áž}œ‹Îµœªœ˜¦¦„¥³ (‹µ„ 4) čo„‘„µ¦‡´—)
a
6) —´Šœ´Êœ x = b
×¥š¸É a, b Áž}œ‹ÎµœªœÁ˜È¤ ¨³ b z 0
(šœ·¥µ¤…°Š‹Îµœªœ˜¦¦„¥³)
c
7) x + y = d
×¥š¸É c, d Áž}œ‹ÎµœªœÁ˜È¤ ¨³ d z 0
(Á¡¦µ³ x + y Áž}œ‹Îµœªœ˜¦¦„¥³)
a
c
8) b + y = d
(‹µ„ 6) ¨³ 7))
c a cb ad
×¥š¸É cd ad, b.d Áž}œ‹ÎµœªœÁ˜È¤ ¨³ bd z 0
9) y = d b = bd
(Á¡¦µ³ a, b, c, d Áž}œ‹ÎµœªœÁ˜È¤ ¨³
b z 0 , d z 0)
10) y Áž}œ‹Îµœªœ˜¦¦„¥³
(‹µ„ 9) ¨³šœ·¥µ¤…°Š‹Îµœªœ˜¦¦„¥³)
Ž¹ÉŠ…´—Â¥oŠ„´ 4) š¸Éªnµ y Áž}œ‹Îµœªœ°˜¦¦„¥³
11) —´Šœ´Êœ ™oµ x Áž}œ‹Îµœªœ˜¦¦„¥³ ¨³ y Áž}œ‹Îµœªœ°˜¦¦„¥³ ¨oª x + y
Áž}œ‹Îµœªœ°˜¦¦„¥³
1
˜´ª°¥nµŠš¸É 4.1.13 ‹Š¡·­¼‹œrªnµ ™oµ x Áž}œ‹Îµœªœ‹¦·Šª„ ¨oª x + x > 2
1
¡·­¼‹œr
Ä®o P šœ ™oµ x Áž}œ‹Îµœªœ‹¦·Šª„ ¨oª x + x > 2
1) ­¤¤»˜·Ä®o aP Áž}œ‹¦·Š
1
2) ‹³Å—oªnµ x Áž}œ‹Îµœªœ‹¦·Šª„ ¨³ x + x t 2
(Á¡¦µ³ a(p o q) { p š aq)
1
3) ‹µ„ 2) Á¦µÅ—o x + x t 2
1
4) œ´Éœ‡º° x + x < 2
1
5) x + x 2 < 0
6) x2 + 1 2x < 0
(‹µ„ 5) ‡¼–š´ÊŠ­°Š…oµŠ—oª¥ x, Á‡¦ºÉ°Š®¤µ¥°­¤„µ¦
ŤnÁž¨¸É¥œ Á¡¦µ³ªnµ x Áž}œ‹Îµœªœ‹¦·Šª„)
7) (x 1)2 < 0
149
Ž¹ÉŠ…´—Â¥oŠ„´ (x 1)2 > 0
(‡»–­¤´˜·…°Š‹Îµœªœ‹¦·Š)
1
—´Šœ´Êœ ™oµ x Áž}œ‹Îµœªœ‹¦·Šª„ ¨oª x + x > 2
˜´ª°¥nµŠš¸É 4.1.14 ‹Š¡·­¼‹œrªnµ (p š q) o p Áž}œ­´‹œ·¦´œ—¦r
¡·­¼‹œr
1) ­¤¤»˜·Ä®o p š q o p ŤnÁž}œ­´‹œ·¦´œ—¦r
2) —´Šœ´Êœ ¤¸„¦–¸š¸É p š q o p ¤¸‡nµ‡ªµ¤‹¦·ŠÁž}œÁšÈ‹
3) ­—Šªnµ p š q ¤¸‡nµ‡ªµ¤‹¦·ŠÁž}œ‹¦·Š
4) ¨³ p ¤¸‡nµ‡ªµ¤‹¦·ŠÁž}œÁšÈ‹
5) ‹µ„ 3) ‹³Å—o p ¤¸‡nµ‡ªµ¤‹¦·ŠÁž}œ‹¦·Š Ž¹ÉŠ…´—Â¥oŠ„´ 4)
—´Šœ´Êœ p š q o p Áž}œ­´‹œ·¦´œ—¦r
˜´ª°¥nµŠš¸É 4.1.15 Ä®o ABC Áž}œ­µ¤Á®¨¸É¥¤¦¼ž®œ¹ÉŠ Ž¹ÉŠ—oµœ AB = —oµœ AC ‹Š¡·­¼‹œrªnµ B = C
¡·­¼‹œr
­·ÉŠš¸É˚¥r˜o°Š„µ¦Ä®o¡·­¼‹œr ‡º° ™oµ—oµœ AB = —oµœ AC ¨oª B = C
Ä®o P šœ ™oµ—oµœ AB = —oµœ AC ¨oª B = C
1) ­¤¤»˜·Ä®o aP
2) ‹³Å—oªnµ—oµœ AB = —oµœ AC
3) ¨³ B z C
4) Á¡¦µ³ªnµ B z C —´Šœ´Êœ B > C ®¦º° B < C
5) ™oµ B > C ‹³Å—o —oµœ AC > —oµœ AB Ž¹ÉŠ…´—Â¥oŠ„´ 2)
6) ™oµ B < C ‹³Å—o —oµœ AC < —oµœ AB Ž¹ÉŠ…´—Â¥oŠ„´ 2)
œ´Éœ‡º° ™oµ—oµœ AB = —oµœ AC ¨oª B = C
150
˜´ª°¥nµŠš¸É 4.1.16 ­Îµ®¦´‹Îµœªœ‹¦·Š x ė Ç
‹Š¡·­¼‹œrªnµ
2x
Áž}œ‹Îµœªœ°˜¦¦„¥³®¦º°
2x
Áž}œ‹Îµœªœ°˜¦¦„¥³
¡·­¼‹œr
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151
 f„®´—š¸É 18
1. „ε®œ—Ä®o x Áž}œ‹ÎµœªœÁ˜È¤
‹Š¡·­¼‹œrªnµ ™oµ x Áž}œ‹Îµœªœ‡¸É ¨oª x + 1 Áž}œ‹Îµœªœ‡¼n Ä®očo„µ¦¡·­¼‹œr×¥„µ¦®µ…o°…´—Â¥oŠ
2. ‹Š¡·­¼‹œrªnµ ™oµ x Áž}œ‹Îµœªœ˜¦¦„¥³ ¨³ y Áž}œ‹Îµœªœ°˜¦¦„¥³ ¨oª y x Áž}œ‹Îµœªœ
°˜¦¦„¥³
3. ‹Š¡·­¼‹œrªnµ ™oµ x Áž}œ‹Îµœªœ˜¦¦„¥³Ž¹ÉŠÅ¤nÁšnµ„´«¼œ¥r ¨³ y Áž}œ‹Îµœªœ°˜¦¦„¥³ ¨oª xy Áž}œ
‹Îµœªœ°˜¦¦„¥³
4. „ε®œ—Á°„£¡­´¤¡´š›r‡º° ÁŽ˜…°Š‹Îµœªœ‹¦·Š
P(x) ‡º° 2x + 1 ¨³ t Áž}œ‹Îµœªœ‹¦·Šª„
t
‹Š¡·­¼‹œrªnµ ™oµ |x| < 2 ¨oª |P(x) 1| < t
5. ‹Š¡·­¼‹œrªnµ (p o q) l ~p › q Áž}œ­´‹œ·¦´œ—¦r ץčo„µ¦¡·­¼‹œr×¥„µ¦®µ…o°…´—Â¥oŠ
6. ‹Š¡·­¼‹œrªnµ A ˆ Ac I
152
4) „µ¦¡·­¼‹œr p › q
Á¦µ°µ‹¡·­¼‹œr p › q ×¥¡·­¼‹œr ~p o q šœš´ÊŠœ¸Ê Á¡¦µ³ p › q
­nªœ„µ¦¡·­¼‹œr ~ p o q „ȗεÁœ·œ„µ¦˜µ¤ª·›¸š¸É„¨nµª¤µÂ¨oª
{
~p
o
q
˜´ª°¥nµŠš¸É 4.1.17 ‹Š¡·­¼‹œrªnµ ™oµŸ¨‡¼–…°Š‹ÎµœªœÁ˜È¤­°Š‹ÎµœªœÁž}œ‹Îµœªœ‡¼n ¨oª ‹ÎµœªœÄ—‹Îµœªœ
®œ¹ÉŠÁž}œ‹Îµœªœ‡¼n
¡·­¼‹œr
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153
5) „µ¦¡·­¼‹œr×¥„µ¦Â‹Š„¦–¸ (Proof by Cases)
„µ¦¡·­¼‹œr p o q ĜµŠ‡¦´ÊŠ p °µ‹ž¦³„°—oª¥„¦–¸˜nµŠ Ç Áž}œ‹Îµœªœ‹Îµ„´— Ánœ p
ž¦³„°—oª¥ p1 › p2 Ĝ„µ¦¡·­¼‹œr p o q „È‹³˜o°Š¡·­¼‹œr (p1 › p2) o q
ÁœºÉ°Š‹µ„ (p1 o q) š (p2 o q) o {(p1 › p2) o q} Áž}œ­´‹œ·¦´œ—¦r
—´Šœ´Êœ Á¦µ­µ¤µ¦™¡·­¼‹œr p1 › p2 o q ×¥¡·­¼‹œr p1 o q ¨³ p2 o q šœ ¨³Äœ„µ¦¡·­¼‹œr
p1 o q ¨³ p2 o q œ´Êœ „ȗεÁœ·œ„µ¦¡·­¼‹œr˜µ¤ª·›¸š¸É„¨nµª¤µÂ¨oªÄœ®´ª…o° 1) – 4)
Ĝ„¦–¸š¸É p ‡º° p1 › p2 › p3 ... › pn „µ¦¡·­¼‹œr p o q „È‹³˜o°ŠÂ¥„„µ¦¡·­¼‹œrÁž}œ n „¦–¸ ‡º°
¡·­¼‹œr p1 o q, p2 o q, p3 o q, ..., pn o q
„µ¦¡·­¼‹œr p1 › p2 o q °¥¼nĜ¦¼ž—´Šœ¸Ê
¡·­¼‹œr „¦–¸š¸É 1 ¡·­¼‹œrªnµ p1 o q
-------—εÁœ·œ„µ¦¡·­¼‹œr˜µ¤ª·›¸š¸É„¨nµª¤µÂ¨oª Ĝ®´ª…o° 1) – 4)
--------
„¦–¸š¸É 2 ¡·­¼‹œrªnµ p2 o q
-------—εÁœ·œ„µ¦¡·­¼‹œr˜µ¤ª·›¸š¸É„¨nµª¤µÂ¨oª Ĝ®´ª…o° 1) – 4)
--------
Á¡¦µ³Œ³œ´Êœ p1 › p2 o q
˜´ª°¥nµŠš¸É 4.1.18 ‹Š¡·­¼‹œrªnµ ™oµ a Áž}œ‹ÎµœªœÁ˜È¤ ¨oª a2 a Áž}œ‹Îµœªœ‡¼n
¡·­¼‹œr
™oµÄ®o a Áž}œ‹ÎµœªœÁ˜È¤ ‹³Å—oªnµ a Áž}œ‹Îµœªœ‡¼n®¦º°‹Îµœªœ‡¸É —´Šœ´Êœ Ĝ„µ¦¡·­¼‹œrªnµ
™oµ a Áž}œ‹ÎµœªœÁ˜È¤ ¨oª a2 a Áž}œ‹Îµœªœ‡¼n Á¦µ‹³˜o°Š¡·­¼‹œr 2 „¦–¸ ‡º°
154
„¦–¸š¸É 1 ¡·­¼‹œrªnµ ™oµ a Áž}œ‹Îµœªœ‡¼n ¨oª a2 a Áž}œ‹Îµœªœ‡¼n
„¦–¸š¸É 2 ¡·­¼‹œrªnµ ™oµ a Áž}œ‹Îµœªœ‡¸É ¨oª a2 a Áž}œ‹Îµœªœ‡¼n
„¦–¸š¸É 1 ­¤¤»˜·Ä®o a Áž}œ‹Îµœªœ‡¼n
‹³Å—oªnµ a = 2n ×¥š¸É n Áž}œ‹ÎµœªœÁ˜È¤
a2 a = (2n)2 2n
(šœ·¥µ¤…°Š‹Îµœªœ‡¼n)
= 4n2 2n
= 2(2n2 n) ×¥š¸É (2n2 n) Áž}œ‹ÎµœªœÁ˜È¤
—´Šœ´Êœ a2 a Áž}œ‹Îµœªœ‡¼n
(šœ·¥µ¤…°Š‹Îµœªœ‡¼n)
œ´Éœ‡º° ™oµ a Áž}œ‹Îµœªœ‡¼n ¨oª a2 a Áž}œ‹Îµœªœ‡¼n
„¦–¸š¸É 2 ­¤¤»˜·Ä®o a Áž}œ‹Îµœªœ‡¸É
‹³Å—oªnµ a = 2n + 1 ×¥š¸É n Áž}œ‹ÎµœªœÁ˜È¤
a2 a = (2n+1)2 (2n+1)
(šœ·¥µ¤…°Š‹Îµœªœ‡¸É)
= (4n2+4n+1) (2n+1)
= 4n2 + 2n
= 2(2n2+n) ×¥š¸É 2n2 + n Áž}œ‹ÎµœªœÁ˜È¤
—´Šœ´Êœ a2 a Áž}œ‹Îµœªœ‡¼n
(šœ·¥µ¤…°Š‹Îµœªœ‡¼n)
œ´Éœ‡º° ™oµ a Áž}œ‹Îµœªœ‡¸É ¨oª a2 a Áž}œ‹Îµœªœ‡¼n
‹µ„„¦–¸š¸É 1 ¨³„¦–¸š¸É 2 ­¦»žÅ—oªnµ ™oµ a Áž}œ‹ÎµœªœÁ˜È¤ ¨oª a2 a Áž}œ‹Îµœªœ‡¼n
155
˜´ª°¥nµŠš¸É 4.1.19 „ε®œ—Ä®o a ¨³ b Áž}œ‹ÎµœªœÁ˜È¤
‹Š¡·­¼‹œrªnµ ™oµ ab Áž}œ‹Îµœªœ‡¸É ¨oª a ¨³ b ˜nµŠÁž}œ‹Îµœªœ‡¸É
¡·­¼‹œr
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156
 f„®´—š¸É 19
„ε®œ—Ä®o a Áž}œ‹ÎµœªœÁ˜È¤ ‹Š¡·­¼‹œrªnµ
1. ™oµ a Áž}œ‹ÎµœªœÁ˜È¤ ¨oª a2 + 3a + 1 Áž}œ‹Îµœªœ‡¸É
2. ™oµ a Áž}œ‹ÎµœªœÁ˜È¤ ¨oª a2 + a Áž}œ‹Îµœªœ‡¼n
3. ™oµ 3 ®µ¦ a2 ¨Š˜´ª ¨oª 3 ®µ¦ a ¨Š˜´ª
4. „ε®œ—Ä®o…o°‡ªµ¤˜n°Åžœ¸ÊÁž}œ‹¦·Š
(„) ‡nµ­´¤¼¦–r (absolute value) …°Š‹Îµœªœ‹¦·Š x čo­´¨´„¬–r |x| ¤¸šœ·¥µ¤—´Šœ¸Ê
­ x , ™oµ x > 0
|x| = ® x , ™oµ x < 0
¯
(…) x, x  R o x < 0 › x = 0 › x > 0 Á¤ºÉ° R ‡º° ÁŽ˜…°Š‹Îµœªœ‹¦·Š
(‡) x  R, x > 0 l x < 0 ¨³ (0) = 0
(Š) x  R, (x) = x
(‹) x  R, x2 > 0
‹Š¡·­¼‹œrªnµ
4.1 ™oµ x Áž}œ‹Îµœªœ‹¦·Š ¨oª |x| = |x|
4.2 ™oµ x Áž}œ‹Îµœªœ‹¦·Š ¨oª |x2| = |x|2
4.3 ™oµ x ¨³ y Áž}œ‹Îµœªœ‹¦·Š ¨oª |xy| = |x||y|
157
6) „µ¦¡·­¼‹œr p l q
Á¦µ¡·­¼‹œr p l q ×¥¡·­¼‹œrÂ¥„Áž}œ 2 ˜°œ ‡º°
˜°œš¸É 1 ¡·­¼‹œr p o q ¨³˜°œš¸É 2 ¡·­¼‹œr q o p
š´ÊŠœ¸ÊÁ¡¦µ³ p l q { ( p o q ) š ( q o p )
Ĝ„µ¦¡·­¼‹œr p o q ¨³ q o p „Èčoª·›¸„µ¦¡·­¼‹œr˜µ¤ª·›¸˜nµŠ Ç š¸É„¨nµª¤µÂ¨oª
„µ¦¡·­¼‹œr p l q °¥¼nĜ¦¼ž
¡·­¼‹œr ˜°œš¸É 1 ¡·­¼‹œr p o q
----------—εÁœ·œ„µ¦¡·­¼‹œr˜µ¤ª·›¸š¸É„¨nµª¤µÂ¨oª
----------˜°œš¸É 2 ¡·­¼‹œr q o p
----------—εÁœ·œ„µ¦¡·­¼‹œr˜µ¤ª·›¸š¸É„¨nµª¤µÂ¨oª
----------Á¡¦µ³Œ³œ´Êœ p l q
˜´ª°¥nµŠš¸É 4.1.20 „ε®œ—Ä®o a Áž}œ‹ÎµœªœÁ˜È¤ ‹Š¡·­¼‹œrªnµ
(a + 1)2 Áž}œ‹Îµœªœ‡¼n „Șn°Á¤ºÉ° a Áž}œ‹Îµœªœ‡¸É
„µ¦¡·­¼‹œrÂ¥„Áž}œ 2 ˜°œ ‡º°
1) ¡·­¼‹œrªnµ ™oµ (a + 1)2 Áž}œ‹Îµœªœ‡¼n ¨oª a Áž}œ‹Îµœªœ‡¸É
2) ¡·­¼‹œrªnµ ™oµ a Áž}œ‹Îµœªœ‡¸É ¨oª (a + 1)2 Áž}œ‹Îµœªœ‡¼n
¡·­¼‹œr
1) ץčo…o°‡ªµ¤Â¥oŠ­¨´š¸É
œ´Éœ‡º° ‹³¡·­¼‹œrªnµ ™oµ a Ťnčn‹Îµœªœ‡¸É ¨oª (a + 1)2 Ťnčn‹Îµœªœ‡¼n
­¤¤»˜·Ä®o a Ťnčn‹Îµœªœ‡¸É
—´Šœ´Êœ a Áž}œ‹Îµœªœ‡¼n
a = 2n ×¥š¸É n Áž}œ‹ÎµœªœÁ˜È¤
158
a + 1 = 2n + 1
(a + 1)2 = (2n + 1)2
= 4n2 + 4n + 1
= 2(2n2 + 2n) + 1 ×¥š¸É 2n2 + 2n Áž}œ‹ÎµœªœÁ˜È¤
(a + 1)2 Áž}œ‹Îµœªœ‡¸É
Á¡¦µ³Œ³œ´Êœ (a + 1)2 Ťnčn‹Îµœªœ‡¼n
œ´Éœ‡º° ™oµ a Ťnčn‹Îµœªœ‡¸É ¨oª (a + 1)2 Ťnčn‹Îµœªœ‡¼n
—´Šœ´Êœ ™oµ (a + 1)2 Áž}œ‹Îµœªœ‡¼n ¨oª a Áž}œ‹Îµœªœ‡¸É
¡·­¼‹œr
2) ץčo„µ¦¡·­¼‹œr p o q ×¥˜¦Š
­¤¤»˜·Ä®o a Áž}œ‹Îµœªœ‡¸É
a = 2n + 1 ×¥š¸É n Áž}œ‹ÎµœªœÁ˜È¤
a + 1 = (2n + 1) + 1 = 2n + 2
(a + 1)2 = (2n + 2)2 = 4n2 + 8n + 4 = 2(2n2 + 4n + 2)
×¥š¸É 2n2 + 4n + 2 Áž}œ‹ÎµœªœÁ˜È¤
œ´Éœ‡º° (a + 1)2 Áž}œ‹Îµœªœ‡¼n
—´Šœ´Êœ ™oµ a Áž}œ‹Îµœªœ‡¸É ¨oª (a + 1)2 Áž}œ‹Îµœªœ‡¼n
159
˜´ª°¥nµŠš¸É 4.1.21 „ε®œ—Ä®o a ¨³ b Áž}œ‹ÎµœªœÁ˜È¤
‹Š¡·­¼‹œrªnµ a + b Áž}œ‹Îµœªœ‡¼n „Șn°Á¤ºÉ° a - b Áž}œ‹Îµœªœ‡¼n
¡·­¼‹œr
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160
 f„®´—š¸É 20
„ε®œ—Ä®o a ¨³ b Áž}œ‹ÎµœªœÁ˜È¤ ‹Š¡·­¼‹œr˜n¨³…o°˜n°Åžœ¸Ê
1. a Áž}œ‹Îµœªœ‡¸É „Șn°Á¤ºÉ° a + 1 Áž}œ‹Îµœªœ‡¼n
2. a Áž}œ‹Îµœªœ‡¸É „Șn°Á¤ºÉ° a2 Áž}œ‹Îµœªœ‡¸É
3. a Áž}œ‹Îµœªœ‡¼n „Șn°Á¤ºÉ° a + 2 Áž}œ‹Îµœªœ‡¼n
4. a2 Áž}œ‹Îµœªœ‡¸É „Șn°Á¤ºÉ° a + 2 Áž}œ‹Îµœªœ‡¸É
5. 3 ®µ¦ a ¨Š˜´ª „Șn°Á¤ºÉ° 3 ®µ¦ a2 ¨Š˜´ª
6. ­Îµ®¦´ A ¨³ B Ä—Ç ‹Š¡·­¼‹œrªnµ A  B „Șn°Á¤ºÉ° B c  Ac
161
7) „µ¦¡·­¼‹œrÂ¥oŠÃ—¥„µ¦¥„˜´ª°¥nµŠ‡oµœ (Disproof by Counter Example)
„µ¦¡·­¼‹œrÂ¥oŠÃ—¥„µ¦¥„˜´ª°¥nµŠ‡oµœ Áž}œ„µ¦‡´—‡oµœ…o°‡ªµ¤ “­Îµ®¦´š»„ Ç ­¤µ·„ÄœÁ°„£¡
­´¤¡´š›r U ­°—‡¨o°Š„´¨´„¬–³š¸É„ε®œ—Ä®o” „¨nµª‡º° Áž}œ„µ¦¡·­¼‹œrªnµ x [ P(x) ] Áž}œÁšÈ‹ ®¦º°
ax [ P(x) ] Áž}œ‹¦·Šœ´ÉœÁ°Š „µ¦¡·­¼‹œr„¦³šÎµÃ—¥„µ¦®µ­¤µ·„®œ¹ÉŠ˜´ª‹µ„Á°„£¡­´¤¡´š›r U ¤µÂšœ
˜´ªÂž¦ x ¨oªšÎµÄ®o P(x) Áž}œÁšÈ‹ (®¦º° aP(x) Áž}œ‹¦·Š) œ´Éœ‡º° Á¦µ­µ¤µ¦™¡·­¼‹œr x [ P(x) ] Áž}œÁšÈ‹
(®¦º° ax [ P(x) ] Áž}œ‹¦·Š) ×¥¡·­¼‹œrªnµ x [ aP(x) ] Áž}œ‹¦·Š Á¦µ¡·­¼‹œrÁnœœ¸ÊŗoÁ¡¦µ³
ax [ P(x) ] { x [ aP(x) ]
„µ¦¡·­¼‹œr x [ P(x) ] Áž}œÁšÈ‹ °¥¼nĜ¦¼ž
¡·­¼‹œr Á¨º°„ a š¸ÉÁ®¤µ³­¤ ץĮo a  U
¨³šÎµÄ®o aP(a) Áž}œ‹¦·Š
------------------------Á¡¦µ³Œ³œ´Êœ aP(a) Áž}œ‹¦·Š
œ´Éœ‡º° x [ aP(x) ] Áž}œ‹¦·Š
—´Šœ´Êœ x [ P(x) ] Áž}œÁšÈ‹
˜´ª°¥nµŠš¸É 4.1.22 ‹Š¡·­¼‹œrªnµ x [™oµ x Áž}œ‹Îµœªœ‹¦·Š ¨oª x2 > x] Áž}œÁšÈ‹
1
¡·­¼‹œr
ÁœºÉ°Š‹µ„ 2 Áž}œ‹Îµœªœ‹¦·Š Áž}œ‹¦·Š
12 1 1
¨³ 2 = 4 ! 2 Áž}œ‹¦·Š
1
12 1
‹¹ŠÅ—o 2 Áž}œ‹Îµœªœ‹¦·Š ¨³ 2 ! 2 Áž}œ‹¦·Š
œ´Éœ‡º° x [x Áž}œ‹Îµœªœ‹¦·Š ¨³ x2 ! x] Áž}œ‹¦·Š
—´Šœ´Êœ x [™oµ x Áž}œ‹Îµœªœ‹¦·Š ¨oª x2 > x] Áž}œÁšÈ‹
162
˜´ª°¥nµŠš¸É 4.1.23 ‹Š¡·­¼‹œrªnµ x [™oµ x Áž}œ‹ÎµœªœÁŒ¡µ³ ¨oª x Áž}œ‹Îµœªœ‡¸É] Áž}œÁšÈ‹ Á¤ºÉ°
Á°„£¡­´¤¡´š›r‡º° ÁŽ˜…°Š‹ÎµœªœÁ˜È¤
¡·­¼‹œr
ÁœºÉ°Š‹µ„ 2 Áž}œ‹ÎµœªœÁŒ¡µ³ Áž}œ‹¦·Š
¨³ 2 Ťnčn‹Îµœªœ‡¸É Áž}œ‹¦·Š
‹¹ŠÅ—o 2 Áž}œ‹ÎµœªœÁŒ¡µ³ ¨³ 2 Ťnčn‹Îµœªœ‡¸É Áž}œ‹¦·Š
œ´Éœ‡º° x [x Áž}œ‹ÎµœªœÁŒ¡µ³ ¨³ x Ťnčn‹Îµœªœ‡¸É] Áž}œ‹¦·Š
—´Šœ´Êœ x [™oµ x Áž}œ‹ÎµœªœÁŒ¡µ³ ¨oª x Áž}œ‹Îµœªœ‡¸É] Áž}œÁšÈ‹
 f„®´—š¸É 21
‹Š¡·­¼‹œrªnµ…o°‡ªµ¤˜n°Åžœ¸ÊÁž}œÁšÈ‹
1. x [™oµ x Áž}œ‹ÎµœªœÁ˜È¤ ¨oª x2 = 1]
2.
3.
x [™oµ x Áž}œ‹Îµœªœ‹¦·Š ¨oª x > 0 ®¦º° x Áž}œ‹Îµœªœ˜¦¦„¥³]
x [™oµ x Áž}œ‹Îµœªœ‹¦·Š ¨oª x2 = x]
4. ž¨µš»„˜´ª¤¸Á„¨È—
5. ™oµ a, b, c Áž}œ‹Îµœªœ‹¦·Š ¨³ a > b ¨oª ac > bc
1 1
6. ™oµ a, b Áž}œ‹Îµœªœ‹¦·Šš¸ÉŤnÁšnµ„´«¼œ¥r ¨³ a > b ¨oª a < b
7. „ε®œ— A = {1,2,3,4} ¨³ B = {1,3,5,7,9} ‹Š¡·­¼‹œrªnµ A Áž}œÁŽ˜¥n°¥ (subset) …°Š B Áž}œÁšÈ‹
163
8) „µ¦¡·­¼‹œrªnµ¤¸ (°¥nµŠœo°¥®œ¹ÉŠ) (Proof of Existence)
„µ¦¡·­¼‹œrªnµ¤¸ Áž}œ„µ¦¡·­¼‹œr…o°‡ªµ¤ž¦³Á£šš¸É˜¦ª‹­°ªnµ­¤µ·„ÄœÁ°„£¡­´¤¡´š›r U ¤¸
°¥nµŠœo°¥ 1 ­¤µ·„š¸É­°—‡¨o°Š„´¨´„¬–³š¸É„ε®œ—Ä®o „¨nµª‡º° Áž}œ„µ¦¡·­¼‹œrªnµ x [ P(x) ] Áž}œ‹¦·Š
Ž¹ÉŠ„µ¦¡·­¼‹œr„¦³šÎµÃ—¥„µ¦®µ­¤µ·„®œ¹ÉŠ˜´ªÄœÁ°„£¡­´¤¡´š›r¤µÂšœ˜´ªÂž¦ x ¨oªšÎµÄ®o P(x) Áž}œ
‹¦·Š
„µ¦¡·­¼‹œr°¥¼nĜ¦¼ž—´Šœ¸Ê
¡·­¼‹œr Á¨º°„­¤µ·„ a š¸ÉÁ®¤µ³­¤ ×¥š¸É a  U
¨³šÎµÄ®o P(a) Áž}œ‹¦·Š
---------------------------Á¡¦µ³Œ³œ´Êœ P(a) Áž}œ‹¦·Š
œ´Éœ‡º° x [ P(x) ] Áž}œ‹¦·Š
„µ¦¡·­¼‹œrªnµ¤¸œ¸Ê¤¸‡ªµ¤‹ÎµÁž}œ¤µ„Äœ‡–·˜«µ­˜¦r¦³—´­¼Š Á¡¦µ³„µ¦¡·­¼‹œr—´Š„¨nµªÁ¡ºÉ°Áž}œ„µ¦
¥ºœ¥´œªnµÁ®˜»„µ¦–rœ´Êœ¤¸‹¦·Š „n°œš¸É‹³—εÁœ·œ„µ¦«¹„¬µ˜n°Åž
1
˜´ª°¥nµŠš¸É 4.1.24 ‹Š¡·­¼‹œrªnµ “¤¸‹ÎµœªœÁ˜È¤ x °¥nµŠœo°¥®œ¹ÉŠ‹Îµœªœ Ž¹ÉŠ x = x”
1
¡·­¼‹œr
˚¥r˜o°Š„µ¦¡·­¼‹œrªnµ x [x Áž}œ‹ÎµœªœÁ˜È¤ ¨³ x = x] Áž}œ‹¦·Š
Ä®o x = 1 —´Šœ´Êœ x Áž}œ‹ÎµœªœÁ˜È¤ Áž}œ‹¦·Š
1 1
¨³ x = 1 = 1 = x Áž}œ‹¦·Š
1
œ´Éœ‡º° x [x Áž}œ‹ÎµœªœÁ˜È¤ ¨³ x = x] Áž}œ‹¦·Š
˜´ª°¥nµŠš¸É 4.1.25 ‹Š¡·­¼‹œrªnµ “¤¸‹Îµœªœ‹¦·Š x š¸É­°—‡¨o°Š„´­¤„µ¦ x2 3x + 2 = 0”
¡·­¼‹œr
˚¥r˜o°Š„µ¦¡·­¼‹œrªnµ x [x Áž}œ‹Îµœªœ‹¦·Š ¨³ x2 3x + 2 = 0] Áž}œ‹¦·Š
Ä®o x = 1 —´Šœ´Êœ x Áž}œ‹Îµœªœ‹¦·Š Áž}œ‹¦·Š
¨³‹³Å—o x2 3x + 2 = 12 3 + 2 = 0 Áž}œ‹¦·Š
®¦º°Ä®o x = 2 —´Šœ´Êœ x Áž}œ‹Îµœªœ‹¦·Š Áž}œ‹¦·Š
164
¨³‹³Å—o x2 3x + 2 = 4 6 + 2 = 0 Áž}œ‹¦·Š
œ´Éœ‡º° x [x Áž}œ‹Îµœªœ‹¦·Š ¨³ x2 3x + 2 = 0] Áž}œ‹¦·Š
˜´ª°¥nµŠš¸É 4.1.26 ‹Š¡·­¼‹œrªnµ “¤¸‹ÎµœªœÁ˜È¤ª„ N Ž¹ÉŠ­Îµ®¦´š»„ Ç ‹ÎµœªœÁ˜È¤ª„ n š¸É n > N
1
¨oª n < 0.001”
¡·­¼‹œr
Ä®o N = 1,000 —´Šœ´Êœ N Áž}œ‹ÎµœªœÁ˜È¤ª„ Áž}œ‹¦·Š
¨³ÁœºÉ°Š‹µ„ n > N —´Šœ´Êœ n > 1,000
1 1
n < 1000
,
1
Á¡¦µ³Œ³œ´Êœ n < .001
Áž}œ‹¦·Š
1
—´Šœ´Êœ ¤¸ N = 1,000 Ž¹ÉŠ­Îµ®¦´š»„ Ç ‹ÎµœªœÁ˜È¤ª„ n š¸É n > N ¨oª n < 0.001
…o°­´ŠÁ„˜
Ĝ„µ¦®µ N Á¦µÅ—o‹µ„„µ¦ª·Á‡¦µ³®r‹µ„­·ÉŠš¸ÉÁ¦µ˜o°Š„µ¦ —´Šœ¸Ê
1
Á¦µ˜o°Š„µ¦ n < 0.001
1 1
œ´Éœ‡º° n < 1000
,
®¦º°
n > 1,000
—´Šœ´Êœ Á¦µ„ε®œ— N = 1,000
Á¦µ°µ‹„ε®œ— N Áž}œ‡nµ°ºÉœ„Èŗo ˜n˜o°Š¤µ„„ªnµ 1,000
165
˜´ª°¥nµŠš¸É 4.1.27 ‹Š¡·­¼‹œrªnµ “¤¸‹Îµœªœ‹¦·Š G > 0 Ž¹ÉŠ­Îµ®¦´š»„ Ç x ™oµ |x2| < G ¨oª
|3x6| < 0.0001”
¡·­¼‹œr
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
166
®¤µ¥Á®˜»
™oµ˜o°Š„µ¦¡·­¼‹œrªnµ ¤¸ x Á¡¸¥Š˜´ªÁ—¸¥ªÁšnµœ´Êœš¸ÉšÎµÄ®o P(x) Áž}œ‹¦·Š
Á¦µ‹³˜o°Š¡·­¼‹œr 2 ˜°œ ‡º°
˜°œš¸É 1 ¡·­¼‹œrªnµ x [ P(x) ] Áž}œ‹¦·Š
˜°œš¸É 2 ¡·­¼‹œrªnµ ­Îµ®¦´ x1 ¨³ x2 ™oµ Px1 Áž}œ‹¦·ŠÂ¨³ Px 2 Áž}œ‹¦·Š
¨oª x1 = x2
˜´ª°¥nµŠš¸É 4.1.28 ‹Š¡·­¼‹œrªnµ ¤¸‹Îµœªœ‹¦·Š x Á¡¸¥Š‹ÎµœªœÁ—¸¥ªÁšnµœ´Êœ š¸É 2 + x = 0
¡·­¼‹œr
˜°œš¸É 1 ¡·­¼‹œrªnµ x [x Áž}œ‹Îµœªœ‹¦·Š ¨³ 2 + x = 0] Áž}œ‹¦·Š
˜°œš¸É 2 ¡·­¼‹œrªnµ ™oµ 2 + x1 = 0 ¨³ 2 + x2 = 0 ¨oª x1 = x2
Áž}œ‹¦·Š
˜°œš¸É 1 Á¡¦µ³ªnµ 2 Áž}œ‹Îµœªœ‹¦·Š
¨³ 2 + (2) = 0
Áž}œ‹¦·Š
—´Šœ´Êœ 2 Áž}œ‹Îµœªœ‹¦·Š ¨³ 2 + (2) = 0
Áž}œ‹¦·Š
œ´Éœ‡º° x [x Áž}œ‹Îµœªœ‹¦·Š ¨³ 2 + x = 0] Áž}œ‹¦·Š
˜°œš¸É 2 Ä®o x1 ¨³ x2 Áž}œ‹Îµœªœ‹¦·ŠÄ— Ç
×¥š¸É 2 + x1 = 0 ¨³ 2 + x2 = 0
—´Šœ´Êœ 2 + x1 = 2 + x2 (˜nµŠ„ÈÁšnµ„´ «¼œ¥r)
x1 = x 2
(„µ¦˜´—°°„…°Š„µ¦ª„)
‹µ„˜°œš¸É 1 ¨³˜°œš¸É 2 ­—Šªnµ ¤¸‹Îµœªœ‹¦·Š x Á¡¸¥Š‹ÎµœªœÁ—¸¥ªÁšnµœ´Êœš¸É 2 + x = 0
1.
2.
3.
4.
5.
6.
 f„®´—š¸É 22
‹Š¡·­¼‹œrªnµ “¤¸‹ÎµœªœÁ˜È¤ x °¥nµŠœo°¥®œ¹ÉŠ‹Îµœªœš¸É x + 2 < 5”
‹Š¡·­¼‹œrªnµ x[5 + x 7 = x] Áž}œ‹¦·Š Á¤ºÉ°Á°„£¡­´¤¡´š›r ‡º° ÁŽ˜…°Š‹Îµœªœ‹¦·Š
‹Š¡·­¼‹œrªnµ “¤¸‹ÎµœªœÁ˜È¤š¸É¥„„ε¨´Š­°ŠÂ¨oªŸ¨¨´¡›r°¥¼n¦³®ªnµŠ 90 „´ 110”
‹Š¡·­¼‹œrªnµ ¤¸‹ÎµœªœÁ˜È¤ x Á¡¸¥Š‹ÎµœªœÁ—¸¥ªÁšnµœ´Êœš¸ÉšÎµÄ®o x + 3 = 10
„ε®œ—Ä®o A = {0, 2, 8, 9} ¨³ a † b ®¤µ¥™¹Š ®¨´„®œnª¥…°Š a + b
‹Š¡·­¼‹œrªnµ
5.1 ¤¸ x Ĝ A Ž¹ÉŠ x † y = y ­Îµ®¦´š»„ y Ĝ A
5.2 ¤¸ z Ĝ A Ž¹ÉŠ z † 8 = 0
‹Š¡·­¼‹œrªnµ ­Îµ®¦´‹Îµœªœ‹¦·Š x ė Ç ‹³¤¸‹Îµœªœ‹¦·Š y Ž¹ÉŠ x + y = 0
167
˜°œš¸É 4.2 °»žœ´¥Á·Š‡–·˜«µ­˜¦r( Mathematical Induction )
­´ŠÁ„˜Ÿ¨ª„…°Š‹Îµœªœ˜n°Åžœ¸Ê
1 + 2
= ……………….
1 + 2 + 4
= ……………….
1 + 2 + 4 + 8
= ……………….
……………………………….
.
.
.
= ……………….
šÎµœµ¥¦¼žÂš´ÉªÅž
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………
„µ¦šÎµœµ¥œ¸Ê°µ‹‹³Ÿ·—„ÈŗoÁ¡¦µ³„µ¦šÎµœµ¥œ¸ÊÁž}œ
„µ¦Ä®oÁ®˜»Ÿ¨Â………………………………………………..
168
Ĝª·µ‡–·˜«µ­˜¦r n°¥‡¦´ÊŠš¸ÉÁ¦µ¡…o°‡ªµ¤®¦º°­¼˜¦Ž¹ÉŠÁ„¸É¥ª…o°Š„´‹ÎµœªœÁ˜È¤ª„ Ánœ
n ( n 1)
1. 1 + 2 + 3 + … + n = 2
2. 1 + 2 + 4 + … + 2n–1 = 2n – 1
3. 2n > 1 + n
4. x2n – y2n ®µ¦—oª¥ x – y ¨Š˜´ª
„µ¦š¸É‹³Â­—Šªnµ …o°‡ªµ¤…oµŠ˜oœÁž}œ‹¦·Š °µ‹„¦³šÎµÅ—o×¥„µ¦Âšœ‡nµ Á¦·É¤˜´ÊŠÂ˜nšœ‡nµ n = 1
—¼ªnµ…o°‡ªµ¤—´Š„¨nµªÁž}œ‹¦·Š®¦º°Å¤n ™oµÁž}œ‹¦·Š„Èšœ‡nµ n = 2 —¼ªnµ…o°‡ªµ¤Áž}œ‹¦·Š°¸„®¦º°Å¤n—´Šœ¸Ê
Á¦ºÉ°¥Åž ™¹ŠÂ¤oªnµ­Îµ®¦´‹ÎµœªœÁ˜È¤ª„ n ˜n¨³‡nµ Á¦µ‹³ÄoÁª¨µÁ¡¸¥Š 1 ª·œµš¸ Ĝ„µ¦Â­—Šªnµ…o°‡ªµ¤
—´Š„¨nµªÁž}œ‹¦·Š„Șµ¤ Á¦µÅ¤n­µ¤µ¦™š¸É‹³Â­—ŠÅ—oªnµ…o°‡ªµ¤œ´ÊœÁž}œ‹¦·Š­Îµ®¦´š»„‡nµ…°Š n ×¥ª·›¸
—´Š„¨nµª Á¡¦µ³ªnµÁŽ˜…°Š‹ÎµœªœÁ˜È¤ª„Áž}œÁŽ˜°œ´œ˜r Á¡ºÉ°Â„ož{®µ—´Š„¨nµªœ´„‡–·˜«µ­˜¦r‹¹ŠÅ—o‡·—
ª·›¸¡·­¼‹œr…¹Êœ¤µ Á¡ºÉ°Â­—Šªnµ…o°‡ªµ¤…oµŠ˜oœÁž}œ‹¦·Š­Îµ®¦´š»„‡nµ…°Š n š¸ÉÁž}œ‹ÎµœªœÁ˜È¤ª„ ª·›¸š¸Éčoœ¸Ê
Á¦¸¥„ªnµ °»žœ´¥Á·Š‡–·˜«µ­˜¦r
®¨´„„µ¦š¸É®œ¹ÉŠ…°Š°»žœ´¥Á·Š‡–·˜«µ­˜¦r (First Principle of Mathematical Induction)
Ä®o P(n) šœ…o°‡ªµ¤š¸ÉÁ„¸É¥ª…o°Š„´‹ÎµœªœÁ˜È¤ª„ n
™oµ
1) P(1) Áž}œ‹¦·Š
¨³ 2) Ä®o k Áž}œ‹ÎµœªœÁ˜È¤ª„Ä— Ç
™oµ P(k) Áž}œ‹¦·Š ¨oª P(k+1) Áž}œ‹¦·Š—oª¥
¨oª‹³Å—oªnµ P(n) Áž}œ‹¦·Š ­Îµ®¦´‹ÎµœªœÁ˜È¤ª„ n š»„‹Îµœªœ
‹³Á®Èœªnµ „µ¦¡·­¼‹œrªnµ P(n) Áž}œ‹¦·Šœ´Êœ‹³˜o°Š„¦³šÎµ­°Š…´Êœ˜°œ ‡º°
…´Êœš¸É 1 ­—Šªnµ P(n) Áž}œ‹¦·Š Á¤ºÉ° n = 1
…´Êœš¸É 2 ¡·­¼‹œrªnµ ™oµ P(n) Áž}œ‹¦·Š Á¤ºÉ° n = k ¨oª‹³šÎµÄ®o P(n) Áž}œ‹¦·Š Á¤ºÉ° n = k + 1 —oª¥
„µ¦š¸ÉÁ¦µ¡·­¼‹œrÁ¡¸¥Š­°Š…´Êœ˜°œœ¸Ê„ÈÁž}œ„µ¦Á¡¸¥Š¡°Â¨oª ˜µ¤®¨´„„µ¦…°Š°»žœ´¥Á·Š
‡–·˜«µ­˜¦r Á¡ºÉ°Ä®oÁ…oµÄ‹ª·›¸„µ¦œ¸Ê—¸¥·ÉŠ…¹Êœ„È°µ‹‹³°›·µ¥Å—o ץčo®¨´„˜¦¦„«µ­˜¦r —´Šœ¸Ê
™oµ P(1)
Áž}œ‹¦·Š
169
P(1) o P(2) Áž}œ‹¦·Š
P(2)
Áž}œ‹¦·Š („‘‹ŠŸ¨˜µ¤Á®˜»)
P(2) o P(3) Áž}œ‹¦·Š
P(3)
Áž}œ‹¦·Š („‘‹ŠŸ¨˜µ¤Á®˜»)
‹³Å—o P(n)
Áž}œ‹¦·Š
Ž¹ÉŠ®¨´„„µ¦—´Š„¨nµª°µ‹°›·µ¥—oª¥£µ¡Áž¦¸¥Áš¸¥Å—o—´Šœ¸Ê
¨³
Á¡¦µ³Œ³œ´Êœ‹³Å—o
¨³
Á¡¦µ³Œ³œ´Êœ‹³Å—o
™oµÁ¦µ®µÃ—¤·Ãœ¤µ‹Îµœªœ®œ¹ÉŠ („¸É¦o°¥˜´ª„Èŗo) ¤µ˜´ÊŠÅªoÄ®o®nµŠ„´œ¦³¥³¡°Á®¤µ³ ×¥¤¸ÁŠºÉ°œÅ…
ªnµ ™oµ˜´ªÄ—˜´ª®œ¹ÉŠ¨o¤‹³Åž„¦³š˜´ª™´—ŞĮo¨o¤¨Š—oª¥ ‹³Á®Èœªnµ‹µ„ÁŠºÉ°œÅ… —´Š„¨nµª ™oµ˜´ªš¸É®œ¹ÉŠ
¨o¤Á¡¸¥Š˜´ªÁ—¸¥ª „È‹³Åž„¦³š˜´ªš¸É­°Š¨o¤ ¨³Áž}œŸ¨˜n°ÁœºÉ°ŠÄ®o˜´ªš¸É­µ¤ ˜´ªš¸É­¸É ¨³˜´ª™´—Åž‹œ™¹Š
˜´ªš¸É n ¨o¤¨Š—oª¥
˜´ª°¥nµŠš¸É 4.2.1 ‹Š¡·­¼‹œrªnµ 1 + 3 + 5 + 7 + ... + (2n–1) = n2 ­Îµ®¦´š»„ Ç ‹ÎµœªœÁ˜È¤ª„ n
¡·­¼‹œr Ä®o P(n) šœ…o°‡ªµ¤ 1 + 3 + 5 + 7 + ... + (2n–1) = n2
…´Êœš¸É 1 ™oµ n = 1
P(1) šœ…o°‡ªµ¤ 2˜1 – 1 = 12 Ž¹ÉŠÁž}œ‹¦·Š Á¡¦µ³˜nµŠ„ÈÁšnµ„´ 1
…´Êœš¸É 2 Ä®o k Áž}œ‹ÎµœªœÁ˜È¤ª„Ä— Ç
­¤¤˜·Ä®o P(k) Áž}œ‹¦·Š ¨oª¡·­¼‹œrªnµ P(k+1) Áž}œ‹¦·Š
(1.1)
„¨nµª‡º° ­¤¤˜·Ä®o 1 + 3 + 5 + 7 + ... + (2k–1) = k2
¨oª˜o°Š¡·­¼‹œrªnµ
1 + 3 + 5 + 7 + ... + {2(k+1) – 1} = (k+1)2
ÁœºÉ°Š‹µ„
170
1 + 3 + 5 + 7 + ... + {2(k+1) – 1} = 1 + 3 + 5 + 7 + ... + (2k–1) + {2(k+1) – 1}
= k2 + {2(k+1) – 1} ×¥ (1.1)
= k2 + 2k + 1
= (k+1)2
Á¡¦µ³Œ³œ´Êœ P(k+1) Áž}œ‹¦·Š
œ´Éœ‡º° ™oµ P(k) Áž}œ‹¦·Š ¨oª P(k+1) Áž}œ‹¦·Š—oª¥
—´Šœ´Êœ ×¥®¨´„„µ¦…°Š°»žœ´¥Á·Š‡–·˜«µ­˜¦r ­¦»žÅ—oªnµ P(n) Áž}œ‹¦·Š ­Îµ®¦´
š»„ Ç ‹ÎµœªœÁ˜È¤ª„ n
˜´ª°¥nµŠš¸É 4.2.2 ‹Š¡·­¼‹œrªnµ n(n2+2) ®µ¦—oª¥ 3 ¨Š˜´ª ­Îµ®¦´š»„ Ç ‹ÎµœªœÁ˜È¤ª„ n
¡·­¼‹œr Ä®o P(n) šœ…o°‡ªµ¤ n(n2+2) ®µ¦—oª¥ 3 ¨Š˜´ª
…´Êœš¸É 1 ™oµ n = 1
P(1) šœ…o°‡ªµ¤ 1(12+2) ®µ¦—oª¥ 3 ¨Š˜´ª Ž¹ÉŠÁž}œ‹¦·Š Á¡¦µ³ªnµ 1(12+2) = 3
…´Êœš¸É 2 Ä®o k Áž}œ‹ÎµœªœÁ˜È¤ª„Ä— Ç
­¤¤˜·Ä®o P(k) Áž}œ‹¦·Š œ´Éœ‡º° k(k2+2) ®µ¦—oª¥ 3 ¨Š˜´ª
‹³¡·­¼‹œrªnµ P(k+1) Áž}œ‹¦·Š œ´Éœ‡º° ‹³¡·­¼‹œrªnµ (k+1){(k+1)2 + 2} ®µ¦—oª¥ 3 ¨Š˜´ª
ÁœºÉ°Š‹µ„ (k+1){(k+1)2 + 2} = k(k+1)2 + (k+1)2 + 2(k+1)
= k(k2+2k+1) + (k2+2k+1) + 2k + 2
= (k.k2+2k) + (3k2+3k+3)
= k(k2+2) + 3(k2+k+1)
‹µ„š¸É­¤¤˜·Ä®o P(k) Áž}œ‹¦·Š Á¦µÅ—oªnµ 3 ®µ¦ k(k2+2) ¨Š˜´ª
Á¡¦µ³ªnµ 3(k2+k+1) ¤¸ 3 Áž}œ˜´ªž¦³„° ‹¹ŠÅ—oªnµ 3 ®µ¦ 3(k2+k+1) ¨Š˜´ª
—´Šœ´Êœ 3 ®µ¦ (k+1) {(k+1)2 + 2} ¨Š˜´ª
Á¡¦µ³Œ³œ´Êœ P(k+1) Áž}œ‹¦·Š
œ´Éœ‡º° ™oµ P(k) Áž}œ‹¦·Š ¨oª P(k+1) Áž}œ‹¦·Š—oª¥
—´Šœ´Êœ ×¥®¨´„„µ¦…°Š°»žœ´¥Á·Š‡–·˜«µ­˜¦r ­¦»žÅ—oªnµ P(n) Áž}œ‹¦·Š ­Îµ®¦´
š»„ Ç ‹ÎµœªœÁ˜È¤ª„ n
…o°­´ŠÁ„˜ 1. Ĝ„µ¦¡·­¼‹œr P(k+1) Áž}œ‹¦·Š˜o°ŠÁ°µ P(k) ¤µÄoĜ„µ¦¡·­¼‹œr—oª¥
171
2. ‹µ„˜´ª°¥nµŠ…oµŠ˜oœ‹³Á®Èœªnµ „µ¦¡·­¼‹œr¤¸ 2 …´Êœ˜°œ …´ÊœÂ¦„¡·­¼‹œr…o°‡ªµ¤Áž}œ‹¦·Š
Á¤ºÉ° n = 1 …´Êœš¸É­°Š¡·­¼‹œrªnµ ™oµ…o°‡ªµ¤Áž}œ‹¦·Š ­Îµ®¦´‹ÎµœªœÁ˜È¤ª„ k ¨oª …o°‡ªµ¤œ´Êœ‹³˜o°Š
Áž}œ‹¦·Š­Îµ®¦´‹ÎµœªœÁ˜È¤ª„ k + 1 —oª¥ š´ÊŠ­°Š…´Êœ˜°œÁž}œ­·ÉŠ‹ÎµÁž}œ ‹³…µ—…´ÊœÄ—…´Êœ®œ¹ÉŠÅ¤nŗo ™oµ
…µ—…´Êœ®œ¹ÉŠ…´ÊœÄ—¨oª…o°‡ªµ¤‹³Å¤nÁž}œ‹¦·Š­Îµ®¦´š»„ Ç ‹ÎµœªœÁ˜È¤ª„ n Ánœ ¡·‹µ¦–µ…o°‡ªµ¤
˜n°Åžœ¸ÊªnµÁž}œ‹¦·Š­Îµ®¦´š»„ Ç ‹ÎµœªœÁ˜È¤ª„ n ®¦º°Å¤n
n 1
„. 1 + 2 + 3 + ... + n = 2
…. 2 + 4 + 6 + ... + 2n = n(n+1) + 2
n 1
„. Ä®o P(n) šœ…o°‡ªµ¤ 1 + 2 + 3 + ... + n = 2
11
…o°‡ªµ¤œ¸ÊÁž}œ‹¦·Š Á¤ºÉ° n = 1 Á¡¦µ³ 1 = 2 Áž}œ‹¦·Š
˜n™oµÁ¦µ­¤¤˜·Ä®o P(k) Áž}œ‹¦·Š Á¤ºÉ° k Áž}œ‹ÎµœªœÁ˜È¤ª„Ä— Ç
k 1
œ´Éœ‡º° 1 + 2 + 3 + ... + k = 2
( k 1) 1
ŤnÁž}œ‹¦·Š
Á¦µ‹³¡ªnµ 1 + 2 + 3 + ... + (k+1) =
2
Á¡¦µ³ªnµ 1 + 2 + 3 + ... + (k+1) = 1 + 2 + 3 + ... + k + (k+1)
k 1
= 2 + (k+1)
3
= 2 (k+1)
( k 1) 1
z
2
œ´Éœ‡º° P(k+1) Ťn‹¦·Š
—´Šœ´Êœ „¦–¸Ánœœ¸ÊÁ¦µ‹¹ŠÅ¤n­µ¤µ¦™Äo®¨´„„µ¦…°Š°»žœ´¥Á·Š‡–·˜«µ­˜¦rŗo
…. Ä®o P(n) šœ…o°‡ªµ¤ 2 + 4 + 6 + ... + 2n = n(n+1) + 2
™oµ­¤¤˜·Ä®o P(k) Áž}œ‹¦·Š Á¤ºÉ° k Áž}œ‹ÎµœªœÁ˜È¤ª„Ä— Ç
œ´Éœ‡º° 2 + 4 + 6 + ... + 2k = k(k+1) + 2
¡·‹µ¦–µ 2 + 4 + 6 + ... + 2(k+1) = 2 + 4 + 6 + ... + 2k + 2(k+1)
= k(k+1) + 2 + 2(k+1)
= (k+1){k+2} + 2
= (k+1){(k+1) + 1} + 2
—´Šœ´Êœ‹³Á®ÈœÅ—oªnµ P(k+1) Áž}œ‹¦·Š
172
˜nÁ¤ºÉ° n = 1 ‹³Å—oªnµ P(1) šœ…o°‡ªµ¤ 2˜1 = 1(1+1) + 2 Ž¹ÉŠÅ¤nÁž}œ‹¦·Š Á¡¦µ³ªnµ
1(1+1) + 2 = 4
—´Šœ´Êœ „¦–¸Ánœœ¸ÊÁ¦µ‹¹ŠÅ¤n­µ¤µ¦™Äo®¨´„„µ¦…°Š°»žœ´¥Á·Š‡–·˜«µ­˜¦rŗoÁnœ„´œ
3. ™oµÁ¦µ˜o°Š„µ¦¡·­¼‹œrªnµ P(n) Áž}œ‹¦·Š­Îµ®¦´š»„‹ÎµœªœÁ˜È¤ª„ n ×¥š¸É
n > m Á¦µ˜o°Š„¦³šÎµ—´Šœ¸Ê
3.1 ­—Šªnµ P(m) Áž}œ‹¦·Š
3.2 Ä®o k Áž}œ‹ÎµœªœÁ˜È¤ª„Ä— Ç š¸É¤µ„„ªnµ®¦º°Ášnµ„´ m
¡·­¼‹œrªnµ™oµ P(k) Áž}œ‹¦·Š ¨oª P(k+1) Áž}œ‹¦·Š
˜´ª°¥nµŠš¸É 4.2.3 ‹Š¡·­¼‹œrªnµ 2n < 2n ­Îµ®¦´š»„ Ç ‹ÎµœªœÁ˜È¤ª„ n š¸É¤¸‡nµ¤µ„„ªnµ®¦º°
Ášnµ„´ 3
¡·­¼‹œr Ä®o P(n) šœ…o°‡ªµ¤ 2n < 2n
…´Êœš¸É 1 ™oµ n = 3
P(3) šœ…o°‡ªµ¤ 2˜3 < 23 Ž¹ÉŠÁž}œ‹¦·Š Á¡¦µ³ªnµ 2˜3 = 6 ¨³ 23 = 8
…´Êœš¸É 2 Ä®o k Áž}œ‹ÎµœªœÁ˜È¤ª„Ä— Ç Â¨³ k > 3
­¤¤˜·Ä®o P(k) Áž}œ‹¦·Š œ´Éœ‡º° 2k < 2k
‹³¡·­¼‹œrªnµ P(k+1) Áž}œ‹¦·Š œ´Éœ‡º° ‹³¡·­¼‹œrªnµ 2(k+1) < 2(k+1)
ÁœºÉ°Š‹µ„ 2(k+1) = 2k + 2
< 2k + 2 (‹µ„š¸É­¤¤˜·Ä®o P(k) Áž}œ‹¦·Š)
< 2k + 2k (Á¡¦µ³ 2 < 2k ­Îµ®¦´š»„‹ÎµœªœÁ˜È¤ª„ k > 3)
= 2˜2k
= 2(k+1)
Á¡¦µ³Œ³œ´Êœ 2(k+1) < 2(k+1)
—´Šœ´Êœ‹³Å—oªnµ P(k+1) Áž}œ‹¦·Š
œ´Éœ‡º° ™oµ P(k) Áž}œ‹¦·Š ¨oª P(k+1) Áž}œ‹¦·Š
—´Šœ´Êœ ×¥®¨´„„µ¦…°Š°»žœ´¥Á·Š‡–·˜«µ­˜¦r ­¦»žÅ—oªnµ P(n) Áž}œ‹¦·Š
­Îµ®¦´š»„ Ç ‹ÎµœªœÁ˜È¤ª„ n š¸É¤¸‡nµ¤µ„„ªnµ®¦º°Ášnµ„´ 3
­¦»ž
Á¦µÄo®¨´„„µ¦…°Š°»žœ´¥Á·Š‡–·˜«µ­˜¦rÁ¤ºÉ°˜o°Š„µ¦Â­—Šªnµ…o°‡ªµ¤š¸ÉÁ„¸É¥ª…o°Š„´
173
‹ÎµœªœÁ˜È¤ª„ n Áž}œ‹¦·Š ­Îµ®¦´š»„ Ç ‹ÎµœªœÁ˜È¤ª„ n ž¦³Á—Èœ­Îµ‡´…°Š„µ¦
¡·­¼‹œrœ´Êœ˜o°Š˜¦ª‹­°š¸É‹»—Á¦·É¤˜oœ (‡nµÁ¦·É¤˜oœ…°Š n) Ž¹ÉŠ…o°‡ªµ¤Áž}œ‹¦·Š ¨³Äœ
„µ¦¡·­¼‹œrªnµ…o°‡ªµ¤Áž}œ‹¦·Š Á¤ºÉ° n = k + 1 œ´Êœ ˜o°ŠœÎµ…o°‡ªµ¤Á¤ºÉ° n = k Áž}œ‹¦·Š¤µ
čoÁ­¤°
®¨´„„µ¦š¸É­°Š…°Š°»žœ´¥Á·Š‡–·˜«µ­˜¦r (Second Principle of Mathematical Induction
®¦º° Strong Induction)
„µ¦¡·­¼‹œrªnµ P(n) Áž}œ‹¦·Šš»„‡nµ…°Š n š¸ÉÁž}œ‹ÎµœªœÁ˜È¤ª„ ×¥®¨´„„µ¦š¸É­°Š…°Š°»žœ´¥Á·Š
‡–·˜«µ­˜¦r ‹³„¦³šÎµ—´Šœ¸Ê
­Îµ®¦´ n š¸ÉÁž}œ‹ÎµœªœÁ˜È¤ª„
Ä®o P(n) šœ…o°‡ªµ¤š¸É°µ‹Áž}œ‹¦·Š®¦º°Å¤n‹¦·ŠÁ„¸É¥ª„´ n
™oµ
1) P(1) Áž}œ‹¦·Š
2) Ä®o k Áž}œ‹ÎµœªœÁ˜È¤ª„Ä— Ç
™oµ P(1), P(2), ..., P(k) Áž}œ‹¦·Š ¨oª P(k+1) Áž}œ‹¦·Š—oª¥
¨oª ‹³­¦»žÅ—oªnµ P(n) Áž}œ‹¦·Š­Îµ®¦´š»„ Ç ‹ÎµœªœÁ˜È¤ª„ n
‹³Á®Èœªnµ„µ¦š¸É‹³¡·­¼‹œrªnµ P(n) Áž}œ‹¦·Šœ´Êœ‹³˜o°ŠšÎµ­°Š…´Êœ˜°œ ‡º°
…´Êœš¸É 1 ­—Šªnµ P(n) Áž}œ‹¦·Š Á¤ºÉ° n = 1
…´Êœš¸É 2 ¡·­¼‹œrªnµ ™oµ P(n) Áž}œ‹¦·Š­Îµ®¦´š»„ Ç n < k ¨oªšÎµÄ®o P(n) Áž}œ‹¦·Š Á¤ºÉ° n = k + 1 —oª¥
…o°­´ŠÁ„˜ ™oµÁ¦µ˜o°Š„µ¦¡·­¼‹œrªnµ P(n) Áž}œ‹¦·Š­Îµ®¦´‹ÎµœªœÁ˜È¤ª„ n ×¥š¸É n > m
Á¦µ˜o°Š„¦³šÎµ—´Šœ¸Ê
(1) ­—Šªnµ P(m) Áž}œ‹¦·Š
(2) Ä®o k Áž}œ‹Îµœªœœ´Ä— Ç š¸É¤µ„„ªnµ®¦º°Ášnµ„´ m
™oµ P(n) Áž}œ‹¦·Š­Îµ®¦´š»„ n < k ¨oª P(n) Áž}œ‹¦·Š Á¤ºÉ° n = k + 1 —oª¥
174
˜´ª°¥nµŠš¸É 4.2.4 „ε®œ—‡ªµ¤­´¤¡´œ›r
an = 6an-1 9an-2
×¥š¸É a0 = 1 ¨³ a1 = 6
‹Š¡·­¼‹œrªnµ an = 3n + n ˜ 3n
¡·­¼‹œr Ä®o P(n) šœ…o°‡ªµ¤ an = 3n + n ˜ 3n
…´Êœš¸É 1 ™oµ n = 2
P(2) šœ…o°‡ªµ¤ a2
=
=
=
=
6a1 9a0
36 9
27
32 + 2 ˜ 32
Ž¹ÉŠÁž}œ‹¦·Š
—´Šœ´Êœ P(2) Áž}œ‹¦·Š
…´Êœš¸É 2 Ä®o k Áž}œ‹ÎµœªœÁ˜È¤ª„Ä— Ç š¸É¤µ„„ªnµ®¦º°Ášnµ„´ 2
P(1) š P(2) š ... š P(k) Áž}œ‹¦·Š ‹³¡·­¼‹œrªnµ P(k+1) Áž}œ‹¦·Š œ´Éœ‡º°‹³˜o°Š¡·­¼‹œrªnµ
ak+1 = 3k+1 + (k+1) . 3k+1
‹µ„
an = 6an-1 9an-2
—´Šœ´Êœ ak+1 = 6ak 9ak-1
= 6(3k+k.3k) 9(3k–1+(k1).3k–1)
(‹µ„š¸É­¤¤»˜·Ä®o P(k) ¨³ P(k1) Áž}œ‹¦·Š)
= 2(3k+1+k.3k+1) 3k+1 (k1).3k+1
ak+1 = 3k+1 + (k+1) . 3k+1
Á¡¦µ³Œ³œ´Êœ P(k+1) Áž}œ‹¦·Š
œ´Éœ‡º° ™oµ P(n) Áž}œ‹¦·Š­Îµ®¦´š»„ n < k ¨oª P(n) Áž}œ‹¦·Š Á¤ºÉ° n = k + 1 —oª¥
—´Šœ´ÊœÃ—¥®¨´„„µ¦š¸É­°Š…°Š°»žœ´¥Á·Š‡–·˜«µ­˜¦r ­¦»žÅ—oªnµ P(n) Áž}œ‹¦·Š­Îµ®¦´š»„ Ç ‹Îµœªœ
Á˜È¤ª„ n
175
 f„®´—š¸É 23
„ε®œ—Ä®o n Áž}œ‹ÎµœªœÁ˜È¤ª„Ä— Ç ‹Š¡·­¼‹œrªnµ
1. 1 + 2 + 4 + 8 + ... + 2n–1 = 2n – 1
1
1
1
1
n
2. 1.2 + 2.3 + 3.4 + ... + n ( n 1) = n 1
a (1 r n )
3. a + ar + ar2 + ... + arn–1 = 1 r , r z 1
n 2 ( n 1) 2
3
3
3
3
4. 1 + 2 + 3 + ... + n =
4
(4 n3 n)
5. 12 + 32 + 52 + ... + (2n–1)2 =
3
n ( n 1)( 2 n 7)
6. 1.3 + 2.4 + 3.5 + ... + n(n+2) =
6
2 2
2
1
7. 1 + 2 + … + n = 1 – n
3 3
3
3
3n
8. (23) – 1 ®µ¦—oª¥ 7 ¨Š˜´ª
9. 32n+2 – 8n – 9 ®µ¦—oª¥ 64 ¨Š˜´ª
10. 2n + (–1)n+1 ®µ¦—oª¥ 3 ¨Š˜´ª
11. x2n – y2n ®µ¦—oª¥ x – y ¨Š˜´ª
12. 2n > n n > 0
13. n! > 2n n > 4
(Á¤ºÉ° n! = n(n–1)(n–2) ...3.2.1)
14. 2n–1 < n!
15. 2n > n3 n > 10
16. (1+x)n > 1 + nx Á¤ºÉ° n > 2 , x > –1 ¨³ x z 0
17. 2n > 1 + n
18. 2n + 1 < 3n
19. 2n > 2+n š»„ n > 3
1
1
1 1
20. 1 + 4 + 9 + … + 2 < 2 – n
n
n
21. (1+h) > 1+nh š»„ n ×¥š¸É h > –1
176
22. n(n+1) Áž}œ‹Îµœªœ‡¼n
23. 2n2 + 4n + 1 Áž}œ‹Îµœªœ‡¸É
24. „ε®œ—Ä®o a1 = 3 ¨³ an = 3an–1 ‹Š¡·­¼‹œrªnµ an = 3n
1 2
n2
Áž}œ‹¦·Š
25. ‹ŠÂ­—Šªnµ™oµ 1 + 2 + 3 + ... + n =
2
­Îµ®¦´ n = k > 1 ¨oª…o°‡ªµ¤œ¸Ê‹³Áž}œ‹¦·Š ­Îµ®¦´ n = k + 1 —oª¥
™µ¤ªnµ …o°‡ªµ¤œ¸Ê‹³Áž}œ‹¦·Š­Îµ®¦´š»„‡nµ n ®¦º°Å¤n
26. „ε®œ—Ä®o 1. a1 = a
2. an+1 = (an)a
‹Š¡·­¼‹œrªnµ anbn = (ab)n Á¤ºÉ° a ¨³ b Áž}œ‹Îµœªœ‹¦·Š
27. |xn| = |x|n Á¤ºÉ° x Áž}œ‹Îµœªœ‹¦·Š
n
n
n
i 1
i 1
i 1
28. ¦ ( a i b i ) = ¦ a i + ¦ b i
29. ™oµÄ®o |a+b| < |a| + |b| Áž}œ‹¦·Šš»„‹Îµœªœ‹¦·ŠÄ— Ç
‹Š¡·­¼‹œrªnµ |x1+x2+…+xn| < |x1| + |x2| + … + |xn|
30. ‹Š®µ‹ÎµœªœÁ˜È¤ª„ n š¸ÉšÎµÄ®o 8n – 3n ®µ¦—oª¥ 5 ¨Š˜´ª ‹µ„œ´Êœ¡·­¼‹œrªnµ‡Îµ˜°š¸É®µ¤µÅ—oÁž}œ
‹¦·Š (™¼„˜o°Š)
31. „ε®œ—Ä®o A , B ¨³ Ai ( i = 1 , 2 , 3 , … , n) Áž}œÁŽ˜Ä—Çš¸ÉŤnčnÁŽ˜ªnµŠ
¨³ A ˆ B c = Ac ‰ B c ‹Š¡·­¼‹œrªnµ ( A1 ˆ A2 ˆ ... ˆ An )c = A1c ‰ A2c ‰ ... ‰ An c
32. „ε®œ—‡ªµ¤­´¤¡´œ›r
an = 2an-1 an-2
×¥š¸É a1 = 2 ¨³ a2 = 3
‹Š¡·­¼‹œrªnµ an = n + 1
33. „ε®œ—‡ªµ¤­´¤¡´œ›r
an = 6an-1 11an-2 + 6an-3
×¥š¸É ao = 2 , a1 = 5 ¨³ a2 = 15
‹Š¡·­¼‹œrªnµ an = 1 2n + 2.3n
34. „ε®œ—‡ªµ¤­´¤¡´œ›r
u(x+y) + u(xy) = 2u(x) + 2u(y)
×¥š¸É u(0) = 0 ¨³ u(1) = ao
‹Š¡·­¼‹œrªnµ u(n) = n2ao
177
®œnª¥š¸É 5
„µ¦Â„ož{®µšµŠ‡–·˜«µ­˜¦r
˜°œš¸É 5.1 „¦³ªœ„µ¦Â„ož{®µ
˜°œš¸É 5.2 „µ¦­¦oµŠ­¦¦‡rž
{ ®µ
œª‡·— 1. „µ¦Â„ož{®µšµŠ‡–·˜«µ­˜¦r¤¸„¦³ªœ„µ¦Â„ož{®µ˜µ¤…´Êœ˜°œ—´Šœ¸Ê
(1) šÎµ‡ªµ¤Á…oµÄ‹ž{®µ
(2) ªµŠÂŸœÂ„ož{®µ
(3) —εÁœ·œ„µ¦˜µ¤ÂŸœš¸ÉªµŠÅªo
(4) ˜¦ª‹­°Ÿ¨ÁŒ¨¥®¦º°‡Îµ˜°
(5) ­¦oµŠ­¦¦‡rž{®µ…¹ÊœÄ®¤n‹µ„ž{®µš¸É¤¸°¥¼nÁ—·¤
2. „µ¦°°„Â„ož{®µ‹³šÎµÄ®oŸ¼o°°„š¦µªnµ‹³˜o°ŠÄo‡ªµ¤¦¼oÁ¦ºÉ°ŠÄ—oµŠ ¨³
­µ¤µ¦™­¦oµŠ­¦¦‡r„µ¦Â„ož{®µÅ—o®¨µ„®¨µ¥ª·›¸
ª´˜™»ž¦³­Š‡r
Á¤ºÉ°«¹„¬µ®œnª¥š¸É 4 ‹Â¨oªœ´„Á¦¸¥œ­µ¤µ¦™
1. šÎµ‡ªµ¤Á…oµÄ‹Ã‹š¥rž{®µÅ—o
2. ªµŠÂŸœÂ¨³°°„„µ¦Â„ož{®µÅ—o
3. „ož{®µÅ—o°¥nµŠ´—Á‹œ ¦´—„»¤ ™¼„˜o°Š
4. ­¦oµŠ­¦¦‡r„µ¦Â„o˚¥rž{®µ ¨³—´—ž¨ŠÃ‹š¥rž{®µÁ—·¤Áž}œÃ‹š¥rž{®µÄ®¤nŗo
178
„·‹„¦¦¤¦³®ªnµŠÁ¦¸¥œ
1. °µ‹µ¦¥r¥„˜´ª°¥nµŠÃ‹š¥rÄ®oœ´„Á¦¸¥œ°£·ž¦µ¥Âœª‡·—…°Šª·›¸„µ¦Â„ož{®µšµŠ‡–·˜«µ­˜¦rªnµ
¤¸„¦³ªœ„µ¦Â„ož{®µ°¥nµŠÅ¦
2. °µ‹µ¦¥r­¦»ž„¦³ªœ„µ¦Â„nž{®µšµŠ‡–·˜«µ­˜¦r ¨oªÂ­—Š˜´ª°¥nµŠ„µ¦Â„ož{®µÁž}œ
…´Êœ˜°œ˜µ¤„¦³ªœ„µ¦Â„ož{®µš¸É­¦»žÅªo
3. nŠœ´„Á¦¸¥œÁž}œ„¨»n¤Ä®o°°„„µ¦Â„ož{®µ ×¥¤¸„µ¦œÎµÁ­œ°ª·›¸„µ¦Â„ož{®µ…°Š
˜n¨³„¨»n¤®œoµ´ÊœÁ¦¸¥œ
4. œ´„Á¦¸¥œœÎµ„·‹„¦¦¤˜µ¤˜´ª°¥nµŠÂ¨³Â f„®´—
5. œ´„Á¦¸¥œž¦³Á¤·œŸ¨¡´•œµ„µ¦…°Š˜œÁ°Š
­ºÉ°„µ¦­°œ
1. Á°„­µ¦„µ¦­°œ
2.  f„ž’·´˜·
3. Á‡¦ºÉ°ŠŒµ¥…oµ¤«¸¦¬³
ž¦³Á¤·œŸ¨
ž¦³Á¤·œŸ¨‹µ„ f„®´—¨³„µ¦š—­°
179
˜°œš¸É 5.1 „¦³ªœ„µ¦Â„ož{®µ
„µ¦Â„ož{®µšµŠ‡–·˜«µ­˜¦r
Ĝ„µ¦Â„ož{®µšµŠ‡–·˜«µ­˜¦rœ´Êœ
…´ÊœÂ¦„‹³˜o°Š°nµœÃ‹š¥rž{®µÄ®oÁ…oµÄ‹„n°œ
Ž¹ÉŠÄœ
ž{®µ°µ‹‹³¤¸‡Îµ«´¡šr ®¦º°šœ·¥µ¤š¸É„ε®œ—Ä®o ‹ÎµÁž}œš¸É‹³˜o°ŠšÎµ‡ªµ¤Á…oµÄ‹„n°œ ¨oª—¼…o°„ε®œ—
š¸ÉÄ®o¤µÂ¨³Ã‹š¥r™µ¤°³Å¦ Á¤ºÉ°Á…oµÄ‹‡¦Â¨oª°¥nµÁ¡·ÉŠÂ„ož{®µš´œš¸ ‡ª¦ªµŠÂŸœ„n°œªnµ˜o°ŠÄo‡ªµ¤¦¼o
Á¦ºÉ°ŠÄ—oµŠ ¨oª—εÁœ·œ„µ¦˜µ¤ÂŸœ­»—šoµ¥˜¦ª‹­°Ÿ¨ÁŒ¨¥ ®¦º°‡Îµ˜°š¸ÉŗoªnµÅ¤n¤¸…o°…´—Â¥oŠ„´š¸É
„ε®œ—Ä®o „Èœnµ‹³Áž}œŸ¨ÁŒ¨¥®¦º°‡Îµ˜°š¸É™¼„˜o°Š ­nªœÄ®nÁ¦µ¤´„‹³®¥»—„´œÂ‡nœ¸Ê ˜n™oµ¦¼o‹´„—´—ž¨Š
ž{®µš¸É¤¸°¥¼nÁ—·¤Â¨oª˜´ÊŠ‡Îµ™µ¤Ä®¤n Á¦µ‹³Å—ož{®µš¸Éªœ‡·—¤µ„…¹ÊœÂ¨³Å—o‡ªµ¤¦¼o¤µ„¤µ¥ —´Š˜´ª°¥nµŠ
˜n°Åžœ¸Ê
ž{®µš¸É 1
‹ŠÁ¦¸¥Š°´œ—´…°Š‹Îµœªœ˜n°Åžœ¸ÊÄ®o™¼„˜o°Š ‹µ„œo°¥Åž¤µ„ 2514, 4258, 8171, 16128 ¨³ 32103
ª·›¸Â„ož{®µ
…´ÊœšÎµ‡ªµ¤Á…oµÄ‹
Á¤ºÉ°«¹„¬µÃ‹š¥r¨oª¡ªnµ ¤¸‹Îµœªœ®oµ‹Îµœªœ Á…¸¥œ°¥¼nĜ¦¼žÁ¨…¥„„ε¨´Š ¨³‹³˜o°ŠÁ¦¸¥Š
°´œ—´‹Îµœªœš´ÊŠ®oµœ¸Ê‹µ„Žoµ¥Åž…ªµ ¨³Á¦¸¥Š‹µ„ œo°¥Åž¤µ„
…´ÊœªµŠÂŸœ
‹µ„‹Îµœªœš´ÊŠ®oµ¡ªnµÁž¨¸É¥œÂž¨Š Áž}œÁ¨…¥„„ε¨´Šš¸É¤¸“µœ Áž}œ 2 ŗoš´ÊŠ®¤— ‹¹Š‹³Äo‡ªµ¤¦¼o
Á¦ºÉ°Š°­¤„µ¦ÄœÁ¨…¥„„ε¨´Š —´Šœ¸Ê
“™oµ a Áž}œ‹Îµœªœ‹¦·Šš¸É¤µ„„ªnµ 1 b ¨³ c Áž}œ‹Îµœªœ‹¦·Šª„ ×¥š¸É b < c ¨oª
‹³Å—o ab < ac ¨³ c Áž}œ‹Îµœªœ‹¦·Šª„¨oª (ab)c = abc ”
…´Êœ—εÁœ·œ„µ¦Â„ož{®µ
¡·‹µ¦–µÁ¨…¥„„ε¨´Š ¨oªÁž¨¸É¥œÂž¨ŠÄ®o¤¸“µœÁž}œ 2 ‹³Å—o
2514
4258
= 22(258) = 2516
8171
= 23(171) = 2513
16128
= 24(128) = 2512
180
32103
= 25(103) = 2515
ÁœºÉ°Š‹µ„ 512 < 513 < 514 < 515 < 516 ‹¹ŠÅ—o
2512 < 2513< 2514< 2515 < 2516
œ´Éœ‡º° 16128 < 8171< 2514< 32103< 4528 ‹¹ŠÁ¦¸¥Š°´œ—´‹Îµœªœš´ÊŠ®oµ‹µ„œo°¥Åž¤µ„Å—o—´Šœ¸Ê
16128 , 8171, 2514, 32103, 4528
ž{®µš¸É 2
ÁŽ˜ A = qp p , q  N ,1 d p d 10 š 1 d q d 10
ª·›¸Â„ož{®µ
^
` ¤¸­¤µ·„š´ÊŠ®¤—„¸É˜´ª
…´ÊœšÎµ‡ªµ¤Á…oµÄ‹
ÁŽ˜…°Šž{®µ‡º° A ­·ÉŠš¸É˚¥r˜o°Š„µ¦ ‡º° ‹Îµœªœ­¤µ·„…°ŠÁŽ˜ A œ°„‹µ„œ¸Ê¡ªnµ­¤µ·„
…°ŠÁŽ˜ A Áž}œ‹Îµœªœš¸ÉÁ…¸¥œÂšœÅ—o—oª¥Á«¬­nªœ ×¥š¸É˜´ªÁ«¬Â¨³˜´ª­nªœ˜nµŠ„ÈÁž}œ‹ÎµœªœÁ˜È¤š¸É¤¸‡nµ
ŗo ˜´ÊŠÂ˜n 1 ™¹Š 10
…´ÊœªµŠÂŸœ
1. ‹„‹ŠÁ«¬­nªœš´ÊŠ®¤— ­¤¤˜·¤¸ a Á«¬­nªœ
2. ®µÁ«¬­nªœš¸É¤¸‡nµÁšnµ„´œ Ĝ˜n¨³‡nµÅªoÁž}œ„¨»n¤Á—¸¥ª„´œ
3. œ´‹Îµœªœ„¨»n¤š¸Éŗoš´ÊŠ®¤—­¤¤˜·Áž}œ b
4. œ´Á«¬­nªœš´ÊŠ®¤—š¸É¤¸‡nµŽÊε„´œ ­¤¤˜·Áž}œ c
5. ®µ‡nµ…°Š (a+b-c) ‹³Å—o‡Îµ˜°š¸É˜o°Š„µ¦
…´Êœ—εÁœ·œ„µ¦
1. Á«¬­nªœš´ÊŠ®¤—¤¸ 10 u10 = 100
2. ‹´—„¨»n¤Á«¬­nªœš¸É¤¸‡nµÁšnµ„´œÅªo—oª¥„´œ
„¨»n¤š¸É 1 11 = 22 = 33 = 44 = 55 = 66 = 77 = 88 = 99 = 10
10
„¨»n¤š¸É 2 12 = 42 = 63 = 48 = 105
„¨»n¤š¸É 3 12 = 42 = 63 = 84 = 105
„¨»n¤š¸É 4 13 = 26 = 93
„¨»n¤š¸É 5 13 = 62 = 93
181
„¨»n¤š¸É 6 14 = 82
„¨»n¤š¸É 7 41 = 82
„¨»n¤š¸É 8 15 = 102
„¨»n¤š¸É 9 15 = 102
„¨»n¤š¸É 10 23 = 46 =
„¨»n¤š¸É 11 23 = 46 =
„¨»n¤š¸É 12 25 = 104
„¨»n¤š¸É 13 25 = 104
„¨»n¤š¸É 14 43 = 86
„¨»n¤š¸É 15 43 = 86
„¨»n¤š¸É 16 35 = 106
„¨»n¤š¸É 17 53 = 106
„¨»n¤š¸É 18 45 = 108
„¨»n¤š¸É 19 45 = 108
6
9
9
6
¡ªnµ¤¸ 19 „¨»n¤ š¸É¤¸Á«¬­nªœŽÊε ¨³Á«¬­nªœš¸ÉŽÊε¤¸š´ÊŠ®¤— 56 ˜´ª —´Šœ´ÊœÁ«¬­nªœš¸É¤¸‡nµ
˜„˜nµŠ„´œ¤¸š´ÊŠ®¤— 100+19-56 = 63 ‹Îµœªœœ´Éœ‡º° ÁŽ˜ A ¤¸­¤µ·„ 63 ˜´ª
ž{®µš¸É 3
­Îµ®¦´‹ÎµœªœÁ˜È¤ª„Ä—Ç n! = n(n-1) (n-2) ... 3.2.1 ˜´ª°¥nµŠ 4! = 4.3.2.1 ™oµ ®µ¦ 2545!
—oª¥ 2546 ¨³ r Áž}œÁ«¬š¸Éŗo‹µ„„µ¦®µ¦œ¸Ê r ¤¸‡nµÁž}œÁšnµÄ—
ª·›¸Â„ož{®µ
…´ÊœšÎµ‡ªµ¤Á…oµÄ‹
˚¥r°„‡ªµ¤®¤µ¥…°Š n! Á¤ºÉ° n Áž}œ‹ÎµœªœÁ˜È¤ª„¤µÄ®o ¡¦o°¤š´ÊŠÂ­—Š˜´ª°¥nµŠ ‹¹ŠšÎµ
Ä®oÁ…oµÄ‹‡ªµ¤®¤µ¥…°Š n! ŗoŠnµ¥ ­·ÉŠš¸É˚¥r˜o°Š„µ¦‡º° ®µ‡nµÁ«¬š¸Éŗo‹µ„„µ¦®µ¦ 2545! —oª¥ 2546
…´ÊœªµŠÂŸœ ¡·‹µ¦–µ 2545! ¡ªnµ
2545! = 2545 u 2544 u 2543 u .... u 3 u 2 u 1
—´Šœ´Êœ™oµ 2546 Â¥„˜´ªž¦³„°š¸É¤µ„„ªnµ 1 ŗo‹³šÎµÄ®o˜´ªž¦³„°…°Š 2546 š»„˜´ª¤¸‡nµ
182
œo°¥„ªnµ 2545 —´Šœ´Êœ˜´ªž¦³„°š»„˜´ª…°Š 2546 š¸ÉŤnčn 2546 ‹³Áž}œ‹ÎµœªœÄœŸ¨‡¼–š¸Éž¦³„°„´œ
Áž}œ 2545! ¥n°¤­nŠŸ¨Ä®o 2546 ®µ¦ 2545! ¨Š˜´ª‹¹ŠšÎµÄ®oÁ«¬ r ¤¸‡nµÁšnµ„´ 0
…´Êœ—εÁœ·œ„µ¦
¡·‹µ¦–µ 2546 = 2 x 1273
Ž¹ÉŠ 2 ¨³ 1273 ˜nµŠ„ÈÁž}œ˜´ªž¦³„°š¸É˜nµŠ„´œ…°Š 2545!
—´Šœ´Êœ 2 u 1273 ®µ¦ 2545! ŗo¨Š˜´ª
œ´Éœ‡º° 2545! ®µ¦—oª¥ 2546 ¨Š˜´ª‹¹ŠÅ¤n¤¸Á«¬
—´Šœ´Êœ r = 0
ž{®µš¸É 4
„ε®œ—Ä®o a ¨³ b Áž}œ‹ÎµœªœÁ˜È¤ Ž¹ÉŠ
2 2 (2 2)
Ášnµ„´ÁšnµÄ—
a b = ab ‡nµ (2 2) 2 2
ª·›¸Â„ož{®µ
…´ÊœšÎµ‡ªµ¤Á…oµÄ‹
˚¥r„ε®œ—„µ¦—εÁœ·œ„µ¦šª·£µ‡ œÁŽ˜…°Š‹ÎµœªœÁ˜È¤¤µÄ®o‡º° a b = ab
2 2 (2 2)
Ánœ 2 3 = 23 = 8 ˚¥r˜o°Š„µ¦®µ‡nµ…°Š (2 2) 2 2
…´ÊœªµŠÂŸœ
‹µ„Á«¬­nªœ®µ‡nµ˜´ªÁ«¬ ×¥šÎµªŠÁ¨ÈÄœ„n°œ°°„¤µš¸¨³…´Êœ ˜n°Åž®µ‡nµ˜´ª­nªœÄœšÎµœ°Š
Á—¸¥ª„´œ ¨oª®µŸ¨®µ¦…°Š˜´ªÁ«¬ ¨³˜´ª­nªœ‹³Å—o‡nµš¸É˜o°Š„µ¦
œ°„‹µ„œ¸ÊÁœºÉ°Š‹µ„ ¤¸ªŠÁ¨ÈŽo°œ„´œ ‹³¡ªnµ
(a b) c = (ab)c = abc
¨³
a (b c) = a
b
c
‹¹Š˜o°ŠœÎµ­¤´˜·š´ÊŠ­°Šœ¸ÊŞčo—oª¥
…´Êœ—εÁœ·œ„µ¦
‹µ„
‹³
‹µ„
2
2 (2 2) = 2
2
= 24 = 16
2 2 (2 2) = 2 2 ( 2 2 ) = 216
(2 2) 2
2
= 22 = 42 = 16
183
2
‹³Å—o (2 2) 2 2 = 162 = 24 = 28
2 2 (2 2) 216
8
—´Šœ´Êœ =
8 = 2 = 256
(2 2) 2 2 2
ž{®µš¸É 5
„ε®œ—Ä®o x Áž}œ‹Îµœªœ‹¦·ŠŽ¹ÉŠ –1 < 2x + 3 < 1 ¨³ y Áž}œ‹Îµœªœ‹¦·ŠŽ¹ÉŠ –1 < y + 6 < 7
‹Š®µnªŠš¸ÉÁž}œÁŽ˜‡Îµ˜°…°Š x + y, x – y, y – x, xy ¨³ xy
ª·›¸Â„ož{®µ
…´ÊœšÎµ‡ªµ¤Á…oµÄ‹
˚¥r°„‡nµ…°Š x ¨³ y Ĝ¦¼ž°­¤„µ¦Á·ŠÁ­oœ ¨oªÄ®o®µnªŠš¸ÉÁž}œÁŽ˜‡Îµ˜°…°ŠŸ¨ª„
Ÿ¨¨ Ÿ¨‡¼– ¨³Ÿ¨®µ¦ ¦³®ªnµŠ x „´ y
…´ÊœªµŠÂŸœ
1. ®µ‡nµ x ¨³ y Ĝ¦¼ž°­¤„µ¦
2. Ä®oÁŠºÉ°œÅ… ­Îµ®¦´‹Îµœªœ‹¦·Š a, b, c, d, x, y, z ¨³ w
(1) ™oµ a < x < b ¨oª – b < – x < – a
(2) ™oµ a < x < b ¨³ c < y < d ¨oª a + c < x + y < b + d
(3) ™oµ 0 < a < x < b ¨³ 0 < c < y < d ¨oª 0 < ac < xy < bd
(4) ™oµ a < x < b < 0 ¨³ c < y < d < 0 ¨oª 0 < bd < xy < ac
(5) ™oµ 0 < a < x < b ¨³ c < y < d < 0 ¨oª bc < xy < ad < 0
(6) ™oµ 0 < a < b ¨oª 0 < 1b < 1a
(7) ™oµ a < b < 0 ¨oª < 1b < 1a < 0 Áž}œ˜oœ
…´Êœ—εÁœ·œ„µ¦Â„ož{®µ
‹µ„
–1 < 2x + 3 < 1
‹³Å—o –1 – 3 < 2x + 3 – 3 <1 – 3
—´Šœ´Êœ –4 < 2x < –2
œ´Éœ‡º° –2 < x < 1
‹µ„
–1 < y + 6 < 7
‹³Å—o –1 – 6 < y + 6 – 6 < 7 – 6
..............c
184
‹³Å—o –7 < y <1
..............d
®µ‡nµ…°Š x + y
‹µ„ c ¨³ d ¨³ÁŠºÉ°œÅ… 2(2)
‹¹ŠÅ—o (–2) + (–7) < x + y < (–1) + 1
œ´Éœ‡º° –9 < x + y < 0
—´Šœ´ÊœÁŽ˜‡Îµ˜°…°Š x + y ‡º° (–9, 0)
®µ‡nµ…°Š x – y
‹µ„ c ¨³ d ‹³Å—o –1 < –y < 7 ..............e
c + e;
(–2) + (–1) < x – y < (–1) + 7
‹³Å—o
–3 < x – y < –6
—´Šœ´ÊœÁŽ˜‡Îµ˜°…°Š x – y ‡º° (–3, 6)
®µ‡nµ…°Š y – x
‹µ„ –3 < x – y < –6
‹¹ŠÅ—o –6 < y – x < 3
—´Šœ´ÊœÁŽ˜‡Îµ˜°…°Š y – x ‡º° (–6, 3)
®µ‡nµ…°Š xy
¡·‹µ¦–µ –7 < y < 1 ‹³Å—o –7 < y < 0 ®¦º° y = 0 ®¦º° 0 < y < 1
¨³‹µ„ –2 < x < –1 < 0 ‹³¤¸„¦–¸š¸É¡·‹µ¦–µÅ—o—´Šœ¸Ê
„¦–¸š¸É 1 –7 < y < 0 ¨³ –2 < x < –1
‹³Å—o (0) (1) < xy < (–7)( –2) ×¥ÁŠºÉ°œÅ… 2(4)
—´Šœ´Êœ 0 < xy < 14
„¦–¸š¸É 2 y = 0 ¨³ –2 < x < –1
‹³Å—o xy = 0
„¦–¸š¸É 3 0 < y < 1 ¨³ –2 < x < –1
‹³Å—o (1) (–2) < xy < (0) (–1) ×¥ÁŠºÉ°œÅ… 2 (5)
—´Šœ´Êœ –2 < xy < 0
185
‹µ„š´ÊŠ­µ¤„¦–¸‹³Å—oÁŽ˜‡Îµ˜°…°Š xy ‡º° (0, 14) ‰ {0} ‰ (–2, 0) ®¦º° (–2, 14)
®µ‡nµ…°Š xy
Ĝ„µ¦®µ‡nµ…°Š xy œ´Êœ‹³˜o°Š˜´—„¦–¸š¸É y = 0 °°„Åž‹¹Š¡·‹µ¦–µ „¦–¸š¸É –7 < y < 0
¨³„¦–¸š¸É 0 < y < 1 Ášnµœ´Êœ
„¦–¸š¸É 1 –7 < y < 0 ‹³Å—o 1y < – 17
‹µ„ –2 < x < –1 ‹³Å—o 17 < xy ×¥ÁŠºÉ°œÅ… 2(4)
„¦–¸š¸É 2 0 < y < 1 ‹³Å—o 1 < 1y
‹µ„ –2 < x < –1 ‹³Å—o xy < –1 ×¥ÁŠºÉ°œÅ… 2(5)
‹µ„š´ÊŠ­°Š„¦–¸‹³Å—oÁŽ˜‡Îµ˜°…°Š xy ‡º°
17 , f ‰ (–f , –1) ®¦º° (–f , –1) ‰ 17 , f
®µ‡nµ…°Š xy
¡·‹µ¦–µ –2 < x < –1 ‹³Å—o –1 < 1x < – 12
ÁœºÉ°Š‹µ„ x z 0 „µ¦®µ‡nµ…°Š xy ‹¹Š¡·‹µ¦–µ„¦–¸…°Š y š´ÊŠ­µ¤„¦–¸ÁnœÁ—¸¥ª„´„µ¦®µ‡nµ…°Š
xy —´Šœ¸Ê
„¦–¸š¸É 1 –7 < y < 0 ¨³ –1 < 1x < – 12
‹³Å—o (0) 12 < xy < (–7) (–1) ×¥ÁŠºÉ°œÅ… 2(4)
œ´Éœ‡º° 0 < xy < 7
„¦–¸š¸É 2 0 < y < 1 ¨³ –1 < 1x < – 12
‹³Å—o (1) (–1) < xy < 0 12 œ´Éœ‡º° –1 < xy < 0
„¦–¸š¸É 3 y = 0 ‹³Å—o xy = 0
‹µ„š´ÊŠ­µ¤„¦–¸ ‹³Å—oÁŽ˜‡Îµ˜°…°Š xy ‡º°
(–1, 0) ‰ {0} ‰ (0, 7) ®¦º° (–1, 7)
186
˜°œš¸É 5.2 „µ¦­¦oµŠ­¦¦‡rž{®µ
„µ¦—´—ž¨Šž{®µ‡–·˜«µ­˜¦r
‹µ„ž{®µš´ÊŠ®oµž{®µš¸É„¨nµª¤µÂ¨oª ™oµÁž¨¸É¥œÂž¨Š‡Îµ™µ¤ ®¦º°—´—ž¨ŠÃ‹š¥r‹³šÎµÄ®o¤¸
ž{®µš¸Éªœ‡·—®¨µ¥®¨µ¥…¹Êœ¤µš´œš¸
‹µ„ž{®µš¸É 1
‹ŠÁ¦¸¥Š°´œ—´…°Š‹Îµœªœ˜n°Åžœ¸ÊÄ®o™¼„˜o°Š‹µ„œo°¥Åž¤µ„ 2514 , 4258, 8171, 16128 ¨³ 32103
—´—ž¨ŠÁž}œ
‹ŠÁ¦¸¥Š°´œ—´…°Š‹Îµœªœ˜n°Åžœ¸ÊÄ®o™¼„˜o°Š‹µ„œo°¥Åž¤µ„
5142 , 2584, 1718, 12816 ¨³ 10332
‹µ„ž{®µ‹³Á®Èœªnµ ÁŠºÉ°œÅ…‡ªµ¤¦¼oÁž¨¸É¥œÅž ‡º° ˜o°ŠÄoÁŠºÉ°œÅ… “­Îµ®¦´‹Îµœªœ‹¦·Šª„
a, b ¨³ c ™oµ a < b ¨oª ac < bc”
2
¡·‹µ¦–µ 2584 = (258)2 = (66564)2
2
1718 = (171)4 = (2221461)2
2
(128)16 = (128)8 = n2 ×¥š¸É n = 1288 ¤µ„„ªnµ 2221461
2
(103)32 = (103)16 = m2 ×¥š¸É m = 10316¤µ„„ªnµ 1288
—´Šœ´Êœ 5142 < 2584 <1718 < 12816 < 10332 °´œ—´…°Š‹Îµœªœ‹µ„œo°¥Åž¤µ„‡º° 5142 , 2584,
1718, 12816 ¨³ 10332
‹µ„ž{®µš¸É 2
ÁŽ˜ A = qp p , q  N,1 d p d 10 š 1 d q d 10¤¸­¤µ·„š´ÊŠ®¤—„¸É˜´ª —´—ž¨ŠÁž}œ
„ε®œ— A = qp p , q  N,1 d p d 10 š 1 d q d 10 ‹Š®µ
‡ªµ¤œnµ‹³Áž}œÄœ„µ¦Á¨º°„­¤µ·„­°Š˜´ªš¸É˜„˜nµŠ„´œ…°Š
A š¸É¤¸Ÿ¨ª„Ášnµ„´ 1
187
‹µ„ž{®µ‹³Á®Èœªnµ œÎµž{®µÁ—·¤¤µ‡·—Á¡·É¤Á˜·¤ —oª¥„µ¦Á¨º°„­¤µ·„Äœ A Ž¹ÉŠÁž}œÁ«¬­nªœ­°Š
‹Îµœªœš¸É˜„˜nµŠ„´œ¤µª„„´œ Ä®o¤¸‡nµÁšnµ„´ 1 Á¤ºÉ°Á¨º°„Å—o¨oª‹¹Š®µ‡ªµ¤œnµ‹³Áž}œÄœ„µ¦Á¨º°„œ´Êœ Ž¹ÉŠ
˜o°ŠÄo‹Îµœªœ­¤µ·„…°Š A š´ÊŠ®¤—‹µ„ž{®µš¸É 2 „n°œ®œoµœ¸Ê Ž¹ÉŠ˜o°ŠÄo‹Îµœªœ­¤µ·„ 63 ˜´ª ®¨´Š‹µ„
œ´ÊœÄoª·›¸‹´—®¤¼nÁ¨º°„­¤µ·„ 2 ˜´ª …°ŠÁŽ˜ A ‹µ„­¤µ·„š´ÊŠ®¤— Á¡ºÉ°Áž}œž¦·£¼¤·˜´ª°¥nµŠÅ—oÁšnµ„´
63 63 u 62
2 = 2 = 63 u 31
®¨´Š‹µ„œ´ÊœÂ‹„‹Š‡¼n­¤µ·„…°Š A š¸É˜„˜nµŠ„´œÂ˜n¤¸Ÿ¨ª„Ášnµ„´ 1 Ž¹ÉŠÂ‹„‹ŠÅ—o—´Šœ¸Ê
^12 , 23`, ^14 , 45`, ^15 , 45`, ^16 , 65`, ^17 , 67`, ^18 , 87`
^19 , 89`, ^101 , 109`, ^25 , 35`, ^27 , 75`, ^29 , 79`, ^73 , 47`
^35 , 85`, ^103 , 107`, ^49 , 95`
Ž¹ÉŠ¤¸š´ÊŠ®¤— 15 ‡¼n ­¤µ·„‹³Á®Èœªnµ„µ¦Â‹„‹Šœ¸Ê Ťn‡·—°´œ—´Á¡¦µ³Áž}œÁ®˜»„µ¦–r…°Šª·›¸‹´—®¤¼n
—´Šœ´Êœ‡ªµ¤œnµ‹³Áž}œÄœ„µ¦Á¨º°„­¤µ·„­°Š˜´ªš¸É˜„˜nµŠ„´œ…°Š A š¸É¤¸Ÿ¨ª„Ášnµ„´ 1 ‡º°
15
15
63 u 31 = 1953
‹µ„ž{®µš¸É 3
­Îµ®¦´‹ÎµœªœÁ˜È¤ª„Ä—Ç n! = n(n – 1) (n – 2) .... 3.2.1
˜´ª°¥nµŠ 4! = 4.3.2.1
™oµ®µ¦ 2545! —oª¥ 2546 ¨³ r Áž}œÁ«¬­nªœš¸Éŗo‹µ„„µ¦®µ¦œ¸Ê r ¤¸‡nµÁž}œÁšnµÄ—
—´—ž¨ŠÁž}œ
‹ÎµœªœÁŒ¡µ³š¸ÉÁ¨È„š¸É­»—®µ¦ 2545! Ťn¨Š˜´ª‡º°‹ÎµœªœÄ—
‹µ„ž{®µš¸É 3 Á—·¤¡ªnµ 2546 ®µ¦ 2545! ŗo¨Š˜´ª ¨³ 2546 Áž}œ‹Îµœªœž¦³„° Á®ÈœÅ—o
´—‹ÎµœªœÁŒ¡µ³˜´ª™´—Åžš¸É¤¸‡nµ¤µ„„ªnµ 2546 ‹³®µ¦ 2545! Ťn¨Š˜´ª š´ÊŠœ¸ÊÁ¡¦µ³‹ÎµœªœÁŒ¡µ³˜´ªœ´Êœ
‹³Å¤nčn˜´ªž¦³„°…°Š 2545!
‹ÎµœªœÁŒ¡µ³˜´ª™´—Åžš¸É¤µ„„ªnµ 2546 ‡º° 2549 Áž}œ‹ÎµœªœÁŒ¡µ³ ˜´ªš¸ÉÁ¨È„š¸É­»—š¸É®µ¦ 2545!
Ťn¨Š˜´ª
®¤µ¥Á®˜» ÁŽ˜…°Š‹ÎµœªœÁŒ¡µ³ÄœnªŠ 2500 – 2599 ‡º°
{2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2591, 2593}
188
‹µ„ž{®µš¸É 4 „ε®œ—Ä®o a ¨³ b Áž}œ‹ÎµœªœÁ˜È¤ Ž¹ÉŠ
2 2 2 2 a b = ab ‡nµ…°Š Ášnµ„´ÁšnµÄ—
22 2 2
—´—ž¨ŠÁž}œ
„ε®œ—Ä®o Áž}œ„µ¦—εÁœ·œ„µ¦œÁŽ˜…°Š‹ÎµœªœÁ˜È¤
ª„ N œ·¥µ¤Ã—¥ a b = ab
(1) ¤¸Á°„¨´„¬–r­Îµ®¦´ œ N ®¦º°Å¤n
(2) ¤¸­¤´˜·„µ¦­¨´š¸Éœ N ®¦º°Å¤n
‹µ„ a b = ab š»„ a, b  N
(1) ‹³Á®ÈœÅ—oªnµ a 1 = a1 = a —¼Á®¤º°œªnµ 1 œnµ‹³Áž}œÁ°„¨´„¬–r­Îµ®¦´ œ N ˜nœ·¥µ¤…°Š
Á°„¨´„¬–r ­Îµ®¦´„µ¦—εÁœ·œ„µ¦ œÁŽ˜ A Ä—Ç ‡º° ­Îµ®¦´ i A ‹³Å—o a i = i a = a ‹µ„ œ
N ¡ªnµ
a 1 = a1 = a
˜n 1 a = 1a = 1 z a
—´Šœ´Êœ 1 ŤnčnÁ°„¨´„¬–r­Îµ®¦´ œ N
­¤¤˜·ªnµ¤¸ n  N Ž¹ÉŠ n Áž}œÁ°„¨´„¬–r­Îµ®¦´ œ A
—´Šœ´Êœ a n = n a = a š»„ a  N
‹³Å—o an = na š»„ a  N ˜n…o°‡ªµ¤œ¸ÊŤnÁž}œ‹¦·Š
‹¹Š­¦»žÅ—oªnµ Ťn¤¸ n ˜´ªÄ—Äœ N š¸ÉÁž}œÁ°„¨´„¬–r­Îµ®¦´ œ N
(2) ÁœºÉ°Š‹µ„ ab z ba Á¤ºÉ° a z b š»„ a, b  y
‹¹Š­¦»žÅ—oªnµ Ťn¤¸­¤´˜·„µ¦­¨´š¸Éœ N
‹µ„ž{®µš¸É 5
„ε®œ—Ä®o x Áž}œ‹Îµœªœ‹¦·ŠŽ¹ÉŠ –1 < 2x + 3 < 1 ¨³ y Áž}œ‹Îµœªœ‹¦·ŠŽ¹ÉŠ –1 < y + 6 < 7
‹Š®µnªŠš¸ÉÁž}œÁŽ˜‡Îµ˜°…°Š x + y, x – y, y – x, xy, xy ¨³ xy
—´—ž¨ŠÁž}œ
„ε®œ—Ä®o x Áž}œ‹Îµœªœ‹¦·Š Ž¹ÉŠ –1 < 2 x + 3 < 1 ¨³ y Áž}œ‹Îµœªœ‹¦·ŠŽ¹ÉŠ –1 < y + 6 < 7
‹Š®µ‡nµ¤µ„š¸É­»— ¨³‡nµœo°¥š¸É­»—…°Š x + y, x – y, y – x, xy, xy ¨³ xy ™oµ¤¸
189
‹µ„ž{®µš¸É 5 Á—·¤ Á¤ºÉ°Áž¨¸É¥œÁŠºÉ°œÅ…‹µ„ “ < ” Áž}œ “ < ”
—´Šœ´Êœ‡nµ…°Š x ¨³ y Á—·¤‹µ„ –2 < x < –1 ¨³ –7 < y < 1
‹¹ŠÁž¨¸É¥œÁž}œ –2 < x < –1 ¨³ –7 < y < 1
­nŠŸ¨Ä®o‹µ„Á—·¤
–9<x+y<0
Áž¨¸É¥œÁž}œ
šÎµÄ®o‡nµ˜Éε­»—…°Š
¨³‡nµ­¼Š­»—…°Š
‹µ„Á—·¤ -3 < x - y < 6
–9< x+y<0
x + y ༡ Р9
x + y ༡ 0
Áž¨¸É¥œÁž}œ
šÎµÄ®o˜Éε­»—…°Š
¨³‡nµ­¼Š­»—…°Š
–3 < x – y < 6
x – y ‡º° –3
x Рy ༡ 6
–6 < y – x < 3
Áž¨¸É¥œÁž}œ
šÎµÄ®o‡nµ˜Éε­»—…°Š
¨³‡nµ­¼Š­»—…°Š
–6 < y – x < 3
y – x ‡º° –6
y Рx ༡ 3
–2 < xy < 14
Áž¨¸É¥œÁž}œ
šÎµÄ®o˜Éε­»—…°Š
¨³‡nµ­¼Š­»—…°Š
–2 < xy < 14
xy ‡º° –2
xy ༡ 14
– f < xy < –1 ®¦º° 17 < x < f
– f < xy < –1 ®¦º° 17 < x < f
‹µ„Á—·¤
‹µ„Á—·¤
‹µ„Á—·¤
Áž¨¸É¥œÁž}œ
‹³Á®ÈœÅ—oªnµ xy Ťn¤¸‡nµ˜Éε­»— ¨³Å¤n¤¸‡nµ­¼Š­»—
‹µ„Á—·¤
–1 < xy < 7
Áž¨¸É¥œÁž}œ
–1 < xy < 7
šÎµÄ®o‡nµ˜Éε­»—…°Š xy ‡º° –1
¨³‡nµ­¼Š­»—…°Š xy ‡º° 7
190
‹µ„ž{®µš´ÊŠ 5 …o° —´Šš¸Éœ³œÎµ¤µœ¸Ê‹³Á®ÈœÅ—oªnµ Á¦µ­µ¤µ¦™—´—ž¨Šž{®µÁ—·¤Áž}œž{®µ
Ä®¤nŗo ×¥š¸Éž{®µš¸É—´—ž¨ŠÄ®¤n °µ‹Äo¡ºÊœ“µœ‡ªµ¤¦¼oÁ—·¤ ‹µ„ž{®µÁ—·¤ ¤µÂ„ož{®µš¸É—´—ž¨Š
¨oª ¨³œ°„‹µ„œ¸Ê ˜o°ŠÄ®o‡ªµ¤¦¼o¡ºÊœ“µœ°ºÉœÁ¡·É¤Á˜·¤„Èŗo °´œÁž}œ„µ¦šÎµÄ®o„¦³ªœ„µ¦Â„ož{®µ ¤¸
„µ¦­¦oµŠ­¦¦‡r¤µ„…¹Êœ
 f„®´—š¸É 24
1. ‹Š°°„„µ¦Â„ož{®µ ¨³Â„ož{®µ˜n°Åžœ¸Ê
„ε®œ—Ä®o
U = {1, 2, 3, ......, 100}
A = {X  U _ 2 ®µ¦¨Š˜´ª}
B = {X  U _ 3 ®µ¦¨Š˜´ª}
_ n(A – B) – n (B – A) _
Ášnµ„´ÁšnµÄ—
2. ‹Š—´—ž¨Šž{®µÄœ…o° 1 Áž}œž{®µÄ®¤n ¨oª°°„„µ¦Â„ož{®µ ¨³Â„ož{®µš¸É—´—ž¨Šœ¸Ê
191
®œnª¥š¸É 6
°­¤„µ¦
˜°œš¸É 6.1 °­¤„µ¦š¸ÉÁ„¸É¥ª„´¡¸‡–·˜
˜°œš¸É 6.2 °­¤„µ¦Á„¸É¥ª„´‡nµ­´¤¼¦–r
œª‡·— 1. ­¤„µ¦Â¨³°­¤„µ¦ Áž}œ­·ÉŠ­Îµ‡´Äœ¦³‡–·˜«µ­˜¦r ­Îµ®¦´°­¤„µ¦œ´Êœ­µ¤µ¦™œ·¥µ¤
ŗo‹µ„­¤„µ¦
2. Ĝ¦³‹Îµœªœ‹¦·Š­¤´˜·…°Š°­¤„µ¦œÎµÅžž¦³¥»„˜rčo°¥nµŠ¤µ„¤µ¥š´ÊŠÄœª·µ
‡–·˜«µ­˜¦r ¨³ª·µ°ºÉœÇ
3. ×¥š´ÉªÇŞŸ¨ÁŒ¨¥…°Š°­¤„µ¦œ·¥¤Á…¸¥œÄœ¦¼žÁŽ˜‡Îµ˜°
4. °­¤„µ¦š¸ÉÁ„¸É¥ª„´‡nµ­´¤¼¦–rÁž}œ¡ºœÊ “µœ­Îµ‡´š¸ÉœÎµÅžÄoĜª·µ ‡¨‡¼¨´­ ¨³„µ¦ª·Á‡¦µ³®r
ª´˜™»ž¦³­Š‡r
Á¤ºÉ°«¹„¬µ®œnª¥š¸É 7 ‹Â¨oª œ´„Á¦¸¥œ­µ¤µ¦™
1. œ·¥µ¤°­¤„µ¦Å—o
2. ¦³»­¤´˜·¡ºÊœ“µœ…°Š°­¤„µ¦…°Š‹Îµœªœ‹¦·ŠÅ—o
3. ®µÁŽ˜‡Îµ˜°…°Š°­¤„µ¦š¸É¤¸®¨µ¥˜´ªž¦³„°Å—o
4. ¡·­¼‹œr …o°‡ªµ¤Á„¸¥É ª„´°­¤„µ¦Á·Š¡¸‡–·˜Å—o
5. ¡·­¼‹œr …o°‡ªµ¤Á„¸¥É ª„´°­¤„µ¦…°Š‡nµ­´¤¼¦–rŗo
„·‹„¦¦¤¦³®ªnµŠÁ¦¸¥œ
1. °µ‹µ¦¥r°›·µ¥‡ªµ¤®¤µ¥Â¨³­¤´˜·…°Š°­¤„µ¦ ­—Š˜´ª°¥nµŠª·›¸„µ¦Â„o˚¥rž{®µ
°­¤„µ¦š¸ÉÁ„¸É¥ª„´¡¸‡–·˜ ¨³°­¤„µ¦Á„¸É¥ª„´‡nµ­´¤¼¦–r
2. œ´„Á¦¸¥œšÎµ„·‹„¦¦¤¦³®ªnµŠÁ¦¸¥œ ¨³Â f„®´—
3. œ´„Á¦¸¥œž¦³Á¤·œ¡´•œµ„µ¦…°Š˜œÁ°Š
­ºÉ°„µ¦­°œ
1. Á°„­µ¦„µ¦­°œ
2.  f„ž’·´˜·
3. Á‡¦ºÉ°ŠŒµ¥…oµ¤«¸¦¬³
ž¦³Á¤·œŸ¨
ž¦³Á¤·œŸ¨‹µ„ f„®´—¨³„µ¦š—­°
192
˜°œš¸É 6.1 °­¤„µ¦š¸ÉÁ„¸É¥ª„´¡¸‡–·˜
Á¦ºÉ°Šš¸É 1 ‡ªµ¤®¤µ¥Â¨³­¤´˜·…°Š°­¤„µ¦
1. ‡ªµ¤®¤µ¥…°Š°­¤„µ¦ ‡ªµ¤®¤µ¥¡ºœÊ “µœ…°Š°­¤„µ¦ œ·¥µ¤Å—o—´Ššœ·¥µ¤˜n°Åžœ¸Ê
šœ·¥µ¤ 1 ­Îµ®¦´‹Îµœªœ‹¦·Š a ¨³ b ėÇ
a ¤µ„„ªnµ b „Șn°Á¤ºÉ° ¤¸‹Îµœªœ‹¦·Šª„ c š¸ÉšÎµÄ®o a = b + c
a ¤µ„„ªnµ b Á…¸¥œÂšœ—oª¥ a > b
šœ·¥µ¤ 2 ­Îµ®¦´‹Îµœªœ‹¦·Š a ¨³ b ėÇ
b œo°¥„ªnµ a „Șn°Á¤ºÉ° a ¤µ„„ªnµ b
b œo°¥„ªnµ a Á…¸¥œÂšœ—oª¥ b < a
šœ·¥µ¤ 3 ­Îµ®¦´‹Îµœªœ‹¦·Š a ¨³ b ėÇ
a ¤µ„„ªnµ®¦º°Ášnµ„´ b „Șn°Á¤ºÉ° a ¤µ„„ªnµ b ®¦º° a Ášnµ„´ b
a ¤µ„„ªnµ®¦º°Ášnµ„´ b Á…¸¥œÂšœ—oª¥ a t b
šœ·¥µ¤ 4 ­Îµ®¦´‹Îµœªœ‹¦·Š a ¨³ b ėÇ
b ¤µ„„ªnµ®¦º°Ášnµ„´ a „Șn°Á¤ºÉ° b œo°¥„ªnµ a ®¦º° b Ášnµ„´ a
b ¤µ„„ªnµ®¦º°Ášnµ„´ a Á…¸¥œÂšœ—oª¥ b d a
šœ·¥µ¤ 2, 3 ¨³ 4 °µ‹Á…¸¥œÄ®¤nŗo—´Šœ¸Ê
šœ·¥µ¤ 2c b < a „Șn°Á¤ºÉ° ¤¸ c R+ š¸ÉšÎµÄ®o b + c = a
šœ·¥µ¤ 3c a t b „Șn°Á¤ºÉ° ¤¸ c R+ š¸ÉšÎµÄ®o a = b + c
šœ·¥µ¤ 4c b d a „Șn°Á¤ºÉ° ¤¸ c R+ 8 {0}š¸ÉšÎµÄ®o b + c = a
2. ­¤´˜·…°Š°­¤„µ¦
„ε®œ—Ä®o a, b, c, x ¨³ y Áž}œ‹Îµœªœ‹¦·Š
1. ™oµ a > b ¨³ b > c ¨oª a > c
2. ™oµ a > b ¨oª a + c > b + c
3. ™oµ a > b ¨³ c > 0 ¨oª ac > bc
4. ™oµ a > b ¨³ c < 0 ¨oª ac < bc
5. ™oµ a > b > 0 ¨³ c > 0 ¨oª ac > bc > 0
6. ™oµ a > 1 ¨³ x > y > 0 ¨oª ax > ay > 0
7. ™oµ 0 < a < 1 ¨³ x > y > 0 ¨oª 0 < ax < ay
8. ™oµ a > 1 ¨³ x > y > 0 ¨oª loga x > loga y
193
9. ™oµ 0 < a < 1 ¨³ x > y > 0 ¨oª loga x < loga y
10. a2n t 0 š»„‹ÎµœªœÁ˜È¤ª„ n
11. a > 0 „Șn°Á¤ºÉ° 1 > 0
a
12. a < 0 „Șn°Á¤ºÉ° 1 < 0
a
13. ­Îµ®¦´‹Îµœªœ‹¦·Š a ¨³ b Ž¹ÉŠ a < b
( x – a )( x – b ) < 0 „Șn°Á¤º°É a < x < b
14. ­Îµ®¦´‹Îµœªœ‹¦·Š a ¨³ b Ž¹ÉŠ a < b
( x – a )( x – b ) > 0 „Șn°Á¤º°É x < a ®¦º° x > b
Á¦ºÉ°Šš¸É 2 ˜´ª°¥nµŠª·›¸„µ¦Â„ož{®µÃ‹š¥r°­¤„µ¦
…o°˜„¨Š „ε®œ—Ä®oÁ°„£¡­´¤¡´œ›r ‡º° ÁŽ˜…°Š‹Îµœªœ‹¦·Š ­Îµ®¦´‹Îµœªœš¸ÉÁ…¸¥œÃ—¥Å¤nÁ‹µ³‹Š
‹ÎµœªœÁ®¨nµœ´œÊ ‡º°‹Îµœªœ‹¦·Š
˜´ª°¥nµŠš¸É 6.1.1 ‹Š¡·­¼‹œrªnµ a2n+1 > 0 „Șn°Á¤ºÉ° a > 0 š»„‹ÎµœªœÁ˜È¤ª„ n
¡·­¼‹œr (1) ‹³Â­—Šªnµ ™oµ a2n+1 > 0 ¨oª a > 0
„ε®œ—Ä®o
a2n+1 > 0
‹³Å—o
a2n a > 0
­¤¤˜· a d 0 ‹³¤¸„¦–¸—´Šœ¸Ê ‡º° a < 0 ®¦º° a = 0
™oµ a < 0 ‹³Å—o 1 < 0
a
2n
—´Šœ´Êœ a ˜ a < 0
a
2n
a < 0 …´—Â¥oŠ„´ a 2n t 0
a < 0 ‹¹ŠÁž}œÅžÅ¤nŗo
™oµ a = 0 ‹³Å—o a2n+1 = 0 …´—Â¥oŠ„´š¸„É 宜—Ä®o a 2n+1 > 0
a = 0 ‹¹ŠÁž}œÅžÅ¤nŗo
œ´Éœ‡º°
a>0
(2) ‹³Â­—Šªnµ ™oµ a > 0 ¨oª a2n+1 > 0
„ε®œ—Ä®o
a>0
‹³Å—o
a 2n > 0 —´Šœ´Êœ a 2n . a > 0
‹¹ŠÅ—o
a 2n+1 > 0
194
‹µ„ (1) ¨³ (2) ‹¹Š­¦»žÅ—oªµn
a 2n+1 > 0 „Șn°Á¤ºÉ° a > 0 š»„‹ÎµœªœÁ˜È¤ª„ n
®¤µ¥Á®˜» ­Îµ®¦´…o°‡ªµ¤ a 2n+1 < 0 „Șn°Á¤ºÉ° a < 0 š»„‹ÎµœªœÁ˜È¤ª„ n
Ä®ošÎµÁž}œÂ f„®´—
˜´ª°¥nµŠš¸É 6.1.2 ‹Š¡·­¼‹œrªnµ ( x – a )2n ( x – b )2m + 1 > 0 „Șn°Á¤ºÉ° x z a ¨³ x – b > 0 š»„‹ÎµœªœÁ˜È¤
ª„ m ¨³ n
¡·­¼‹œr (1) ‹³Â­—Šªnµ ™oµ ( x – a )2n ( x – b )2m + 1> 0 ¨oª x – a z 0 ¨³ x – b > 0
„ε®œ—Ä®o ( x – a )2n ( x – b )2m + 1 > 0
¡·‹µ¦–µ x – a
™oµ x – a = 0 ‹³šÎµÄ®o ( x – a )2n ( x – b )2m + 1 = 0
Ž¹ÉŠ…´—Â¥oŠ„´š¸É„ε®œ—Ä®o ( x – a )2n ( x – b )2m + 1 > 0
—´Šœ´Êœ x – a z 0
…c
2n
‹³Å—o ( x – a ) > 0
—´Šœ´Êœ
(x a) 2n (x b) 2m 1
(x a) 2n
>0
­—Šªnµ
( x – b )2m + 1 > 0
×¥˜´ª°¥nµŠš¸É 6.1 ¡·­¼‹œr¤µÂ¨oª ‹³Å—o x – b > 0 …d
‹µ„ c ¨³ d ‹¹ŠÅ—o x – a z 0 ¨³ x – b > 0
(2) ‹³Â­—Šªnµ ™oµ x – a z 0 ¨³ x – b > 0 ¨oª ( x – a )2n ( x – b )2m + 1 > 0
ÁœºÉ°Š‹µ„ x – a z 0 ¨³ x – b > 0
‹³Å—o ( x – a )2n > 0 ¨³ ( x – b )2m + 1 > 0
—´Šœ´Êœ ( x – a )2n ( x – b )2m + 1 > 0
‹µ„ c ¨³ d ‹¹Š­¦»žÅ—oªnµ
( x – a )2n ( x – b )2m + 1 > 0 „Șn°Á¤ºÉ° x – a z 0 ¨³ x – b > 0
®¤µ¥Á®˜» ­Îµ®¦´…o°‡ªµ¤ ( x – a )2n ( x – b )2m + 1 < 0 „Șn°Á¤ºÉ° x z a ¨³ x – b < 0
š»„‹ÎµœªœÁ˜È¤ª„ m ¨³ n Ä®ošÎµÁž}œÂ f„®´—
195
101
4
˜´ª°¥nµŠš¸É 6.1.3 ‹Š®µÁŽ˜‡Îµ˜°…°Š (x 3) (x32 7)
(x 4)
d0
101
4
ª·›¸šÎµ (x 3) (x32 7) d 0 „Ș°n Á¤ºÉ° (x – 3)101 (x + 7)4 d 0 ¨³ x + 4 z 0
(x 4)
„Șn°Á¤ºÉ° (x – 3 d 0 ®¦º° x = –7 ) ¨³ x z –4
„Șn°Á¤ºÉ° (x d 3 ®¦º° x = –7 ) ¨³ x z –4
„Șn°Á¤ºÉ° x < –4 ®¦º° –4 < x d 3
—´Šœ´ÊœÁŽ˜‡Îµ˜°‡º° { x  R | x < –4 ®¦º° –4 < x d 3 }
 f„®´—š¸É 25
1. ‹Š¡·­¼‹œrªnµ a2n+1 < 0 „Șn°Á¤ºÉ° a < 0 š»„‹ÎµœªœÁ˜È¤ª„ n
2. ‹Š¡·­¼‹œrªnµ ( x – a )2n ( x – b )2m + 1 < 0 „Șn°Á¤ºÉ° x z a ¨³ x – b < 0
š»„‹ÎµœªœÁ˜È¤ª„ m ¨³ n
3. ‹Š®µÁŽÈ˜‡Îµ˜°…°Š °­¤„µ¦˜n°Åžœ¸Ê
(1) (x – 3)20 (x + 4)33 < 0
(2) (x – 4)21 (x + 2)11 > 0
(X 1) 45
t0
(X 5) 8 (X 4) 21
27
16
(4) (X 2) (X18 3) d 0
(X 4)
(3)
˜°œš¸É 6.2 °­¤„µ¦Á„¸É¥ª„´‡nµ­´¤¼¦–r
Á¦ºÉ°Šš¸É 1 °­¤„µ¦¡ºÊœ“µœ…°Š‡nµ­´¤¼¦–r…°Š‹Îµœªœ‹¦·Š
šœ·¥µ¤ 5 ­Îµ®¦´‹Îµœªœ‹¦·Š x Ä—Ç ‡nµ­´¤¼¦–r…°Š x Á…¸¥œÂšœ—oª¥ | x | „ε®œ—×¥
­° x Á¤ºÉ° x t 0
|x|= ®
°¯ - x Á¤ºÉ° x 0
‹µ„šœ·¥µ¤…oµŠ˜oœ ­µ¤µ¦™œÎµ¤µ­¦»žÁž}œ­¤´˜·Á„¸É¥ª„´‡nµ­´¤¼¦–r …°Š‹Îµœªœ‹¦·Š ¨³¡·­¼‹œrŗo—´Šœ¸Ê
„ε®œ—Ä®o x ¨³ y Áž}œ‹Îµœªœ‹¦·ŠÄ—Ç a Áž}œ‹Îµœªœ‹¦·Šª„
196
‹Š¡·­¼‹œr…o°‡ªµ¤˜n°Åžœ¸Ê
(1) x d | x |
(2) ™oµ | x | d a ¨oª – a d x d a
(3) ™oµ | x | t a ¨oª x d a ®¦º° x t a
(4) x + y d | x | + | y |
(5) x + y d | x + y |
(6) | x + y | d | x | + | y |
(7) | x | – | y | d | x – y |
(8) | | x | – | y | | d | x – y |
¡·­¼‹œr (1)
x d| x |
¡·­¼‹œr „¦–¸š¸É 1
x t0
‹³Å—o
x=|x|
„¦–¸š¸É 2
x<0
‹³Å—o
0 < –x ¨³ –x = | x |
˜n
x < –x
—´Šœ´Êœ
x<|x|
‹µ„„¦–¸š´ÊŠ2 ‹¹Š­¦»žÅ—oªnµ x d | x |
(2) ™oµ | x | d a ¨oª –a d x d a
¡·­¼‹œr Ä®o | x | d a
„¦–¸š¸É 1
x t0
‹³Å—o
x = | x | da
—´Šœ´Êœ
0 d x da
„¦–¸š¸É 2
x<0
‹³Å—o
0 < –x ¨³ –x = | x |
˜n
| x | da
—´Šœ´Êœ
0 < –x d a
­—Šªnµ
–a d x < 0
‹µ„„¦–¸š´ÊŠ 2 ‹¹Š­¦»žÅ—oªnµ –a d x d a
(3) ™oµ | x | t a ¨oª x d –a ®¦º° x t a „µ¦¡·­¼‹œršÎµÁž}œÂ f„®´—
(4) ™oµ x + y d | x | + | y | „µ¦¡·­¼‹œršÎµÁž}œÂ f„®´—
197
(5) x + y d | x + y | „µ¦¡·­¼‹œršÎµÁž}œÂ f„®´—
(6) | x + y | d | x | + | y |
¡·­¼‹œr „¦–¸š¸É 1
x + y t0
‹³Å—o
x+y=|x+y|
‹µ„…o°(4) ‹³Å—o x + y d | x | + | y |
—´Šœ´Êœ
|x+y|d|x|+|y|
„¦–¸š¸É 2
x+y<0
‹³Å—o
–( x + y ) = | x + y |
—´Šœ´Êœ
( –x ) + ( –y ) = | x + y |
‹µ„…o° (4) ‹³Å—o ( –x ) + ( –y ) d | x | + | y |
˜n
| –x | = | x | ¨³ | –y | = | y |
—´Šœ´Êœ
( –x ) + ( –y ) d | x | + | y |
œ´Éœ‡º°
|x+y|d|x|+|y|
‹µ„„¦–¸š´ÊŠ 2 ‹¹Š­¦»žÅ—oªnµ | x + y | d | x | + | y |
(7) | x | – | y | d | x – y |
¡·­¼‹œr ÁœºÉ°Š‹µ„
| x | = | (x – y ) + y |
‹µ„…o° (6) ‹³Å—o | (x – y ) + y | d | x – y | + | y |
—´Šœ´Êœ
|x|d|x–y|+|y|
œ´Éœ‡º°
|x|–|y|d|x–y|
(8) || x | – | y || d | x – y |
...c
¡·­¼‹œr ‹µ„…o° (7) ‹³Å—o | x | – | y | d | x – y |
¨³
|y|–|x|d|y–x|
—´Šœ´Êœ
–( | x | – | y | ) d | y – x |
˜n
|x–y|=|y–x|
‹¹ŠÅ—o
–( | x | – | y | ) d | x – y |
­—Šªnµ
–| x – y | d | x | – | y | …d
‹µ„ c ¨³ d ‹³Å—o –| x – y | d | x | – | y | d | x – y |
×¥ …o° (2) ‹¹ŠÅ—o
|| x | – | y || d | x – y |
198
 f„®´—š¸É 26
1. „ε®œ—Ä®o x ¨³ y Áž}œ‹Îµœªœ‹¦·Š ¨³ a Áž}œ‹Îµœªœ‹¦·Šª„ ‹Š¡·­¼‹œrªnµ
(1) ™oµ | x | t a ¨oª x d a ®¦º° x t a
(2) x + y d | x | + | y |
(3) x + y d | x + y |
2. ‹Š®µÁŽ˜ ‡Îµ˜°…°Š °­¤„µ¦ | x3 – 1 | > 1 – x
Á¦ºÉ°Šš¸É 2 °­¤„µ¦‡nµ­´¤¼¦–rš¸É¤¸ÁŠºÉ°œÅ…
Ĝ„µ¦Á¦¸¥œ¦³—´­¼Š ×¥ÁŒ¡µ³„µ¦¡·­¼‹œr ¨·¤·˜…°Š¢{Š„r´œ ‹ÎµÁž}œ˜o°ŠÄo°­¤„µ¦‡nµ­´¤¼¦–r
š¸É¤¸ÁŠºÉ°œÅ… —´Š˜´ª°¥nµŠ˜n°Åžœ¸Ê
˜´ª°¥nµŠš¸É 6.2.1 „ε®œ—Ä®o x Áž}œ‹Îµœªœ‹¦·ŠÄ—Ç
‹Š¡·­¼‹œrªnµ ™oµ | x – 4 | < 2 ¨oª | x2 + x – 20 | < 22
¡·­¼‹œr Ä®o x Áž}œ‹Îµœªœ‹¦·Š ¨³ | x – 4 | < 2
‹³Å—o –2 < x – 4 < 2
2<x<6
7 < x + 5 < 11
—´Šœ´Êœ | x + 5 | < 11
‹³Å—o | x – 4 || x + 5 | < 2 u 11
| ( x – 4 )( x + 5 ) | < 22
œ´Éœ‡º° | x2 + x – 20 | < 22
˜´ª°¥nµŠš¸É 6.2.2 „ε®œ—Ä®o x Áž}œ‹Îµœªœ‹¦·ŠÄ—Ç
‹Š¡·­¼‹œrªnµ ™oµ | x – 5 | < 1 ¨oª | x - 3 – 2 | < 1
2
x-4
¡·­¼‹œr Ä®o x Áž}œ‹Îµœªœ‹¦·Š ¨³ | x – 5 | < 1
2
‹³Å—o – 1 < x – 5 < 1
2
2
1 <x–4< 3
2
2
—´Šœ´Êœ 1 < | x – 4 | < 3
2
2
‹³Å—o 2 < 1 < 2
3 |x 4 |
199
x 3 2 = x 3 2 x 8
x 4
x 4
= x 5
x 4
x 5
=
x 4
¡·‹µ¦–µ
=
x 5
x 4
‹µ„ | x – 5 | < 1 ¨³ 1 < 2
2
—´Šœ´Êœ
|x 4 |
x 5
<1
x 4
œ´Éœ‡º° x 3 2 < 1
x 4
 f„®´—š¸É 27
„ε®œ—Ä®o x Áž}œ‹Îµœªœ‹¦·ŠÄ—Ç ‹Š¡·­¼‹œrªnµ
1. ™oµ | x – 3 | < 1 ¨oª | x2 – x – 6 | < 6
2. ™oµ | x + 4 | < 1 ¨oª | x3 + x2 – x | < 90
3. ™oµ | x – 4 | < 1 ¨oª 2x 7 1 < 1
4
x 3
3
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