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趣味の写真を投稿していきます。昆虫好きな長男と一緒に昆虫を追いかけています。最初の年はセミやカマキリ、次の年はカブトムシ、トンボ、そして今年は…

(a+b)(a-b)=a^2-b^2 型

数学・幾何学

カテゴリー[ 昆虫| 田園| 花| 街| 数学・幾何学| 寺院| 城| 祭り| 鉄道| 海| 風力発電]

イメージ 1

20190120日 雨 8⇔5 北西2

 

2変数の和と差の積が、2変数のそれぞれの2乗の差(要、引く順番)に等しいことを示した公式です。

 

( \alpha - \beta ) ( \alpha + \beta ) = \alpha ^ {2} - \beta ^{2}

 

実際の数(自然数)を入れた例を示します。特に「これなら、暗算でいける！」という例です。

 

101 \times 99 = ( 100 + 1 ) ( 100 - 1 ) = 100 ^ {2} - 1^{2} = 9999
102 \times 98 = ( 100 + 2 ) ( 100 - 2 ) = 100 ^ {2} - 2^{2} = 9996
103 \times 97 = ( 100 + 3 ) ( 100 - 3 ) = 100 ^ {2} - 3^{2} = 9991

 

91 \times 89 = ( 90 + 1 ) ( 90 - 1 ) = 90 ^ {2} - 1^{2} = 8099
92 \times 88 = ( 90 + 2 ) ( 90 - 2 ) = 90 ^ {2} - 2^{2} = 8096
93 \times 87 = ( 90 + 3 ) ( 90 - 3 ) = 90 ^ {2} - 3^{2} = 8091

 

81 \times 79 = ( 80 + 1 ) ( 80 - 1 ) = 80 ^ {2} - 1^{2} = 6399
82 \times 78 = ( 80 + 2 ) ( 80 - 2 ) = 80 ^ {2} - 2^{2} = 6396
83 \times 77 = ( 80 + 3 ) ( 80 - 3 ) = 80 ^ {2} - 3^{2} = 6391

 

71 \times 69 = ( 70 + 1 ) ( 70 - 1 ) = 70 ^ {2} - 1^{2} = 4899
72 \times 68 = ( 70 + 2 ) ( 70 - 2 ) = 70 ^ {2} - 2^{2} = 4896
73 \times 67 = ( 70 + 3 ) ( 70 - 3 ) = 70 ^ {2} - 3^{2} = 4891

 

61 \times 59 = ( 60 + 1 ) ( 60 - 1 ) = 60 ^ {2} - 1^{2} = 3599
62 \times 58 = ( 60 + 2 ) ( 60 - 2 ) = 60 ^ {2} - 2^{2} = 3596
63 \times 57 = ( 60 + 3 ) ( 60 - 3 ) = 60 ^ {2} - 3^{2} = 3591

 

51 \times 49 = ( 50 + 1 ) ( 50 - 1 ) = 50 ^ {2} - 1^{2} = 2499
52 \times 48 = ( 50 + 2 ) ( 50 - 2 ) = 50 ^ {2} - 2^{2} = 2496
53 \times 47 = ( 50 + 3 ) ( 50 - 3 ) = 50 ^ {2} - 3^{2} = 2491

 

41 \times 39 = ( 40 + 1 ) ( 40 - 1 ) = 40 ^ {2} - 1^{2} = 1599
42 \times 38 = ( 40 + 2 ) ( 40 - 2 ) = 40 ^ {2} - 2^{2} = 1596
43 \times 37 = ( 40 + 3 ) ( 40 - 3 ) = 40 ^ {2} - 3^{2} = 1591

 

31 \times 29 = ( 30 + 1 ) ( 30 - 1 ) = 30 ^ {2} - 1^{2} = 899
32 \times 28 = ( 30 + 2 ) ( 30 - 2 ) = 30 ^ {2} - 2^{2} = 896
33 \times 27 = ( 30 + 3 ) ( 30 - 3 ) = 30 ^ {2} - 3^{2} = 891

 

21 \times 19 = ( 20 + 1 ) ( 20 - 1 ) = 20 ^ {2} - 1^{2} = 399
22 \times 18 = ( 20 + 2 ) ( 20 - 2 ) = 20 ^ {2} - 2^{2} = 396
23 \times 17 = ( 20 + 3 ) ( 20 - 3 ) = 20 ^ {2} - 3^{2} = 391

 

11 \times 9 = ( 10 + 1 ) ( 10 - 1 ) = 10 ^ {2} - 1^{2} = 99
12 \times 8 = ( 10 + 2 ) ( 10 - 2 ) = 10 ^ {2} - 2^{2} = 96
13 \times 7 = ( 10 + 3 ) ( 10 - 3 ) = 10 ^ {2} - 3^{2} = 91
LibreOffice 数式(Math) のソース

 

( %alpha + %beta ) ( %alpha - %beta ) = %alpha ^ { 2 } - %beta ^{ 2 }

 

101 times 99 = ( 100 + 1 ) ( 100 - 1 ) = 100 ^ { 2 } - 1 ^{ 2 } = 10000 - 1 = 9999