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1 Hiroshi Yuki c 005, All rights reserved. 1

2 1

3 3

4 (a + b)(a b) = a b (x + y)(x y) = x y a b x y a b x y 4

5 5

6 6

7 3 r r r r r 0 r 0 r 0 r = 0 r r r 0 r r 0 r r r 0 r 0 r r = 0 7

8 a b (a b) 0 a b 0 r 0 r a b r ( real number) x x w w a, b, c x, y, z n n OK (a b) 0 (a b) 0 8

9 (a b) = (a b)(a b) = a(a b) b(a b) = aa ab ba + bb = a ab + b (a b) 0, (a b) = a ab + b a b a ab + b 0 (a b) 0 0 ab 9

10 a + b ab a + b ab a b a b a +b a b ab a b ab a b ab = a b 10

11 ab a = b = ab = ( ) = 4 a b = ( ) = 16 = 4 ab 0 ab = a b ab 0 a +b ab a + b a b ab 0 ab 0 ab ab < 0 a = b = = a + b = + ( ) = 4 11

12 = a b = ( ) = 16 = 4 ab 0 ab < 0 a b a > 0 b < 0 c = b c c > 0 a +b ab a c a + c ac = a + c = a + ( b) = a + b c = b 1

13 = ac = a c = a ( b) = a b ac > 0 c = b a + b a b a > 0 b < 0 a b a b a b a + b a b a b a b a b a x b y x y 0 x 0 y 0 x + y xy x 0 y 0 13

14 x+y xy 0 r 0 4 ( x y ) 0 ( ) ( ) ( ) x y = x x y + y = x x y + y = x xy + y 0 x xy + y 0 14

15 xy x + y xy x 0 y 0 x 0 y 0 x y x + y = xy x y x y 0 x = 0 y = x = y x = 3 y = 3 x+y = 3+3 = 3 xy = 3 3 = 3 x = y = x+y = ( )+( ) = xy = ( ) ( ) = x 0 y 0 15

16 (a b) 0 a = b x = y x + y x y

17 (x + y) 1 (x y) 1 (x 0, y 0) (x 1 + x + + x n ) 1 n (x 1 x x n ) 1 n (x k 0) ( n k=1 x k ) 1 n ( n k=1 x k ) 1 n (x k 0) x = y 5 17

18 1 18

19 19

20 0

21 6 1

22

23 1 3 3

24 4

25 PDF URL Copyright (C) 005 Hiroshi Yuki ( ) All rights reserved L A TEX 5

(a + b)(a b) = (a + b)a (a + b)b = aa + ba ab bb = a 2 b 2 (a + b)(a b) a 2 b 2 2 (1 x)(1 + x) = 1 (1 + x) x (1 + x) = (1 + x) (x + x 2 ) =

(a + b)(a b) = (a + b)a (a + b)b = aa + ba ab bb = a 2 b 2 (a + b)(a b) a 2 b 2 2 (1 x)(1 + x) = 1 (1 + x) x (1 + x) = (1 + x) (x + x 2 ) = 2005 0 (a + b)(a b) = (a + b)a (a + b)b = aa + ba ab bb = a 2 b 2 (a + b)(a b) a 2 b 2 2 ( )( + ) = ( + ) ( + ) = ( + ) ( + 2 ) = + ( ) 2 = 2 (a + b)(a b) = a 2 b 2 ( )( + ) ( + ) ( + + 2 ) http://www.hyuki.com/story/genfunc.html

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