4 4.0 2000 ) 9 81
10 4.1 natural numbers 1, 2, 3, 4, 4.2, 3, 2, 1, 0, 1, 2, 3, integral numbers integers 1, 2, 3,, 3, 2, 1 1 4.3 4.3.1 ( ) m, n m 0 n m 82
rational numbers m 1 ( ) 3 = 3 1 4.3.2 3 5 = 2 5 3 = 5 3 1 2 1 2 83
4.3.3 1 4.4 ( ) c a, b a 2 + b 2 = c 2 a = 1, b = 1 c c 2 = 2 3 3 84
c 2 = 2 c 2 = 2 2 2 irrational numbers real numbers 4 4.5 4.5.1 5 + x = 3 c 2 = 2 4 I p.457 467 85
x = 3 5 3 5 5 + x = 3 x = 2 5 + x = 3 3x = 5 5 3 3x = 5 yes ax + b = 0 5 a, b x = b a 5 2x + 3 x 2 2 3 = 5 86
ax + b = 0 a 0 a, b 4.5.2 ax 2 + bx + c = 0 ( a 0) c 2 = 2 x 2 2 = 0 x 2 + 1 = 0 2 0 ax + b = 0 ax + b = 0 ( a 0) 87
x 2 + 1 = 0 4.6 4.6.1 1 4 = 0.25 0.375 = 375 1000 = 3 8 1 3 = 0.33333 0.25 0.33333 (1) 38 111 (2) 1 7 (3) 53 165 10 88
6 7 4.6.2 0.101001000100001 6 0 < = r < b 7 10 x = 0.9999 10x = 9.9999 9x = 9 x = 1 1 = 0.9999 89
8 0.101001000100001 ( ) ( ) 4.7 2 ( ) a > = 0, b > = 0 a < b a < b 1 < 2 < 2 8 90
2 1 2 2 = 1. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 2 1.0, 1.1, 1.2,, 1.9 (1.0) 2 = 1, (1.1) 2 = 1.21,, (1.4) 2 = 1.96, (1.5) 2 = 2.25 1.4 < 2 < 1.5 2 = 1.4 2 = 1.41421356 6 0 9 2 2 4 2 91
2 = 3 = 5 = 1. 1. 2. 4 1 4 2 1 3 5 7 3 2 0 5 0 8 2 3 6 06 7 6 9 7 ( ) 1, 2, 3, 0 { 1, 2, 3, 2 3 2, π 1 2 = 2 2 1 2 = 1.41421356 2 = 0.707106 14 1 3 1 2 = 1.414 2 1 3 + 1 3 = 1.7 2.7 2 92
1 3 1 = 3 + 1 2 = 1.7 + 1 2 = 1. 1 ( ) ( ) 46 7 a a + 1 a (01 ( )) 4.8 4.8.1 ( ) 0 ( 0 9 9 0 0 a 0 86 ax + b = 0, (a 0) a 0 93
6 3 6 6 3 0 6 0 6 0 0 6 0 a b = x b x = a a b x b x a 0 0 x = a x 0 x = 0 a 0 x a = 0 0 x = 0 0 10 a b = x b + x = a a b a + ( b) a b = a + ( b) b b a b = a 1 b 10 0 0 0 94
( ) 1 a, b a + b = b + a, ab = ba 2 a, b, c (a + b) + c = a + (b + c), (ab)c = a(bc) 3 a a + 0 = a, a 1 = a 11 4 a a + ( a) = 0, a 1 a = 1 ( a 0 ) 12 5 a, b, c a(b + c) = ab + ac 13 11 23 1 a = a 0 + a = a, 1 a = a 0 + a = a + 0 = a a + 0 = a 12 a + ( b) = a b a + ( a) = a a 13 (a + b)c = ac + bc 24 a(b + c) = ab + ac (a + b)c = ac + bc 95
15 a 0 a = 0 a = b = a c = b c 2 3 0 + 0 = 0 (0 + 0) a = 0 a 0 a + 0 a = 0 a 0 a 96
0 + 0 = 0 (0 + 0) a = 0 a 0 a + 0 a = 0 a 0 a 0 a = 0 14 ( ) 25 ( 1)a = a 1 + ( 1) = 0 26 ( 1) ( 1) = 1 1 + ( 1) = 0 ( ) (1) ab = 0 a = 0 b = 0 (2) a 0 b 0 ab 0 (1) 2 (2) 2 1 (3) ( ) 97
4.8.2 ( ) a a > 0, a = 0, a < 0 ( ) a > 0 a a < 0 a ( ) 0 a b 0 ( ) a b > 0, a b = 0, a b < 0 a > b, a = b, a < b 15 16 a > b a = b a > = b a < b a = b a < = b 17 ( ) 14 0 a 15 a > b a b b a 16 a = b = a c = b c (a c) (b c) = a c b + c = a b a b = 0 (a c) (b c) = 0 a c = b c 17 a > = b a b a < = b a b 98
( ) a > 0, b > 0 a + b > 0, ab > 0 ( ) 1 a > b, b > c a > c 2 a > b a + c > b + c, a c > b c 3 a > b, c > 0 ac > bc, a c > b c 4 a > b, c < 0 ac < bc, a c < b c 5 a > b, c > d a + c > b + d 6 a > = b a < = b a = b 2 3 4 1 a > b, b > c a b > 0, b c > 0 (a b) + (b c) > 0 a c > 0 a > c 3 a > b a b > 0 (a b)c > ac bc > 0 ac > bc 1 c 0 1 c = 0 c 1 = 1 0 c < 0 c 1 < 0 c > 0 1 c > 0 18 a c = a 1 c 19 a c > b c ( ) 27 a > 0 a < 0 28 18 A A A 1 c = 0 B 1 c < 0 C 1 > 0 c 19 99
( ) a a 2 0 3 4.8.3 20 O E O E E O O E O 0 E 1 OE 21 O 20 21 OE OE 1cm OE 1cm cm m cm m 100
1 1 22 ( ) P x P P(x) O ( ) 47 A, B, C D(2.5), E( 2) D, E -2-1 0 1 2 3 4 C B A 4.8.4 ( ) a { a ( a > a = = 0 ) a ( a < 0) a ( ) (1) 1 (2) O a A a 22 1 1 101
(3)? a 2 { a ( a > a 2 = = 0 ) a ( a ) ( ) a a = a 2 a 2 = a 2 ( ) 3 = 3, 4 = 4, 0 = 0 ( ) 48 (1) 5 (2) 2.4 (3) 3 4 x = 2 x x2 = 2 x 2 = 4 x = ±2 x = 2 x 2 2 2 ( ) 102
( x = a ) a > = 0 x = a x ±a x = a x x = a x = ±a 0 a < 0 ( ) 16 x 2 = 3 x 2 = ±3 x 2 = 3 x 2 = 3 x 2 = ±3 x = 5, 1 ( ) ( ) 49 (1) x + 3 = 5 (2) 3x 1 = 6 ( ) 1 a > = 0 1 a = 0 a = 0 2 ab = a b, a = a b b b 0 3 a = a 4 a + b < = a + b 2 ab = (ab) 2 = a 2 b 2 = a 2 b 2 = a b 103
( ) 29 4 4 4 ( ) ( ) 2 A(a) B(b) d d = b a ( ) AB b a a b 3 a b = (a b) = a + b = b a ( ) A(2) B(5) d C(3) D( 1) d d = 5 3 = 2 = 2 d = 1 3 = 4 = 4 ( ) 50 A(3) B(9) C( 3) AB BC CA 4.9 2 19 20 104
1961 1 ) ( 1961 815 1972 43 1978 1976 105
1976 1968 106