3 10 = 0.3 17 100 = 0.17 257 1000 = 0.257 2
3 I 1 2 2 3 3 4 2 5 5 6 4 7 7 8 4 9 3 10 6 11 7 12 11 16 8 25 31 40 17 250 1 a
1 a 1 3 2 = 0.5 = 0.75 = 0.4 2 4 5 7 3 11 8 = 0.875 10 = 0.3 16 = 0.6875 8 31 17 25 = 0.32 40 = 0.775 250 = 0.068 4
1 b 1 2 3 4 2 5 7 8 3 10 11 16 8 25 31 40 17 250 2 = 2 4 = 2 2 5 = 5 8 = 2 2 2 10 = 2 5 16 = 2 2 2 2 25 = 5 5 40 = 2 2 2 5 250 = 2 5 5 5 5
1 c 2 = 2 4 = 2 2 5 = 5 8 = 2 2 2 10 = 2 5 16 = 2 2 2 2 25 = 5 5 40 = 2 2 2 5 250 = 2 5 5 5 2 5 6
2? 7 32 1 1250 3 20 5 36 1 420 147 625 7
1 3 3 = 0.333 74 = 0.0405405405 8
1 II 1 2 3 10 2 3 6 11 3 4 7 12 2 5 11 16 5 6 8 25 4 7 31 40 7 8 4 9 17 250 9
1 1 2 2 3 3 4 2 5 5 6 4 7 7 8 4 9 3 10 6 11 7 12 11 16 8 25 31 40 17 250 a 10
1 a 2 3 0.6666 6 5 6 0.8333 3 47 0.57142857 571428 4 9 0.4444 4 6 11 0.545454 54 7 0.583333 3 12 11
1 b 1 3 = 0.333 3 74 = 0.0405405405 1 3 = 0.3. 3 74 = 0.0405.. 12
1 b 2 3 = 0.6666 = 0.6. 5 6 = 0.8333 = 0.83. 4 7 = 0.571428571428.. = 0.517428 13
1 b 4 9 = 0.4444 = 0.4. 6 11 = 0.5454 = 0.54.. 7 12 = 0.583333 = 0.583. 14
7 2 7 22 = 0.318.. 22 0.3181 22 7.000 6 6 40? 22 180 176 4 422 0 18 15
3 a c 7 9 = 5 0.7. b 12 = 0.416. 5 6 2 = 2.45.. d 5 = 5.162.. 11 37 16
III 1 1 7 = 0.142857.. 2 7 = 0.285714.. 3 a 7 =.. 4 b 7 =.. 0.428571 0.571428 5 c 7 =.. 6 0.714285 d 7 =.. 0.857142 17
2 1 3 11 = 0.27.. 11 = 0.09.. 2 11 = 0.18.. a 4 11 =.. b 5 11 =.. c 6 11 =.. 0.36 0.45 0.54 d 7 11 =.. e 8 11 =.. f 9 11 =.. 0.63 0.72 0.81 g 10 11 =.. 0.90 18
19
a a b > 0 b a a b b 0 1 2 3 b 1 20
0 a b a b 21
a b b 1 0 a b 22
2 5 23
a b a b > 0 b = 2 m 5 n m n 0 a a b = 2 m 5 a 2 n = n 5 m 2 m 5 n 2 n 5 m 2 = n 5 m a ( 2 5 ) m+n = 2n 5 m a 10 m+n a b 24
2 5 25
(Contrapositive) a 2 5 a ( ) a ( b b 10 b m > 0 m ( 2 5 ) m ) 26
a = b ( 2 5 ) m = c d 2 5 d 27
1 3 = 0.3. 1 7 = 0.142857.. 28
1 6 = 0.16. 7 22 = 0.318.. 29
1 IV 4 9 7 24 7 12 9 70 4 15 8 27 1 22 7 36 5 21 10 33 30
1 IV a 4 9 7 24 7 12 9 70 4 15 8 27 1 22 7 36 4 5 9 8 10 21 27 33 5 21 10 33 31
1 IV b 4 9 7 24 7 12 9 70 4 15 8 27 1 22 7 36 7 4 9 12 1 15 7 7 22 24 70 36 5 21 10 33 32
2 4 5 8 10 9 21 27 33 7 4 1 7 9 7 12 15 22 24 70 36 33
2 a 4 5 8 10 9 21 27 33 9 = 3 3 21 = 3 7 27 = 3 3 3 33 = 3 11 34
2 b 7 4 1 7 9 7 12 15 22 24 70 36 12 = 2 2 3 15 = 3 5 22 = 2 11 24 = 2 2 2 3 70 = 2 5 7 36 = 2 2 3 3 35
3 9 = 3 3 21 = 3 7 27 = 3 3 3 33 = 3 11 12 = 2 2 3 15 = 3 5 22 = 2 11 24 = 2 2 2 3 70 = 2 5 7 36 = 2 2 3 3 36
9 = 3 3 21 = 3 7 27 = 3 3 3 33 = 3 11 3 a 3 7 11 37
12 = 2 2 3 15 = 3 5 22 = 2 11 24 = 2 2 2 3 70 = 2 5 7 36 = 2 2 3 3 3 b 2 3 5 7 11 2 5 38
3 a b c 3 7 11 2 3 5 7 11? 2 5 39
1 V. x = 0.2 = 0.222 10x = 2.222 10x x = 2.222 0.222 9x = 2 x = 2 9. 0.2 = 2 9 40
1 V.. y = 0.13= 0.1313 100y = 13.1313 100y y = 13.1313 0.1313 99y = 13 y = 13 99.. 0.13 = 13 99 41
1 a. 0.4 4.. 0.23 23 9... 99 0.5 5 9 0.37 37 99? 9 99 ( / ) 2 5 42
1 a b. 0.4 4.. 0.23 23 9... 99 0.5 5 0.37 37 9. 7.. 25.. 311 0.7 = 9 0.25 = 99 0.311 = 999 99 43
2. x = 0.12 = 0.1222 10x = 1.222 100x = 12.222 100x 10x = 12.222 1.222 90x = 11 x = 11. 90 0.12 = 11 90 44
2.. y = 0.249 = 0.24949 10y = 2.4949 1000y = 249.4949 1000y 10y = 249.4949 2.4949 990y = 247 y = 247.. 990 0.249 = 247 990 45
2 a. 0.25 23.. 0.320 317 90... 990 0.45 41 90 0.431 427 990? 90 990 ( / ) 2 5 46
2 a b. 0.25 23.. 0.320 317 90... 990 0.45 41 0.431 427 90 990. 61.. 1111.. 4133 0.67 = 90 0.1112 = 9990 0.4174 = 9900 47
2 5 48
.. x = 0.18.. 100x = 18.18.. 100x x = 18.18 0.18.. 99x = 18 x = 18 99 = 2 11 49
.. 0.18 = 2 11 9 99 999 2 5 50
2 5 51
.. x = 0.312.. 10x = 3.12.. 1000x = 312.12.. 1000x 10x = 312.12 3.12.. 990x = 309 x = 309 990 = 103 330 52
.. 0.312 = 103 330 90 990 9990 900 9900 99900 2 5 53
( Continuous )? 1 3 2 ( 1 8 3 2 4 ) 00 1 2 1 13 3 2 2 ( 1 42 + 4 1 5 54
( Continuous )? 2 1 0 1 2 3 4 5 55
( Continuous )? 0 1 2 3 4 5 56
( Continuous )? 0 1 2 3 4 5 57
( Continuous )? 0 1 2 3 4 5 58
( Continuous )? 0 1 2 3 4 5 59
( Continuous )? 0 1 2 3 4 5 60
( Continuous )? 0 1 2 3 4 5 61
( Continuous )? 0 1 2 3 4 5 62
( Continuous )? 0 1 2 3 4 5 63
( Continuous )? 0 1 2 3 4 5 64
( Continuous )? 0 1 2 3 4 5 65
( Continuous )? 0 1 2 3 4 5 66
( Continuous )? 0 1 2 3 4 5 67
( Continuous )? 0 1 2 3 4 5 68
( Continuous )? 0 1 2 3 4 5 69
( Continuous )? 0 1 2 3 4 5 70
( Continuous )? 0 1 2 3 4 5 71
( Continuous )? 2 = 1.41421356 0 1 2 3 4 5 2 72
( Rational Numbers ) ( Irrational Numbers ) 2 ( π ) 1 ( e = lim ( 1 + ) n ) n n 73
I II III IV V