2011 10 26 1. 1627 27, 682, 574, 402 = 2 7 7 7 7 7 7 7 7 7 7 7 7 2 2 7 2 7 7..., 100 100 99 2 100 100 two to the hundred n n, n 2, n 3,... n
10... 10 10 1 = 10 10 2 = 10 10 = 100 10 3 = 10 10 10 = 1000 10 12 = 1, 000, 000, 000, 000 2 1 = 2 2 2 = 2 2 = 4 2 3 = 2 2 2 = 8 2 10 = 1024 2 7 12 a, aq, aq 2,... 2, 2 7, 2 7 2,... 10 3 10 25 7 10 21 t 10 13 km 500 = 5 10 23 km 2 10 = 1024 1000 = 10 3 0.2 2 10, 1000 (M 9.0) (M 7.9) M 8.4
1 0.02g) 2 2 2 = 2 2 = 4 2 2 2 = 2 3 = 8 2 10 = 1, 024 20g) 2 20 = 1, 048, 576 21kg) 2 30 = 1, 073, 643, 872 22t) 2 40 22,000 t) 100 2 99 10 22 t % (1.1) 7 = 1.9487... (1.1) 8 = 2.1435... 7 95% 114% 12% 14%
50%, 1364% 2. n = 1, 2, 3,... 2 n 2 5 = 32 2 4 = 16 2 3 = 8 2 2 = 4 2 1 = 2 2 0 =? 2 1 =? 2 2 =? 2 3 =?? 2 3 = 8 2 2 = 4 2 1 = 2 2 0 = 1 2 1 = 1 2 2 2 = 1 4 = 1 2 2 a n a 0 = 1, a n = 1 a n m, n a m a n = a m+n (a m ) n = a mn
m n m, n 1µ = 10 3 mm = 10 6 m 0.1 mµ = 10 10 m = 0.0000000001 m 10 15 m 1.6 10 24 g 3 10 22 g 10 27 g 2 10 11 10 10 1 10 2 10 2 10 3 10 3 10 6 10 6 10 9 10 9 10 12 10 12 10 15 10 15 g b g b 2 g, b 3 g,..., n b n g
b 2 1 1/2 2 2 1/4 = 10 2 10 1/1000 ( 20 2 20 1/1000000 3. % 1/2 1/3 1/2 x (1 + x) 2 1.1 (1 + x) 2 = 1.1 1/n 1/n x (1 + x) n = 1.1 b n = a b n a a n p 1/n n 1 + p 1 m/n n (1 + p) m 1 1/3 1/ 3 2 m/n 1/ n 2 m n a x 0 x n 0 > a a > 1 x 0 = a x 1 = (n 1)xn 0 + a nx n 1 0 x 2 = (n 1)xn 1 + a nx n 1 1 x 3, x 4, x 5,... n a y = x n
(x, x n ) x n = 2, a = 2 x 0 = 2 x 1 = x2 0 + 2 2x 0 = 3 2 = 1.5 x 2 = x2 1 + 2 2x 1 = 17 12 = 1.416... x 3 = x2 2 + 2 2x 2 = 577 408 = 1.414215... x 4 2 1.4142135623... x 2, x 3, x 4 x 5 x 6 y = x n n m π a n n a n 1 = 1 0 < a < b n a < n b n a m = ( n a) m n 1/a = 1/ n a a > 1, m > n m a < n a n m a = mn a n a n = a a 1/n = n a = a 1/n, a m/n = n a m a a q q = m/n) a p a q a pq = a p+q = (a p ) q a
q = m/n a q a p a q = a p+q (a p ) q = a pq a 0 = 1, a p = 1/a p a > 1, p > q a p > a q. 4. a > 0 x y = a x a = 2 y = a x a > 1 a = 2 x = 0 y = 1 x y = a x x y = a x x x + h a x+h = a x a h a x a x (a h 1) x h a x a x (x, a x ) (x + h, a x+h ) y a x 0 < a < 1 a x x a x x x + h a x a x a = 1/b b > 1 a x = 1/b x a = 1 a x 10 % 5 % b 1 + b b/2 ( 1 + b ) 2 = 1 + b + b2 2 4 > 1 + b 1/4 b/4 1/n b/n n
b x a = 1 + b > 1 a x x x + h h a x (a h 1) h a x ah 1 h h a x h a h 1 h h a h 1 h a = 1 + b > 1, h = 1/n h = 1/n a h 1 a 1 = b n(a 1/n 1) b b a < n(a1/n 1) < b y = a x h = 1/mn 1/mn 1/n 1/n 1/mn n(a 1/n 1)a 1/n n(a 1/n 1) h h (0, 1) (h, a h ) a (0, 1) b a y = a x x = 0 a, h a h 1 h a e
n 1/n 1/n a ( 1 + 1 ) n n ( e = lim 1 + 1 ) n n n ( e = lim 1 + 1 ) n n n e e n n! = 1 2 n n 1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, 6! = 720, 7! = 5040, 8! = 40320, 9! = 362880, 10! = 3628800 e e = 1 + 1 1! + 1 2! + 1 3! + 1 4! + 1 5! + = 2.71828... n! n 1 + + 1 10! e 3 10 9 1 + + 1 8! e e 1 b a = 1 + b = e c c c a log e a 1/n n(a 1/n 1) = n(e c/n 1) = c(n/c)(e c/n 1) h = c/n c eh 1 h
h 0 c c = log e (1 + b) H h 2 h/h 2 = e c c = 0.693... log e 2 2 h/h = e ch/h 1 e ch/h h = e ch/h c H ech/h 1 ch/h ch/h c/h c H = log e 2 H = 0.693... H 137 30 365 24 60 60 0.693 30 365 24 60 60 7 10 10 3 10 12 137 137 1g 3 10 12 3