1991 707 708 1972 36 1990 2
126 130
21
1656 1742 1705
1972 523
334-420 342-423
1433 1435
1975 205
= + = +
= 1 2 ( ) 2 2 = 2
2 2 2 2 2 2 2 = 1 4 [ + ( ) ] 2 1 2 2 2 2 2 2 ( ) = = 2 2 2 2 2 2 2 2 2 + 2 = = ( ) 2 1 ( )( )( )( ) 4 a + b + c b + c a c + a b a + b c
1957 62
(1) = 1 2 (2) = 1 3 D B Wagner AnEarlyChineseDerivationoftheVolumeofaPyramid LiuHui ThirdCenturyA.D. HistoriaMathematica 1979 164-188
= 1 6 ( + + ) 1 8
1 8 1 8 = 4 1 8
730 3.1466 232 142 10 3.1622 (228 266) 3.1556 45 10
S 3 14 64 192 = 3.141024 625 = 3.14 = 157 50 3927 3.1416 1250 = 22 3.14 = 355 3.1415929 7 113 = 22 7 π = 355 1000 113
= 355 113 355 113 1 3 sinx = x x + 1 5! x 5 3! = 1+ 1 2 + 1 2 2 π 3 2 + 3 4 3! 4 5!
2 P 2 πα 1 2 1 (1 3 4 2 e 2 2 e ) 2 2 4 e = a 2 2 2 b 2 a b e a 2 1 1 1 1 3 (1 2 2 2 π 2 2 2 4 )
3 70 2 5 7 1(mod3) 3 3 21 1 5 7 1(mod5) 5 3 15 1 5 7 1(mod7) 7 1592 A B m A B m A R modm m
M 3 5 7 R1 k1 = R 1 2 R 1(mod 3) 3 3 M R2 k 2 = R 2 1 3 5 7 R 2 (mod 5) 5 5 M R3 k 3 = R 3 1 3 5 7 R 3(mod 7) 7 7 R1 2 3 5 7 + R2 1 3 5 7 3 5 + R3 1 3 5 7 R 1(mod 3) 7 R (mod 5) 2 R (mod 7) 3 3 5 7 R1 2 + R2 1 3 5 7 3 5 + R3 1 3 5 7 pm R 1(mod 3) 7 R (mod 5) 2 R (mod 7) M ki 1(mod a i )(i 1 2 n) a i N ( R k M R k M R k M R k M 1 1 + 2 2 + 3 3 + + n n ) pm a a a a 1 2 3 (P M = a a a ) 3 1 2 3 n
M k 1(mod a ) k i i i a i M a i ai G i gi ai a i N a N 60 60 an an R1 mod60
4617 p l35 mod1728 p x 4617 p l2=q2 l3=q3l2+1 l4=q4l3+l2 ln=qnln-1+ln-2 r1=ai-giq1=ai-c1gi
g i-r1q2=gi-(ai-c1gi)q2=c2gi-l2ai
N r 1-r2q3=(ai-c1gi)-(c2gi-l2ai)q3=l3ai-c3gi = 1 n n rn-1=ln- 1ai-cn-1gi rn=cngi-lnai=1 cngi 1 modai k5 ai ti; m=t1 t2 tn ai ti ti ki
2 x y xy z + xyz = 0 ( 1) 2 x x y + z + xz = 0 ( 2) 2 2 2 x + y z = 0 ( 3)
n n( n 1) f x nh f x f x 2 ( + ) = ( ) + ( ) + f( x) 1! 2! n + + f ( x) ( h > 0) f( x) = f( x ) f( x ) 2 1 2 f ( x) = f( x ) f( x ) 2 1 n n n f( x) = 1 1 f( x ) f ( x ) 2 1
2 s 1 + 2 s s f( n! + s) = f( n!) + ( ) + ( 1 2 ) 2 ( 1 2) l 2 l 2l s f( t + s) = f( t) + s + + s( 2 ) ( 1 2 1 2 1 2 )( l l s < ) l + l l l 1 2 1 l + l l l 1 2 1 2 1 2 F(t) = f(t) = a + bt + ct 2 t 1 2 1884-1956 1927-1947
f(n) = n + 1 2! n(n 1) + 1 2 3 n(n 1)(n 2) 3! + 1 4 n(n 1)(n 2)(n 3) 4! n r = 1 n(n + 1) 1 2 n 6 (c a) s = ab + (a +1)(b+1) + + cd = n (2b + d)a+ (2d + b)c + n (c a) 6 6 xn N
n 1 2 2 2 2 s = 1 + 2 + 3 + + n = ( n + 1)( n + ) (1) 3 2 n d a s = a + a + + a + + + d = a + d + ad + 2 2 2 2 2 2 ( 1) ( 2) ( ) (2) 3 2 n( n + 1) 1 s = 1+ 3+ 6+ 10+ + = n( n + 1)( n + 2) (3) 2 6 n 2 r r 1 n r( r + 1) n( n + 1)( n + 2) = 1 2! 3! n 1 1 1 2 1 1 2 1 p r r r r p n n n n p! ( + )( + ) ( + ) = ( + )( + ) ( + ) ( p + 1)! ( p = 1 2 3 6) (1) n 2 1 r = n( n + 1)( 2n + 1) 1 3! n 1 2 1 1 1 1 2 1 1 1 1 3! r( r )( r ) n 3 1 2! r( r ) r n + + = + + 3 1 2! r( r + ) n 1 1 2! r (r + 1) r n 1 1 2 1 1 p! r( r + )( r + ) ( r + p ) r 1 = n( n + 1)( n + 2) ( n + p)[( p + 1) n + 1] ( p + 2)! n 1
L r S = 1 2 Lr
n lims = S 6 2 n n lim[ S + 2( S 1 s )] = S n n n+ n Lr = 2S S = 1 2 Lr 314 S S = 1 2 Lr 3 4 d S = 1 12 l S = 157 200 d S = 25 2 2 2 314 l 2 314 4 S = 1 25 2 Lr 628 8 3927 1250 = 3.1416 25 V y = 1 abk (1) 3 V b = 1 abh (2) 6
V V y b = 2 1 (3) 1 abh (1) (2) 2 3 (3) (3) 4 1 (3) 4 3 4 1 3 (3) (3) + 4 4 1 3 1 1 n 4 4 4 4 1 1 (3) 4 4 = 0 n lim n n
1 2 3 3 r (r ) r 3 3 2 3 d (d ) 3 π 2 3 π 3 V = d = d 4 3 6 1 = 3 d 3 2
2 n+ 1 ah h h ah 2 ax dx n + 1 = 0 n + 1 h h h h 0 1 0 2 0 n 0 1 2 2 n 2 n a xdx + a x dx + + a x dx = ( a x + a x + a x ) dx BD = 1 2 BC = 1 DE = 1 2 4 DC = 1 EF = 1 6 EC = 3 FG = 1 2 8 FC = 3 5 AB 2 4 4 6 2 4 6 8 P PR BC Q x = AP PQ = PR QR = 1 = 1 2 x + 1 3 2 6 1 x 2 x 4 + + x 2 4 2 4 6 S = 1 ABCD 2 1 1 1 1 3 1 + + + 2 4 5 2 4 6 7 = 4 4( 1 1 + 1 1 1 1 + + ) 2 3 2 4 5 2 4 6 7 n
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602 596 600
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1975 786
1958 36 1983
1982 1958 6 1988 317-311
1916 484 841 1177 1122 1066 1057 1027
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1983 1963
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