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 1 2 2 + [ ( )] 4 2 1 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
1 6 D. B. wagner An Early Chinese Derivation of the Volume of a Pyramid Liu Hui Third Century A. D. Historia Mathematica 1979 164 188
1 8 1 8 1 8 4 1 8
730 232 10 3.1466 3.1622 228 266 142 45 10
S192 3.14 64 3.141024 625 157 3.14 50 3927 3.1416 1250
22 355 3.14 3.1415929 7 113 22 7 355, 1000 113 355 113 355 113
1 1 3 5 sinx = x x + x 31 5! 2 2 2 π 1 1 3 = 1+ + 2 + 3 4 3! 4 5
2 1 2 1 3 4 P = 2π a( 1 2 e 2 2 e ) 2 2 4 2 2 2 a b e 2 a b e a 2 1 1 2 1 1 1 3 2 2 2 π = ( 2 2 4 )
1592
N = ( R k M R k M R k M R k M 1 i + 2 2 + 3 2 + + n n n ) pm a i a 2 a 2 a N a N 60 60 an an R1 mod60 145 4617 p 135 mod 1728 P x 4617 p
G i ai ai Gi gi ai Μ Μ Μ Μ
l2 q2 L3 q3l2 1 L4 q4l3 L2 Ln qnln-1 ln-2 r1=ai-giq1=ai-c1gi, r2=gi-r1q2=gi-(ai-c1gi)q2=c2g4-l2a4,r3=r1-r2g4=(ai-c1gi)-(c2gi-l2ai)q3=l3ai-c3gi, rn 1 n n rn-1=li-1 rn=cngi-liai=1 cngi(modai) Cn K3
ai ti m t1 t2 tn ai ti ti ki
1884 1956 1927 1947 xn N
Μ Μ Μ Μ Μ Μ Μ
Μ Μ Μ Μ Μ Μ Μ 2 x y xy z xyz = 0, ( 1) 2 x x y + z + xz = 0, ( 2) 2 x + y z = 0, ( 3)
n n( n 1) 2 f( x + nh) = f( x) + f( x) + f ( x) 1! 2! n + + f( x) ( h 0)
2 s 1 + 2 s s f( nl + s) = f ( nl) + ( ) + ( 1 2 ) 2 ( 1 2 ) l 2 l 2l 1 + 2 1 2 f( t + s) = f( t) + s + s( ) l1 + l 2 l 1 l 2 s 1 2 ( ) ( l 1 l 2, s 1) l + l l l 1 2 1 2
f( t) F( t) = = a + bt + ct 2 t 1 2 1 3 f( n) = n + n( n 1) + n( n 1)( n 2) 2! 3! 1 4 + n( n 1)( n 2), 4! a a2 a3 b b2 b3 ab a2b ab2 f t 1964 191 197 1957 136 137
n 1 r = n( n + 1) 1 2 n c a 6 n n s = ab + ( a + 1)( b + 1) + + cd = [( b + d ) a + ( d + b ) c ] + ( c a) 6 2 2 6 2 2 2 n 1 s = 1 + 2 + 3+ + n = ( n + 1)( n + ) ( 1) 3 2 2 2 2 2 s = a + ( a + 1) + ( a + 2 + + d n d a = a + d + ad + ) 2 2 ( ) ( 2) 3 2 n( n + 1) s = 1+ 3+ 6+ 10 + + 2 1 = n( n + 1)( n + 2) ( 3) 6
n 2 r r 1 n r( r + 1) n( n + 1)( n + 2) =, 1 2! 3! n 1 1 2 1 p! r( r + )( r + ) ( r + p ) 1 = n( n+ )( n+ ) ( n + p) ( p + 1)! ( p = 1 2 3 6) ( 1) n 1 2 r = n( n + 1)( 2n + 1) 1 3! n n n 1 2 1 1 1 1 2 1 1 1 1 3! r( r + )( r + ) = 3 1 2! r( r + ) r + 3 1 2! r( r + ) n 1 1 2! r r 1 r n 1 1 1 2 1 2 1 p r r r r p r n n n n p! ( + )( + ) ( + ) = ( + )( + ) ( + ) ( p + 2)! [( p + 1) n + 1] ( p = 1 2 3 4 5) ( 2) n 1
L r S Lr 1 2
lim S S 6 2 n n n lim S 2 S S S n n n 1 n 1 2 Lr 1 314 S S Lr 2 3 2 3.14 S d S 4 1 2 157 2 25 2 S d S 12 200 314 314 4 1 S Lr 628 25 2 5 25 3927 1250 3.1416
Vr = 1 abk 1 3 V = 1 b abh 2 6 1 2 abh 1 2 3 3 4 1 3 3 4 3 1 3 3 4 4
3 4 1 4 3 4 1 4 7 1 n 3 1 4 n lim n n 1 4 0
1 3 2 2 r r r 3 3 2 3 d d 3 π 2 3 π 3 V = d = d 4 3 6 1 3 3 d 2
2 n+ 1 ah h h n ah ax dx n + 1 = 0 n + 1 h h h h n 2 n a 1xdx + a 2xdx + + a nx dx = ( a1x + a 2x + a nx ) dx 0 0 0 0 1 1 1 1 1 3 1 BD = BC = DE = DC = EF = EC = FG = FC = 2 2 4 2 4 6 2 4 6 8 3 5 AB P PR BC Q x = AP 2 4 6 8 1 1 1 3 2 2 2 4 6 PO = PR QR = 1 1 x = x + x + x + + x 2 2 4 2 4 2 4 6 1 1 1 1 1 1 S ABCD = + + + 2 3 2 4 5 2 4 6 7 π = 4 4 1 1 1 1 1 1 ( + + + ) 2 3 2 4 5 2 4 6 7
í
è ù
á
á
1972 523
1984 1 34
í
í
1963 120 121
1989
ì
1971
ù
1972 154
1972 206 1971 182
1972 523
á á à
ò
á
1990 7
á 1587 1586
602 596 600
á
95
TA SR = TP = AP SN RN = PR NC h b d l = = b a1 a 2 a 1 bl h = + b a2 a1
d a l a a = 1 2 1
ù
1975 786
1958 36 1983
1982 1958 6
1988 317 331
1976 484
841 1177 1122 1066 1057 1027 1990
1958 1980 79
1957
21 1992 1983 1963
1980 105 1934
1982
1958 9
ó ù í
í é í
á
ú 1980
391
à
1977 469
á
1958
623 632 702
1975 318
í ì ì ú
à
ì
ó
é
18 3 100 = 1 54 100
ù
í
à 1991 622