Algebra

  1. Identical equation

    ( a ± b ) 2 = a 2 ± 2 a b + b 2 ( a ± b ) 3 = a 3 ± 3 a 2 b + 3 a b 2 ± b 3 ( a ± b ) 4 = a 4 ± 4 a 3 b + 6 a 2 b 2 ± 4 a b 3 + b 4 ( a + b + c ) 2 = a 2 + b 2 + c 2 + 2 a b + 2 a c + 2 b c ( a + b - c ) 2 = a 2 + b 2 + c 2 + 2 a b - 2 a c - 2 b c ( a - b - c ) 2 = a 2 + b 2 + c 2 - 2 a b - 2 a c + 2 b c ( a + b + c ) 3 = a 3 + b 3 + c 3 + 6 a b c + 3 ( a 2 b + a b 2 + b 2 c + b c 2 + a c 2 + c a 2 ) ( a 1 + a 2 + ... + a n ) 2 = a 1 2 + a 2 2 + ... + a n 2 + 2 ( a 1 a 2 + a 1 a 3 + ... + a n-1 a n ) a 2 - b 2 = ( a - b ) ( a + b ) a 3 + b 3 = ( a + b ) ( a 2 - a b + b 2 ) a 3 - b 3 = ( a - b ) ( a 2 + a b + b 2 ) a 4 + b 4 = ( a 2 + b 2 ) 2 - 2 a 2 b 2 = ( a 2 + 2 a b + b 2 ) ( a 2 - 2 a b + b 2 )

  2. Pow

    am an = am-n am * an = am+n a 1 n m = a n ( a b ) m = am bm , ( b0 ) a 0 = 1 , ( a0 ) a -m = 1 a m , ( a0 ) a m n = a m n

  3. Logarithm

    log b ( N 1 N 2 ) = log b | N 1 | + log b | N 2 | , ( N 1 N 2 > 0 ) log b N1 N2 = log b | N1 | - log b | N2 | , ( N1 N2 > 0 ) log b N α = α log b N , ( N > 0 ) log b N α = 1α log b N , ( N > 0 ) log b N = log b a log a N , ( a > 0 , a 1 , N > 0 ) log b a = 1 log a b , ( a > 0 , a 1 ) a log a b = b = log a a b log 10 N = lg N ( b = 10 )

  4. Sum

    1 + 3 + 5 + ... + ( 2 n - 3 ) + ( 2 n - 1 ) = n 2 2 + 4 + 6 + ... + ( 2 n - 2 ) + 2 n = n ( n + 1 ) 1 2 + 2 2 + 3 2 + ... + ( n - 1 ) 2 + n 2 = n ( n + 1 ) ( 2 n + 1 ) 6 1 3 + 2 3 + 3 3 + ... + ( n - 1 ) 3 + n 3 = n 2 ( n + 1 ) 2 4 1 2 + 3 2 + 5 2 + ... + ( 2 n - 3 ) 2 + ( 2 n - 1 ) 2 = n ( 4 n 2 - 1 ) 3 1 3 + 3 3 + 5 3 + ... + ( 2 n - 3 ) 3 + ( 2 n - 1 ) 3 = n 2 ( 2 n 2 - 1 ) 1 4 + 2 4 + 3 4 + ... + ( n - 1 ) 4 + n 4 = n ( n + 1 ) ( 2 n + 1 ) ( 3 n 2 + 3 n - 1 ) 30

  5. Complex number

    i = -1 i2 = - 1 i3 = i2 * i = - i i4 = i3 * i = 1 i 4n = 1 i 4n+1 = i i 4n+2 = - 1 i 4n+3 = - i ( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i ( a + b i ) - ( c + d i ) = ( a - c ) + ( b - d ) i ( a + b i ) ( a - b i ) = a 2 + b 2 ( a + b i ) ( c + d i ) = ( ac - bd ) + ( ad + bc ) i a + b i c + d i = a + b i c + d i c - d i c - d i = a c + b d c 2 + d 2 + ( b c - a d c 2 + d 2 ) i

  6. Arithmetic sequence

    a 2 = a 1 + d a 3 = a 1 + 2 d a n = a 1 + ( n - 1 ) d S n = a 1 + a 2 + a 3 + ... + a n = ( a 1 + a n ) d 2 = [ 2 a 1 + ( n - 1 ) d ] n 2

  7. Geometric sequence

    a 2 = a 1 q a 3 = a 1 q2 a n = a 1 q n - 1 S n = a 1 + a 2 + a 3 + ... + a n = a 1 * q n - 1 q - 1 , ( q 1 )