1. Two complex numbers,  z1 = x1 + iy1  and  z2 = x2 + iy2, are equal to each other if and only if  x1 = x2  and  y1 = y2:

2. Complex numbers are added (or subtracted) by adding (or subtracting) the real and imaginary parts, correspondingly:

( x1 + i y1 ) ± ( x2 + i y2 ) = ( x1 ± x2 ) + i ( y1 ± y2 ).

Complex numbers have the same addition properties as real numbers:

z1 + z2 = z2 + z1,
( z1 + z2 ) + z3 = z1 + ( z2 + z3 ),
z + 0 = z,
z + ( – z ) = 0.

3. To multiply complex numbers, expand the product and then substitute  (–1)  for  i2:

( x1 + i y1 ) ( x2 + i y2 ) = x1 x2 + i ( x1 y2 + x2 y1) + i2 y1 y2 =
= ( x1 x2 – y1 y2 ) + i ( x1 y2 + x2 y1 ).

Complex numbers have the same multiplication properties as real numbers:

z1 z2 = z2 z1,
( z1 z2 ) z3 = z1 ( z2 z3 ),
z1 ( z2 + z3 ) = z1 z2 + z1 z3,
z · 1 = z,
z · 0 = 0.

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