Complex Variables
Properties of Complex Numbers. The symbol for a complex number z is z = x + iy, where x and y are real numbers and i satisfies i2 = −1. The real number x is called the real part of z and is denoted by x = Re z. The real number y is called the imaginary part of z and is denoted by y = Im z. We now state several important properties of complex numbers.
1. Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal if and only if x1 = x2 and y1 = y2. In particular, z = 0 if and only if Re z = 0 and Im z = 0.
2. Addition of two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 is defined by
z1 + z2 = x1 + x2 + i(y1 + y2),
and multiplication by a real number c is defined by
cz1 = cx1 + icy1.
Subtraction is then defined by
z1 − z2 = z1 + (−1z2) = x1 − x2 + i(y1 − y2).
3. The complex conjugate of z = x + iy is the number 
, that is, the imaginary part of z is multiplied by −1. Thus 
. Note that z is real if and only if 
. Addition and subtraction of a complex number and its complex ...
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