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

z1z2 = z1 + (−1z2) = x1x2 + i(y1y2).

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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