Complex Numbers

The theory of complex numbers was developed by Jean le Rond d′Alembert in 1746. The vector representation of complex numbers was introduced by Casper Wessel in 1798.

A complex number consists of a real and imaginary part.

For example, if;

where a and b are real numbers and i is imaginary. The real part of z, denoted Re z or re z, is a. The imaginary part of z, denoted Im z or im z, is b.

Addition & Subtraction

To add and subtract complex numbers the real and imaginary parts are treated as separate entities.

For example;

Multiplication

To multiply complex numbers, both terms in the first complex number are multiplied by both terms in the second complex number.

For example;

Division

To divide complex numbers a numerical "trick" must be performed involving the denominator's complex conjugate.

For example;

Two complex numbers are said to be equal when their real parts are equal and when their imaginary parts are equal.

Modulus or Magnitude

Argument

See also: Complex Conjugate, Complex Numbers, Exponential Form, Complex Numbers, Logarithms of, Complex Numbers, Polar Form, de Moivre, Imaginary Axis, Imaginary Number, Imaginary Part.