The square-root of -1 is called *i*. An imaginary number is the product of a real number and *i *. A common convention for writing imaginary numbers is to write numbers in front of *i*, and to write variables after *i*.

Examples: 2p*i*, *i*y, -7*i*q.

A complex number can be written as the linear sum of a real number and an imaginary number, z = x + *i*y. It can also be expressed as a function of complex-polar coordinates, *i*q),

R = absolute value = (x^{2} + y^{2})^{1/2}

q = imaginary argument = + Arccos (x/R) if y __>__ 0, and q = - Arccos (x/R) if y < 0

x = R Cos(q)

y = R Sin(q)

Sometimes it is easier to operate on z in the linear form, and sometimes it is easier in the polar form. For example, it is easier to differentiate z in the linear form than in the polar form:

versus

dz = Exp (*i*q) dR + *i*R Exp (*i*q) dq

On the other hand, it is easier to express a power of z in the polar form:

versus

z^{4} = x^{4} - 6x^{2}y^{2} + y^{4} + *i*(4x^{3}y - 4xy^{3})

Every complex number z has a complex-conjugate, often written z*. A complex number and its conjugate have the same real coordinate, but their imaginary coordinates have opposite signs:

z = x + *i*y = R Exp(*i*q)

z* = x - *i*y = R Exp(- *i*q).

The product of a complex number and its conjugate is a real number equal to R^{2}:

or in polar coordinates,

R Exp(*i*q) · R Exp(- *i*q) = R^{2} Exp(*i*q - *i*q) = R^{2}

This property is useful when resolving fractions with complex denominators.

If a complex number is the root of a polynomial with real coefficients and exponents, then the complex-conjugate of that number is also a root of the polynomial. So if you find one complex root of a real-parametered polynomial, then you automatically know a second root, the complex conjugate.

URL http://members.aceweb.com/patrussell/approximations/Complex.htm

Unpublished Work. © Copyright 2001 Pat Russell. Updated April 17, 2009.