1. Imaginary Numbers
  2. Complex Numbers
  3. Complex Conjugates


1. Imaginary Numbers

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: 2pi, iy, -7iq.


2. Complex Numbers

A complex number can be written as the linear sum of a real number and an imaginary number, z = x + iy. It can also be expressed as a function of complex-polar coordinates, z = R Exp(iq), where R is called the absolute value of z, and q is called the imaginary argument of z. The complex-polar coordinates are related to the linear (Cartesian) coordinates:

R = absolute value = (x2 + y2)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:

dz = dx + idy


dz = Exp (iq) dR + iR Exp (iq) dq

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

z4 = R4 Exp(4iq)


z4 = x4 - 6x2y2 + y4 + i(4x3y - 4xy3)


3. Complex Conjugates

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 + iy = R Exp(iq)

z* = x - iy = R Exp(- iq).

The product of a complex number and its conjugate is a real number equal to R2:

(x + iy)(x - iy) = x2 - ixy + ixy + y2 = x2 + y2

or in polar coordinates,

R Exp(iq) · R Exp(- iq) = R2 Exp(iq - iq) = R2

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.



Unpublished Work.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