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.
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,
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 = 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 = x4 - 6x2y2 + y4 + i(4x3y - 4xy3)
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:
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Unpublished Work.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.