Download Complex numbers Numbers of the form a + bi, where i2 =

Complex numbers
Numbers of the form
a + bi, where i2 = -1,
allow one to solve all polynomial equations, including those such as x2 + 1 = 0 and
x2 + x + 1 = 0, which do not have real roots. The two parts of a complex number a and bi
are called its real and imaginary parts respectively. The real numbers are a subset of the
complex: the complex numbers whose imaginary part is zero. The imaginary numbers
are also a subset of the complex: the complex numbers whose real part is zero.
Complex numbers can be represented as points on a “complex plane”: the rectangular x-y
plane, in which the x-axis corresponds to the real numbers, and the y-axis corresponds to
the imaginary numbers. A point’s x coordinate (a) is its real part, its y-coordinate (b) is
its imaginary part.
Why does a biomechanic need to know about complex numbers?
It is sometimes useful to know the frequency content of a signal. This is also called
Fourier analysis of a signal, and will be explained elsewhere. Fourier analysis cannot be
fully understood without explanations involving complex numbers. Complex numbers
are also useful for understanding how filters (to be discussed later) are used to process or
manipulate signals. Complex numbers are also useful in describing the mechanical
properties of systems with viscoelastic behavior, such as muscle, other soft tissue, and
blood and blood vessels. (Viscoelastic behavior is behavior which is spring-like (i.e.
elastic) under some conditions, and friction-dominated (i.e. viscous) under other
conditions.)
A complex number can also be written as
a + bi = rcosθ + i*rsinθ ,
which corresponds to a polar coordinate representation of the same point on the x-y
plane: r is the distance from the origin, and θ is the angle CCW from the positive x-axis
(i.e. from the positive real axis).
The rectangular and polar representations are related as follows:
a = r cosθ, b = r sinθ
r = sqrt(a2 + b2) = “the magnitude”, θ= tan-1(b/a) = “the argument”

It is also true that
a + ib = r cosθ + i*r sinθ = reiθ ,
because of a remarkable fact:
eiθ = cosθ + i sinθ. (Also true: e-iθ = cosθ - i sinθ.)
The truth of this fact can be demonstrated by considering the power series expansions for
exponential, sine, and cosine. A special case of this eqn is
eiπ = -1,
which is notable because it relates e and π, numbers which do not have an obvious
connection with one another. These relationships show that there is a deep connection
between exponential and trigonometric functions.
Addition, subtraction, multiplication, division of complex numbers
Add:
Subtract:
Multiply:
(a+ib) + (c+id) = (a+c) + i(b+d)
(a+ib) - (c+id) = (a-c) + i(b-d)
(a+ib)(c+id) = (ac-bd) + i(ad+bc)
a  ib  a  ib   c  id 
 ac  bd   bc  ad 

i 2
Divide:

  ...   2
2 
2 
c  id  c  id   c  id 
 c d   c d 
Multiplication and division are simpler in polar coordinates.
Note that r1eiθ = a+bi: r1=sqrt(a2 + b2), θ=tan-1(b/a).
Likewise, r2eiθ = c+di: r2=sqrt(c2 + d2), θ=tan-1(d/c)
Multiplication of complex numbers, done in polar coordinates: (r1, θ)(r2, θ) = r1eiθ r2eiθ
= (r1r2)ei(θθ = (r1r2, θ+θ)
Division of complex numbers, done in polar coordinates: (r1, θ)/(r2, θ) = r1eiθ /(r2eiθ) =
(r1/r2)ei(θ−θ = (r1/r2, θ -θ)
Complex conjugate
z  a  bi (sometimes called “z-bar”) is the complex conjugate of the number z=a+bi.
The real part is the same but the imaginary part of the complex conjugate is the negative
of the imaginary part of the original number. It is useful because multiplying a complex
number by its complex conjugate gives a real number, the squared magnitude of z:
2
z  zz   a  ib  a  ib   a 2  b 2 .
Powers
Easiest to do powers in polar coordinates: z = reiθ = r cosθ + i r sinθ
z n   rei   r n ein ,
n
where n is any real power. The preceding relation can also be written as
(r, θ)n = (rn, nθ)
Copyright © 2013 William C. Rose