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Complex numbers
Complex numbers arise naturally in the solution to algebra problems such as x2+1=0.
They were discovered by mathematicians interested in such problems. Mathematicians
knew that the real numbers were not enough to solve all algebra problems. Complex
numbers were the answer. At first mathematicians were suspicious of complex numbers
but eventually they became recognized as essential for mathematics. They play a key
role in nature. For example, the wave function, a key object in quantum mechanics that
describes how a particle behaves, is complex-valued. A possibly-more-familiar example
is the differential equation
.
(1)
This equation describes motion of an object such as a body segment subject to frictional
and spring-like forces. (The numbers a0, a1, and a2, by the way, are real, not complex
numbers. Their values depend on the system we are studying. For example, in a masson-a-spring-with-friction problem, we might have a2=mass, a1=coefficient of friction, and
a0=spring constant.) Equation 1 also describes electric circuits with inductance,
resistance, and capacitance. It also describes fluid mechanical systems such as arteries or
veins. If you don’t have access to complex numbers, you learn that the differential
equation above has three very different looking possible solutions. Which solution is
applicable depends on the relative values of the coefficients a0, a1, and a2. If you have
complex numbers, you find that there is really just one solution:
(2)
where z1 and z2 are the solutions to the equation
.
The key thing here is that z1 and z2 may be complex or real. If they are complex, they
will always be complex conjugates of one another, as is always the case for complex
roots of a quadratic equation. And if they are real, they may be different or they may be
the same as each other (“repeated roots”). By considering complex numbers we have
been able to see a degree of simplicity or “unity” in this problem which was not
otherwise apparent.
Complex numbers have the form a + bi, where i2 = -1. Comlex numbers 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
numbers: the reals are complex numbers with an imaginary part that is zero. The
imaginary numbers are also a subset of the complex: the imaginaries are 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?
(3)
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 equation is
eiπ +1=0.
Some people say this equation, credited to Euler, is the most wonderful equation in math,
because it links five numbers which are the building blocks for applying mathematics to
the real world: 0, 1, π, e, and i. It is especially fascinating because e and π are 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:
(a+ib) + (c+id) = (a+c) + i(b+d)
Subtract:
(a+ib) - (c+id) = (a-c) + i(b-d)
Multiply:
(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 © 2016 William C. Rose