Complex Numbers



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

[pic] . (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 mass-on-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:

[pic] (2)

where z1 and z2 are the solutions to the equation

[pic] . (3)

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 [pic]a and bi [pic]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?

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)

Divide: [pic]  [pic]

Multiplication and division are simpler in polar coordinates.

Note that r1eiθ1 = a+bi: r1=sqrt(a2 + b2), θ1=tan-1(b/a).

Likewise, r2eiθ2 = c+di: r2=sqrt(c2 + d2), θ2=tan-1(d/c)

Multiplication of complex numbers, done in polar coordinates: (r1, θ1)(r2, θ2) = r1eiθ1 r2eiθ2 = (r1r2)ei(θ1+θ2) = (r1r2, θ1+θ2)

Division of complex numbers, done in polar coordinates: (r1, θ1)/(r2, θ2) = r1eiθ1 /(r2eiθ2) = (r1/r2)ei(θ1−θ2) = (r1/r2, θ1 -θ2)

Complex conjugate

[pic][pic](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:

[pic].

Powers

Easiest to do powers in polar coordinates: z = reiθ = r cosθ + i r sinθ

[pic],

where n is any real power. The preceding relation can also be written as

(r, θ)n = (rn, nθ)

Copyright © 2016 William C. Rose

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