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13.2  Application of Complex Numbers 

While imaginary numbers are introduced in algebra, the courses needed to understand how they're used in science and engineering usually aren't taught until college.  In college, you learn how the imaginary number is closely related to trigonometry and the natural logarithm.  One of its major uses is to help mathematicians solve problems that would be much more difficult to solve without it.

Fortunately, the imaginary number can be largely explained by one of the most simple phenomena in sports - a ball bouncing to a rest.  When you drop a ball from a certain height, it starts out with a certain amount of potential energy.  As it falls it accelerates, gaining kinetic energy.  If the ball is well inflated, it bounces several times before coming to a rest.

There are different forces involved that cause it to bounce and then come to a rest.  The combination of the air in the ball and the material of the ball causes it to act like a spring, storing and then releasing energy.  Using technical terms, we say the ball oscillates.  But the material of the ball also acts like a cushion, absorbing energy.  Air friction slows the ball down as it moves both upward and downward, also absorbing energy.  If the air friction and material didn’t absorb energy, the ball would bounce indefinitely. In technical terms, we say the ball's oscillation is damped. 

Let's look at the bouncing motion of three balls with varying amounts of inflation.  A well-inflated ball will nearly regain its previous height with each bounce.  It acts more like a spring than a cushion.  In technical terms, we call this response underdamped oscillation.

A poorly inflated ball will act nearly as much like a cushion as it does a spring.  The kinetic energy is not stored very well but rather is quickly absorbed by the ball's material.  Its oscillation will look like Figure 13.5.

Finally, the ball may be so poorly inflated that it doesn't bounce at all.  It doesn't act like a spring at all but only a cushion.  All of the kinetic energy is absorbed by the ball’s material.  We say this response is overdamped.

It is the conflict between the spring effect and the damping that creates conditions where imaginary numbers occur.  But to show how, let's look at a similar system where it's easier to assign values to the different forces involved.

A shock absorber on a mountain bike has spring and damping characteristics similar to a bouncing ball.  Shock absorbers have a spring to store and release energy and oil-filled chambers that absorb kinetic energy.  The bike acts as a mass connected to the shock absorber.  The different parts of the shock absorber are configured like this:

Figure 13.7 lists the different forces on the shock absorber.  Any force, F_m, on the bike (like when the bike goes over a bump) equals the mass, m, of the bike times the acceleration, a, up or down.  The damping force, F_d, equals a friction coefficient, c, times the velocity, v, of the motion.  The force from the spring, F_s, equals a spring constant, k, times the displacement, x, of the motion.

The values of m,c, and k can vary in ways that create different responses of the system to motion, just as varying the inflation of ball will affect its bouncing.  If the spring effect is greater than the damping, we get underdamped oscillation.  The response over time will look like this:

If the damping is greater than the spring effect, we get an overdamped oscillation.  The response over time will look like this:

We can use the sum of the forces listed in Figure 13.7 to develop a formula for the position of the bike as a function of time.

                                                     (13-5)

The physics of this equation are accurate and the units of the terms are consistent.  But if you applied the coefficients to the equation and then measured the motion of the system, you’d find that the equation just doesn’t work.  This is because the equation has terms with different relationships between displacement and time (like acceleration, velocity, and displacement).  An equation like this is known as a differential equation and can’t be solved with straight algebra.  Unfortunately, solving a differential equation requires taking courses in calculus and differential equations.  So we need to skip several steps here to see how we get the imaginary number. When a differential equation is transformed, we use a different variable, usually s. After transforming Equation 13-5, we come up with the following equation.

                                                                             (13-6)

You may recognize this as a quadratic equation.  If we use the quadratic formula, we get the following roots:

                                                                           (13-7)

If you look inside the square root, you can see the conflict between the damping (c) and the spring effect (k).  Since our main purpose here is to see how non-real numbers can occur in a real world, look at the square root part of the equation like this:

                                                                               (13-8) 

In both the ball and the shock absorber examples, we saw how the damping could be either greater than or less than the spring effect.  If the damping is greater, we get the square root of a positive number and our roots s₁ and s₂ are real.  The real roots indicate an overdamped response.  If the damping is less than the spring effect, we get the square root of a negative number and our roots are complex.  The imaginary numbers indicate that the system is oscillatory.  (If the damping equals the spring effect, we get one root and a condition known as critical damping, c_c, that is non-oscillatory.)

The ball and the shock absorber examples show why many mathematicians regret that the number  was given the name “imaginary”.  A bouncing ball is just as real as a flat one.