What Use Are Complex Numbers?

In an over-simplified context, one can talk about a sine wave or a cosine wave.  Relative to a coordinate system, there are three quantities that pin down the graph:   the amplitude, the frequency, and the phase.  For engineers, the first two can be measured for a single "wave" while the last is a "relative" concept.  

One can now state that complex numbers are a useful way of keeping track of the first two.  

Even without sophisticated equipment (such as signal generators and oscilloscopes), one can try to present simplified descriptions:

For example, draw the graph of the usual sine function on the blackboard:  y = sin t.  Use two pieces of paper to form a vertical slit to cover up the graph [so] one sees a vertical line.  Move the slit down to illustrate the idea of a point moving up and down in simple harmonic motion.  This provide a platform to explain amplitude and frequency.  Moving the entire graph to the right or left to describe the idea of phase.

As a challenging question, draw the graph x = sin 2t and repeat with the slit horizontal moving up and down (t is now the vertical coordinate).  

Ask the class to investigate what happens if we can arrange it to have the x- and y- coordinates of a "point" to depend on t as described in the two cases.  Namely, what does the point trace out in the plane as its locus.

If you do have access to signal generators and oscilloscope, do a demonstration and consider more generally: 
x = sin t, y = sin ct, c a real number.  

Ask the student to draw some conclusions depending on c is rational or irrational.  

If you have the device to freeze the screen, do a demonstration for the case of 
	x = cos t, y = sin t, with t = (r)*2(pi), r = 1/6, 2/6, 3/6, etc.

This now demonstrates the product of complex numbers:
	and the famous Euler identity:  exp(it) = cos t + i* sin t.

There are other things one could do.
			H. Sah, Long Island

An Application of Complex Numbers

A pirate wants to bury his treasure on a deserted island, but he wants to find it easily when he returns in a few years. The island has only two major landmarks: a palm tree and an oak tree.

He takes two of his shipmates to the island with some supplies, and they build a gallows.  When they are done, he asks the first shipmate to walk from the gallows to the palm tree while counting
off his steps, turn right 90 degrees, then walk the same number of steps away from the tree.  The second shipmate heads for the oak tree while counting off his steps, turns left 90 degrees, then walks the same number of steps away from the oak.

The pirate buries the treasure at the midpoint of the segment between the two shipmates.  Then, of course, he hangs his men at the gallows so that he is the only one who knows how to find the treasure.

Years later, he returns to the island to claim his reward.  Since he was last there, a major storm has hit, destroying both trees--- just stumps are left where they had been---and removing all trace of his gallows.  Can he still find his treasure without digging up the whole island?  How?

Michelle Manes
Connected Geometry Project
Education Development Center, Inc.
michelle@edc.org

~~~~~~~~~~
I think this is one of the famous problems.  We can begin with a common sense answer:

Common sense answer.  

If the answer depended on the location of the gallow, then it is hopeless for the pirate.  So the pirate will go on to the next step: 

If the answer does not depend on the location of the gallow, then we can try to follow the instruction and start from the half way point between the stumps.  It is then immediate that there are just two possible locations. Namely, from the midpoint between the two stumps turn either left or right by 90 degree and mark off half the distance between the two tree stumps. Dig in one of these and then the other.  The pirate will find the treasure and worry about getting off the island without being caught.

On the other hand, judging from the fact that the two mates would build a gallow and not ask "why are we building a gallow?", it would seem to me that we are not dealing with pirates with common sense.  Of course, the captain may have a gun aimed on the two mates all these time.  

Proof that the treasure must be buried at one of the two other corners of a square with the line segment joining 	the two stumps forming a diagonal.

The easiest way to show that this is correct involves the use of complex numbers.  Namely, let the gallow be the unknown origin with u and v be the complex coordinates of the two tree stumps.  The instruction is to find the midpoint of the two complex numbers:  u + iu and v -iv  which is:
		(1+i)u/2 + (1-i)v/2

The midpoint of the two tree stumps is:
		(u+v)/2

The difference between these two number is i(u-v)/2.  Namely, from the stump marked by v, the vector pointing to the midpoint between the tree stumps is (u-v)/2.  Multiplication by i means rotation to the
left by 90 degrees.  In other words:  the location of the treasure is 
		(u+v)/2 + i(u-v)/2
is obtained by starting from the midpoint of the two tree stumps, turn 90 degrees (to either side) and mark off the distance that is half way between the distance of the two stumps.  One of the two possible locations
is where the treasure is buried.  The reason that we have two possibilities rather than one comes about because we can exchange the labelling of the two stumps.  In this case, i is replaced by -i.  In particular, the pirate could have started anywhere and repeat the instruction but keep in mind that there are two possibilities because it is not possible to tell from the stumps which is the oak and which is the palm tree unless it is still
possible to figure out from the stump which is which. 

One can also tackle the problem using things like the geometry sketch pad.  Using coordinate geometry would also work but the formulas may discourage the students.  The complex numbers have the advantage
relating arithmetic to geometry: 
	Addition <---> translation
	Multiplication <---> rotation plus similarity

Han Sah, Long Island

~~~~~

It turns out that I was pissed about the same question all throughout high school and beginning calculus.  My high school math teachers had (of course) absolutely NO CLUE as to what the answer was and why complex numbers are viable solutions.  It merely *was* the deal.

The first major break occured for me in differential equations, ie, calc IV.  That course gave me a very tangible answer to this question.   It turns out that the equation for the motion of a spring oscilliating is:

    my'' + gy' + ky = 0 

m = mass
g = damping constant (kind of like friction)
k = spring constant
y'' = second derivative of the position (acceleration)
y'  = first derivative of the position (velocity)
y   = position

The way to solve this is to pretend the solution is e (2.71828....) raised
to some power r.  Then, you simply replace the y'' with r^2 and y' with r
and so on.  The equation then becomes

mr^2 + gr + k = 0

You solve this, and boom, you plug these two values back in for the power
the e.  This gives you the equation of motion.

If those r's are real, COOL, and we're done.

However, if they are complex (THIS IS THE KEY), then you would have y="e" raised to some COMPLEX power.  It turns out that "e" raised to a complex power is EQUIVALENT TO SINE and/or COSINE without complex numbers!  BEAUTIFUL!

So, for a real spring with a mass on the end, you KNOW that it osciallates back and forth in a sinusoidal fashion.  Thus, when you solve this equation, you normally get complex roots for r.  Thus, y = some trig function, which is exactly the expected behavior of a spring (assuming reasonable constraints, if the spring was like, really friction-ized, let's say, submerged in oil, then the powers could easily be real, and you wouldnt get any osciallation, just a return to the equilibrium position).

So the key is that exponentials to complex powers turn out to be equivalent to 'normal' trig functions.  When I learned this, I felt a 10 year quest to solve this idea of complex numbers finally came to an end.

Now, this thing about springs I am discussing is a topic in differential equations, which is like, HARDER THAN HELL, so I am obviously oversimplifying the deal, but, I have essentially given all the necessary information.

If you are really interested, any undergraduate who has had some calculus could easily pick up the book "Introduction to Differential Equations" by Drs. Boyce and DiPrima and read the chapter on "Mechanical Vibration and Oscillation" or whatever it is, and get understanding out of it.

I hope this clears it up.  Complex numbers don't really seem to be cool until differential equations.  That's just how it is.  You have to take their usefulness on biblical faith most of the time.  

~~~~~

One doesn't really need to understand differential equations to see why complex numbers are cool.  It's true that differential equations make it very clear that exponential growth and sinusoidal oscillation are two
aspects of the SAME THING, and that complex numbers are useful to see HOW they are the same.  But one can see it more simply as follows.

Think of the good old real number line.  Think of multiplying by 2 as stretching the number line by a factor of 2, multiplying by 1/2 as squishing it down by a factor of 2, multiplying by -1 as reflecting it about the origin, etc..  This "explains" such facts as -2 x -2 = 4 (first reflect and stretch by a factor of 2, then do the same thing
again, and one has stretched by a factor of 4.)  

Now someone asks you to find the square root of -1.  What can you do so that if you do it TWICE you reflect about the origin.  Answer: nothing, unless you break out into 2 dimensions!  Then you can think of the real line as sitting in the plane, and then the answer is: ROTATE A QUARTER TURN.  (Either clockwise or counterclockwise will do!)  So we call this operation "multiplying by i" and say i x i = -1.  

Now we can think of multiplying by any complex number as a combination of rotation and stretching (or, more precisely, dilation).  It should be clear then that rotating round and round and stretching out and out
are two special cases of a general thing, i.e.,  exp(it)   and    exp(t)  are not all that different.  But the first one gives you trig functions.