In the preceding instructional unit, we explored the differences between rectangular and polar graphing systems. We learned that the polar system is often favored in physics due to the nature of observational data. In the web-page for EMAT6680, we explored polynomials and trigonometry, but only touched on exponentials. In this essay, we will explore some of the applications of polar graphing, and introduce a truly fascinating subject tying together many threads of mathematics.
We ended the last lesson with simply a statement of the relation eπiθ = cos θ + isin θ. In many senses, this is among the most beautiful and useful of formulas, but it often is buried in an avalanche of technicalities. We must first introduce its native element, the imaginary (or complex) number system. Then, we will show that this system may be realized as a polar graph application. Finally, we will develop a few tools that truly unlock the potential for this 'new' system, and point to further explorations.
We can develop the numbers that we know like this. Start with the numbers you can count on your fingers. These are the natural numbers. You can add them, but no two added together can be less than either of the two: a+b < a, and a+b < b are never true. So we come up with negative numbers, which form the integers together with the natural numbers, and can move comfortably in either direction through addition. When we introduce multiplication, we have the same problem: |a*b| < |a| is never true. No problem... just introduce numbers called inverses so that a*a-1 = 1, and call all these numbers rational numbers. We run into the same problem when we start taking roots (square, cube, etc) of numbers, so we extend to the irrational numbers, and life is just fine with this real number system. Mostly. We've come up with a number system that can handle any sum, product, and positive roots, but relations like x2 = -1 are still a problem.
So mathematicians invent this 'number' called i, and i2 = -1. Problem solved!
As glib as this process is, there is an underlying power to it we must explore. The fascinating thing about complex numbers, and the reason why we needed to see that they came to birth through the need to solve certain algebraic problems, is that we are now (in theory, at least) able to solve any polynomial, of any degree, and find every solution to it, and this is only possible within the field of complex numbers. Polynomial-wise, the only operations we can perform on numbers are addition, subtraction, multiplication, division, exponentiation (raising to a power), and root extraction. From the natural numbers, we are able to solve subtraction problems by expanding to the integers. From the integers, we expand to the rationals to solve division problems. From the rationals, we expand to the reals to solve root extractions, but we are limited to positive roots. Somewhere along the way, mathematicians agreed that (x)2n = (-x)2n, so even powers are always positive numbers. If someone needed the root of a negative number, they would shrug their shoulders and mumble that it wasn't defined. We now call those roots imaginary, even though they are as real as any other number, and the system of imaginary numbers is called the complex numbers, though they make a lot of mathematics easier.
First, we note the rules of these new numbers. We represent our new numbers as z = a + bi, where a is the real part, and b is the imaginary part. Notice that the real numbers are complex numbers with a zero imaginary part. We say that the reals are embedded in the complex. Addition works like we'd expect: (a+bi) + (c+di) = (a+c) + (b+d)i. Multiplication is only a little trickier: (a+bi)*(c+di) = a*c + c*bi + a*di + bi*di. Collecting terms, we get (a*c) + (a*d+c*b)i + b*d*i2 = (a*c-b*d) + (a*d+c*b)i.
Graphically, we can picture the complex system as a coordinate system with the real numbers as the horizontal axis and the imaginary as the vertical, and so we graph the number a+bi as (a,b). This appears to be similar to our familiar rectangular system, but with this very important difference: each point in this system is not a pair of numbers, but a single number (from which the moniker complex is derived). It seems like we've moved backwards a bit... whereas we were once able to plot the independent variable of a function against the dependent variable of a function, now we can only display one of those numbers in a plane. If our function's co-domain happens to be complex, we must resort to displaying our results in four dimensions (!). How's that supposed to make life easier?
This is where the polar system comes in. In dealing with complex numbers, it is sometimes more useful to identify a number by its polar coordinates. This offers no immediately obvious advantage, since to identify a number a+bi, we 'rename' it (√(a2+b2), tan-1(b/a)). Yikes!
Except that there is another way, and this is where genius sneaks in and rearranges the furniture while we're sleeping.
We can see for ourselves that a = r*cosθ and b = r*sinθ. In 1714, Roger Cotes published the result that iθ = ln(cosθ + isinθ). So we can deduce that iθ = ln(a/r + ib/r). In 1722, Abraham DeMoivre noted that (cosθ + isinθ)n = (cos(nθ) + isin(nθ))!
Why the exclamation point? Well, a + bi = r*cosθ + r*isinθ, so (a+bi)n = (cosθ + isinθ)n = (cos(nθ) + isin(nθ)). What this means is that we can find all the roots of any complex number, and that is exciting, since it opens up an entire world to sleepy eyes blurred by easy electronic calculations.
We are too often blinded by our reliance upon the real number system. For example, if I ask you what the square root of 16 is, you'll tell me that it's ±4. When I ask you for the cube root of 27, you'll tell me it's 3, and only 3. If I ask for the fourth root of 16, though, you'll again give me two solutions, 2 and -2. Now we can be very naive about the world and just pretend that even exponents always have two roots, while odd exponents always have just one. Even our much vaunted calculators only give us one root, so why bother looking for any others?
What is (-3/2 + (3i√3)/2)3 ? (-3/2 - (3i√3)/2)3 ? Both are 27, so -3/2 ± (3i√3)/2 are two cubic roots of 27, along with 3. And you tried to tell me that three was the only root! Actually, it is the only root in the real numbers, so it is understandable why you would think that it is the only root. What we will soon learn, though, is that in the expanded realm of the complex numbers, every number has exactly n nth roots (i.e., there are 3 cube roots, 7 7th roots, etc.) If a number is real, and we extract an even number of roots, then exactly two of those roots are real. If the number is real and the number of roots is odd, then only one root is real.
This is why DeMoivre's theorem is so powerful and exciting. By using it, we can find all of the roots of a number, provided that we work in polar coordinates. So we've got a number, a+bi. We convert it to the polar system, through trig: a + bi = r*cosθ + r*isinθ, where θ is tan-1(b/a). Then we take the root, remembering that the nth root of x is the same as x1/n. So we have (a + bi)1/n = (r*cosθ + r*isinθ)1/n = r*cos(θ/n) + r*isin(θ/n). Clearly, this is only one root, and I promised that there would be n roots.
To find the other roots, we divide a circle equally into n parts, and add one of these parts to θ for each successive root. For example, if we are calculating the fourth root of a number, and we find the principle root at (r1/n, θ/n), then the other roots are (r1/n,θ/n + 2π/4), (r1/n, θ/n + 2*2π/4), and (r1/n, θ/n + 3*2π/4). The length r1/n is the same for each root, and represents the one or two real roots we can extract from a real number. So without resorting to a lot of dealings with Pythagoras and tedious calculations, we can quickly find n roots for any number, and plot them on a polar graph.
I hope that I make some blood boil and fists clench when I bow out from proving the extensions and applications of DeMoivre's Theorem. Any studious reader should have read the above paragraph about dividing a circle into equal parts and adding the parts to our angle to find all the roots of our number, and should have scratched a head, crumbled some paper, and wondered why aloud. It's not obvious why this works, but narration must sometimes take the wheel while faith rides shotgun. It is a very important result, but the proof of it is more well-suited to a textbook.
Accept this theorem, though, and we can see some interesting implications. Suppose we're given a real number, x, and asked to find it roots. We already know the calculator root: x1/n, a real number. Most calculators work only in real numbers, which, as we saw, were designed so that every real number has at least one root in the real numbers. Calculators missed the memo about complex numbers, though, and are unaware that there are more roots to be found. Since we have this one root and it is on the real number line, and we are using DeMoivre's theorem for the other roots, we know that the roots occur on evenly spaced points on a circle of radius 1/n centered at the origin. In other words, we are crating an n-agon incribed within this circle. Looking for the 17th roots of a real number? Find the one real root, then construct a regular 17-agon (septadecagon?) centered at the origin and with this one root as a vertex to find the other roots at each vertex.
Above, I cited the result due to Cotes: iθ = ln(cosθ + isinθ). In his 1748 ground-breaking work, Introductio in analysin infinitorum, Euler restated this as eiθ = cosθ + isinθ. This amendment is itself just a suitable re-arrangement, and detracts nothing from Cotes' discovery, but it does make manipulations easier, and opens the complex field to very suitable methods of computaion. For example, we can look above to remember that (a+bi)*(c+di) = (a*c-b*d) + (a*d+c*b)i. When we use DeMoivre's theorem, we convert to polar a+bi = r*cosθ + r*isinθ, and c+di = s*cosφ + s*isinφ. then (a+bi)*(c+di) = (r*cosθ + r*isinθ) * (s*cosφ + s*isinφ) = r*s*(cos(θ + φ) + isin(θ + φ)) = r*s*cisθ + φ. Now this is exciting. Where we once had to multiply term by term, sort out the like terms, and simplfy, under this system, we multiply the magnitudes and add the angles. In Euler's notation, it looks even cleaner: r*s*e(θ + φ)i. So now we have a very elegant method for multiplying vectors, a way many times simpler than in the rectangular system.
This only a sampling of two well related subjects, complex analysis and vector analysis. As stated in the instructional unit, polar data is native to observations, so physicists must be adept in the usage of polar systems and the resulting vectors, and the thurough study of these fields are beyond the scope of a mere web-page.
I would be remiss, however, if I neglected to relate what has been called the most beautiful formula in all of mathematics. We start the the Euler equation: eiθ = cosθ + isinθ. Now we use the value θ = π : eiπ = cosπ + isinπ = -1.
eiπ + 1 = 0. Entire books have been written about this. No-one could ever have guessed that five seemingly unrelated numbers (and not just any numbers, but the most important numbers in ALL of mathematics) are related in so elegant a formula.