Complex Numbers

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In Imaginary Numbers I presented one way of thinking about complex numbers - numbers like the square root of negative one, which do not "exist" in the "real" world, but which are nevertheless quite useful in many scientific applications.

That article motivated imaginary numbers as the solution to word problems like "find a number whose squareis -1", just as negative numbers solve word problems like "find anumber which when added to 5 gives 2". Today I'd like to show you a different way of thinking about them.

Instead of dealing with individual numbers, we will workwith pairs. I will write (a, b) to mean the pairformed by a two ordinary ("real") numbers aand b. The simplest pair is (0, 0), and we canwrite down many others, such as (-3.14, 5.67).

The order of the numbers will matter: we willtreat (a, b) and (b, a) as two different things. Ifyou like, you can think of the pair as an object with two slots (Left and Right) or two colors(Yellow and Green). So for example the pair (0,1) has a Yellow zero and aGreen one, which is not the same as the pair (1,0), in whichthe zero is Green and the one is Yellow.

So far, I have not said why we should care about these pairs. We willget there eventually, but first we need to ways to do calculationswith these pairs.

Let's start by defining a method for "merging" two pairstogether. This operation takes two pairs as inputs and combines themto produce a single pair as an output. It will act in a way veryreminiscent of ordinary addition, so I will use a plus sign (+)as the notation for it. Here is the definition:

(a, b) + (c, d) = (a+c, b+d)

This is a rule for combining the pair (a, b) with thepair (c, d), producing a new pair that we might call (e, f). The "Left" element in the new pair is given by therule e = a+c, which you can remember as "Left adds toLeft". Similarly, the right element f is just the sum of thetwo right inputs, b+d, so "Right adds to Right".

This is a useful way to define "addition" of pairs, but it is not theonly way. If I wanted to, I could have defined "addition" of pairs bythe "alternative" rule

(a, b) + (c, d) =?= (a+c, b+2*d+a)

However, this rule does not turn out to be very useful. For one thing,it is not as symmetric as our "official" rule, and symmetry in math isoften closely connected with being useful in the real world.

Our "official" rule has some nice properties -- in fact, all the sameones that ordinary addition has. For example, itis commutative, meaning the order of addition does not matter:

x + y = y + x

This "order does not matter" property applies to adding ordinarynumbers: for example, 5+2 and 2+5 are both equal to 7. Because our"official" rule is so symmetrical, the "order does not matter"property also applies to adding pairs. Write it out:from our official rule,

(c, d) + (a, b) = (c+a, d+b)

but from the commutative property of ordinary arithmetic, c+a = a+c and d+b = b+d, so in fact, the result is actually the same as (a+c, b+d). So indeed, order does not matter when adding pairs.

I want to emphasize that this "commutative" property of addition onpairs is a consequence of the nice symmetric definition I selected.If we had used the "alternative" (non-symmetric) rule to defineaddition, then addition of pairs would not have the"commutative" property. Try a few examples to see this: for instance,try adding (0,1) and (1,0) using the "alternative"definition. The results will not match.

I also want to emphasize that we are talking about the order of addingtwo different pairs, not about swapping the order withina single pair: as mentioned already above, the pair (0,1) isnot the same thing as the pair (1,0).

At any rate, it turns out that the "official" (symmetric) rule foradding pairs is quite useful, mainly because it satisfies the usualrules for addition, such as being commutative. Naturally, we wonder ifwe can also define a kind of "multiplication" of pairs that would alsosatisfy the usual rules. For instance, it would be nice ifmultiplication were also commutative.

It would also be nice ifmultiplication was a kind of "repeated addition". This iscalled the distributive property: at least for ordinary (real) numbers,

a*(b+c) = a*b + a*c

Can we achieve something like this with pairs?

One possible rule would be to define

(a, b) * (c, d) =?= (a*c, b*d).

This is symmetric, and leads to all the usual properties, but it isalso a bit too bland. With this rule, arithmetic on pairs does notbring us any new insights into the world of numbers, because the leftand right sides are completely decoupled. To use the color analogy, weget ordinary arithmetic on yellow numbers, and completely separately,we get ordinary arithmetic on green numbers. We could have savedourselves the trouble and just stuck with ordinary arithmetic onsingle numbers.

Instead, for our purposes today, we will use a different definition ofmultiplication, one which jumbles up the colors a little:

(a, b) * (c, d) = (a*c - b*d, a*d + b*c).

It is not immediately obvious that this will produce a kind ofmultiplication that is commutative and distributive. We have to lookcarefully at the formula to see that it has the sort of internalsymmetry that makes it commutative. But indeed, it does. Similarly, youhave to do a little algebra to convince yourself that it has thedistributive property, but indeed it does. Instead of boring you withthe algebra, let us revisit the earlier question of "why should wecare?"

Well, this "multiplication" operation turns out to have someremarkable and quite unexpected new properties. Let us look at somesimple examples.

If we multiply any pair (a,b) by the pair (1,0), weget back (a,b). Try it and see. More generally,with d=0 we have

(a, b) * (c, 0) = (a*c, b*c).

In other words, we are just multiplying each slot by c. As aresult, the pair (c,0) acts essentially just like theordinary ("real") number c.

Not only can we replicate ordinary arithmetic in this way, but we canget a brand new effect. Try multiplying (0,1) by itself,i.e. squaring it. We get

(0, 1) * (0, 1) = (-1, 0).

But we just saw that (-1, 0) acts like the ordinary number-1. So, we have found a "square root of minus one": a number whichwhen multiplied by itself gives us -1.

In other words, we have discovered a representation for the"imaginary number" called i, which we "invented" backin ImaginaryNumbers to fill a conceptual gap.

Try playing around with adding and multiplying pairs using theserules. You will find that the pair (a, b) acts just like theexpression a + b*i in ordinary algebra, as long as we assumeone extra rule: that i*i = -1. So now we have two ways tounderstand "complex" numbers (numbers that result from adding real andimaginary pieces). The method we used in ImaginaryNumbers involved "making up" a hypothetical number i withthe property that i*i = -1, and then agreeing to use it inordinary algebra just like any other number. The new method todayinvolves doing arithmetic on pairs, and noticing that somepairs act just like ordinary real numbers while others act just likethese "imaginary" ones.

This is pretty typical of what mathematicians call abstract algebra: you invent a new kind of "number" by specifying operations like addition and multiplication for it, then see what it can do. Sometimes you only have one operation instead of two. Sometimes you can find a way to "invert" one or both operations, analogous to doing subtraction and division on ordinary numbers. Sometimes none of the usual properties hold; for instance, when working with three-dimensional rotations, commutativity breaks down - try rotating a book first around its spine, then around the front cover; then try the same two rotations but in the opposite order. In every case, the pure mathematician is interested in what kinds of patterns arise, whether familiar or new, and how the new objects relate to older one.

Of course, the applied mathematician is more interested in whatthese new "numbers" can do for us. As I mention briefly in ImaginaryNumbers, these new numbers are actually central in QuantumMechanics, which is the branch of physics responsible for all modernelectronics (computers and cell phones). They are also useful in manyengineering situations, including the study of waves, oscillations,sound, music, the stability of bridges, and lots of othersituations. Pretty amazing for just taking pairs of ordinary numbersand using a slightly fancy rule for how to multiply them.

If you enjoyed this article, you might also like Counting and Number Systems, or Logarithms.

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