Complex Numbers

We now have a bit of a bootstrapping problem. Complex numbers arise out of the reals when we consider the exponentiation process, which we won't really talk about until later, in the chapter on Algebra. If you have no idea what $b = a^n$ or $a = b^{1/n}$ mean and aren't at all familiar with rules such as $c = a^n b^n = (ab)^n$ and $c = a^n a^m = a^{(n+m)}$ then you should probably pause here and take a quick trip over to where this is covered and come back. We'll wait for you.

Fine. If you are now reading on, I will assume, dear reader, that you are now passingly familiar with the concept of the square root $a^{1/2} = \sqrt{a}$. Consider the square root operation applied to the real numbers. The square root operation (and many other non-integer-power operations) do not close under the reals. Since any number (positive, negative, or zero) squared is a non-negative number, we cannot find the square root of e.g. -1 among the reals.

We then have a choice - we can say fine, we will simply restrict the square root operation so that its domain is non-negative and call attempts to take the square root of a negative number undefined - as you probably did in your introductory algebra class back in high school - or we can be a bit more sophisticated and imagine a number (that clearly isn't ``real'') whose square is negative.

Since algebra is going to lead us to expressions containing square roots of arguments that can be negative quite a lot in science, it turns out to be useful to do the latter. Indeed, we will find it more than just ``useful'' - it will be essential later on down the road, even in introductory physics or math classes to understand this natural extension of the real numbers.

You are probably familiar with the naive definition of the unit imaginary number^2.7 as the square root of -1:

$$i = +\sqrt{-1}$$ (2.10)

This definition is common but slightly unfortunate. If we adopt it, we have to be careful using this definition in algebra (using the rules for exponentiation developed below) or we will end up proving any of the many variants of the following:

$$-1 = i \cdot i = \sqrt{-1}\cdot\sqrt{-1} = \sqrt{-1 \cdot -1} = \sqrt{1} = 1$$ (2.11)

Oops.

A better definition for $i$ that it is just the algebraic number such that:

$$i^2 = -1$$ (2.12)

and to leave the square root bit out. Thus we have the following cycle:

$$i^0 = 1$$

$$i^1 = i$$

$$i^2 = -1$$

$$i^3 = (i^2)i = -1 \cdot i = -i$$

$$i^4 = (i^2)(i^2) = -1 \cdot -1 = 1$$

$$i^5 = (i^4)i = i$$

$$...$$ (2.13)

where we can use these rules to do the following sort of simplification:

$$+\sqrt{- \pi b} = +\sqrt{i^2 \pi b} = +i\sqrt{\pi b}$$ (2.14)

where we never actually write $i = \sqrt{-1}$ except in the sense of $i = \sqrt{-1} = \sqrt{i*i}$.

This sounds really complicated and awful but isn't. It is actually really easy to avoid the ``bad step'' by simply never working backwards from $i \to \sqrt{-1}$ in algebra. One can always pull an $i$ out from under a radical (and change the sign of its contents) but it is very, very dangerous to put an $i$ back in under a radical as a change in sign.

With this definition, we can define an arbitrary complex number $z$ as:

$$z = x + iy = z_r + i z_i$$ (2.15)

where $x$ and $y$ (or $z_r$, the ``real part of $z$'' and $z_i$, the ``imaginary part of $z$'') are both real numbers. We will work out the details of algebra for complex numbers later in this review. Here we only wish to note that they satisfy exactly the same basic properties with respect to arithmetic operations as the reals.

To add two complex numbers $a = a_r + i a_i$ and $b = b_r + i b_i$, we use the rule:

$$a + b = (a_r + b_r) + i (a_i + b_i)$$ (2.16)

That is, we add the real and imaginary parts separately! To multiply them we just use the usual product rule for sums:

$$c = a b$$

$$= (a_r + i a_i)*(b_r + i b_i)$$

$$= a_r b_r + i a_i b_r + i a_r b_i + (i*i)a_i b_i$$

$$= (a_r b_r - a_i b_i) + i (a_i b_r + a_r b_i)$$

$$= c_r + i c_i$$ (2.17)

where $c_r = a_r b_r - a_i b_i$ and $c_i = a_i b_r + a_r b_i$.

We define the complex conjugate of a complex number $z = x + iy$ as $z^* = \bar{z} = x - iy$ (both notations are used, so I indicate them both). The point of defining the complex conjugate of a complex number is that:

$$\vert z\vert^2 = zz* = z*z = x^2 + y^2$$ (2.18)

is a real number that is called the amplitude or magnitude of a complex number - its ``size''. It is a measure of the distance from the complex origin of the complex number. We'll examine the geometry of this below when we examine the algebraic properties of complex numbers in more detail.

For addition, if $a$, $b$, and $c$ are arbitrary complex numbers:

$$a + (b + c) = (a + b) + c \quad {\rm associativity}$$ (2.19)

$$a + b = b + a \quad {\rm commutativity}$$ (2.20)

$$a + 0 = a \quad {\rm identity}$$ (2.21)

$$a + (-a) = 0 \quad {\rm inverse}$$ (2.22)

where e.g. $a = a_r + i a_i$ implies $-a = -a_r - i a_i$.

Similarly, for multiplication:

$$a*(b*c) = (a*b)*c \quad {\rm associativity}$$ (2.23)

$$a*b = b*a \quad {\rm commutativity}$$ (2.24)

$$a*1 = a \quad {\rm identity}$$ (2.25)

$$a*\frac{1}{a} = a*a^{-1} = 1 \quad {\rm inverse}$$ (2.26)

just as before.

The only tricky part of division is that one has to perform a little ritual to put the inverse of a complex number back into the form of a complex number. We'll go over this when we do algebra, but it doesn't hurt to put it here as well. It is:

$$a = \frac{1}{z}$$

$$= \frac{1}{x + iy}$$

$$= \frac{1}{x+iy}\frac{x-iy}{x-iy}$$

$$= \frac{x-iy}{x^2 + y^2}$$

$$= \frac{x}{x^2 + y^2} - i \frac{y}{x^2+y^2}$$

or $a_r = \frac{x}{x^2 + y^2}$, $a_i = -\frac{y}{x^2+y^2}$. This clever trick (multiplying by $z^*/z^*$) makes the denominator real so that the expression can be simplified into real and imaginary parts cleanly.

For combinations of multiplication and addition, we obviously still have

$$a*(b+c) = a*b + a*c \quad {\rm distributivity}$$ (2.27)

as well.

The reals inherit their properties (as you can see) from the complex numbers. Later we'll see just how powerful complex numbers can be in helping to solve problems.

At this point you should begin to have the feeling that this process of generating supersets of the number system(s) we already have figured out up to any given point will never end. You would be right. You are seeing the first steps of a hierarchy of number systems and algebras. Complex numbers are all one usually needs in introductory courses in physics or engineering (so you can relax, we're going to stop here) but further extensions (e.g. quaternions^2.8 or generalized geometric algebras^2.9 the extensions are actually very useful in advanced physics.

This concludes our discussion of numbers per se. When introducing and discussing numbers, we found it very convenient to let various symbols stand for ``example'' numbers of each different type. In English we might say: ``We can always add two real numbers in either order and get the same result'' but that's a whole lot of writing. It was much more compact to say: $a + b = b + a$ where $a$ and $b$ can stand for any real number. We can then extend this same relation to complex numbers by saying that $a$ and $b$ can be complex (or quaternionic, or whatever).

This is an example of the power of algebra. Algebra is basically the process of letting symbols stand for numbers, and then deducing and defining rules and processes that let us take the simple definitions of numbers of different kinds above and work all sorts of symbolic magic on them. There is something sublime, almost ethereal about algebra. It is at once a system of reason - a way of connecting things we already ``know'' to new things that we discover by means of its rules - and the natural language of science.

Science, after all, isn't satisfied with knowledge expressed in English as ``If you let go of a penny, it will fall down''. It wants an algebraic Law of Universal Gravitation that simultaneously explains (after a fashion) why the penny falls down while defining what ``down'' is, while providing you with relations that permit you to compute quite exactly what the behavior of the penny is as it falls. One is qualitative, and doesn't tell you what the penny would do on the Moon, where down may well be up relative to the Earth, or how fast it falls, or why it falls. The other (if accepted as a good approximation to truth) is general and quantitative - it works for pennies, nickels, dimes, and rocks.

Let us therefore take a look at algebra and review everything about it that you are likely to need to know.