Easy Complex Numbers
Why they're natural, necessary, and not nearly so imaginary as you might think.
I love the complex numbers. They are where I think math starts to get interesting and fun. Not to say that the foundation stuff isn’t interesting or fun. Math is amazing everywhere you look, but I find an added pizzaz in the magic of the complex numbers.1
Part of that might come from complex numbers — specifically the “square root of minus one” — being a bridge too far for some. That same pizzaz is, for some, a disqualifier from reality. I can sympathize because, on first contact, √-1 can seem a bit hard to swallow.
What I hope to do in the next two posts is explain why complex numbers are as natural and necessary as the more familiar numbers like 42 or 3.14159. I’ll show you how to understand √-1 in a way that also explains why multiplying two negative numbers results in a positive number.2
The story begins with the most basic numbers of all, the “natural” numbers…
Inventing Numbers
This is the story of the power of our mathematical invention.
The Natural Numbers
The story of the natural numbers is worthy of its own post. Suffice here to say we needed a way to remember, talk about, and work with things such as sheep, ships, bushels of wheat, and so on. We started with direct one-to-one representation: a bag of small pebbles, one for each sheep. Or knots in string. Eventually, we codified the notion of counting things into the natural numbers — so called because they came naturally as soon as humanity began classifying things.3
We denote these with the blackboard bold ℕ symbol.
The natural numbers start at zero or one — depending on your religion — and count upwards (0, 1, 2, 3…) forever.4 We’ve only begun, and already math has the notion of infinity. Right from the start, math was abstract.
These numbers work great. Adding two of them always gives us another natural number. Also true of multiplying two. That’s a nice property; we don’t want anything weird happening when we do math. We can also subtract natural numbers and… huh… well, usually get another natural number but sometimes we don’t. For example, 5-2=3 works fine, but what do we do with 2-5=? Our numbers are broken!
The Integers
To fix them, we invent a new kind of number, the integers. These include the natural numbers but add copies of them with minus signs — effectively creating new numbers by multiplying the ones we have by -1. These are the negative versions of the natural counterparts.
We denote these with the blackboard bold ℤ symbol.5
We fixed our numbers. Now we can answer 2-5 with -3. Numeric life is good. We can add, multiply, and subtract.
Eventually we need to divide something. A dozen apples is easy to divide evenly between four children. Each gets three apples. But what if we have ten apples? Now what? Our numbers are broken again!
The Rational Numbers
Inventing new numbers worked last time, let’s try that again. We’ll invent the rational numbers, which have the form p/q where both p and q are integers.6 This lets us express any kind of fraction we need.
We denote these with the blackboard bold ℚ symbol.7
We fixed our numbers. Now we can answer 10÷4 with 10/4 (and reduce it to 5/2 — two-and-a-half apples for each child). Numeric life is good. We can add, multiply, subtract, and divide.
The rational numbers turn out to be pretty useful, and it is possible reality uses rational numbers under its hood, because the next type of number, despite its name, might be unreal. But let’s continue.
Eventually we look closely enough at circles and squares to discover something troubling. The ratio between a circle’s circumference and its diameter, or the diagonal of a square, gives us a number we can’t express in any rational p/q number. The ancient Greeks (and others) proved there is no form of p/q, where p and q are integers, that exactly equals C/D, where C is the circle’s circumference and D its diameter.
Those ancient Greeks were horrified. So much so that people got killed over this.
And our numbers are broken again.
The Real Numbers
Well, we know what to do, invent new numbers. We call these, perhaps ironically, the real numbers. They are the numbers like pi (the aforementioned ratio between a circle’s circumference and diameter) with fraction digits that never repeat and never end.
We denote these with the blackboard bold ℝ symbol.
A rational number such as 1/3 has a decimal representation with 3s that go on forever, but that’s a repeating pattern. All rational numbers give a (sometimes long) repeating pattern. Technically speaking, even a number like 1.0 has a repeating pattern of infinite zeros.
One of my all-time favorite quotes:
“God made the integers, all else is the work of man.” (“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.”) ~Leopold Kronecker
It addresses a kind of spooky difference between the natural/integer/rational numbers and the real numbers. When looked at closely, the real numbers present some severe problems with regard to physical reality.8
But we’re done now, right? When we math, we can math all the math that maths?
Oops
Well… as it turns out, our numbers are still broken. To quote the great Homer, “D’oh!”
There is a fundamental algebra theorem we want to always be true because it always being true allows us to solve certain algebra problems. The theorem says (essentially) that all polynomials have at least one root. Leaving aside what polynomials or roots are, this means an equation like this:
Must be true for at least one value of x. We can see that x=2 and x=-2 make this equation true. Both +2 and -2 give us 4 when squared, which is what we need.
So, we’re okay on that one, but what about:
How can we make this true? By definition, the square of any number is a positive number. (At least any number we know of so far.) We need a negative number to add to four to get zero. There is no way to make this equation true.
Once again, it appears our numbers are broken.
The Complex Numbers
We know what to do. Back to the drawing board. This time we invent complex numbers, which brings us to today’s numerical destination. There are more numbers to invent, but today the bus stops here.
We denote these with the blackboard bold ℂ symbol.
Let’s take another look at the problem equation:
We need something that squared gives us minus four. We can see two must be involved, because two-squared is four. To get the minus we need something that squared gives us minus one. We can get numbers that square to any negative number if we can just get one that squares to minus one.
We need:
If we take the square root of both sides, we have:
Which is exactly what we need, but √-1 is… weird. Imaginary. And hence its name, the imaginary unit.9 This is a special (slightly magical) new number we invent to fix math. Its full name is imaginary unit, but we call it i for short.
Now we can solve the problem equation:
And all others related to it. Algebra is saved!
This (slightly magical) number i leads to some interesting consequences.
The Imaginary Axis.
The first is that, as in the example above, we can create any negative square by multiplying its square root times i, as above with 2i (2 being the square root of 4).
This means there is an axis of numbers created by multiplying the entire real number line by i. For every real number x on the real axis, there is an imaginary number xi on the imaginary axis. Note that i itself is an imaginary number (in addition to being the imaginary unit) because 1i = i.
The Complex Plane.
We have two axes of numbers, the real number line and the imaginary number line. With two axes, the obvious thing to do is make them orthogonal to each other. That is, to orient them at a right angle to each other. This is similar to XY graphs, which have two orthogonal real axes.10
When we make the real number line the X axis and the imaginary number line the Y axis, we have the complex plane. We started with the imaginary unit, i, and multiplied the real numbers by it to get the imaginary numbers and imaginary number line (or axis). Combining the real and imaginary axes to make a complex plane gives us, at long last, the complex numbers.
Complex numbers are inherently two-dimensional. A complex number always consists of two numbers. In this case, one from the real axis and one from the imaginary axis. The canonical way to write a complex number is:
Where a is a number (a coordinate) from the X axis, and b is from the Y axis. Note that the implied sum, a+bi, can never be completed because the two numbers come from different axes. Note also that, because i appears in the second part, both a and b are real numbers. The plus sign helps remind that a complex number is a combination of two numbers.11
The plus is more important when it comes to complex number math. As a simple example, adding two complex numbers, (a+bi) and (c+di):
For any real numbers a, b, c, and d. The result, on the right, is another complex number. When we operate on complex numbers, we end up with complex numbers.
The two-dimensional nature and operational rules make complex numbers surprisingly useful. And quantum mechanics seems to require complex math, which raises some interesting questions about reality.
Multiplying the horizontal real axis by i to get the vertical imaginary axis, and then having them be 90° from each other, has an important consequence. Multiplying a real number by i rotates it 90°. By extension, multiplying any complex number by i rotates it 90°.
Complex Number Representation
One thing I enjoy about the complex numbers is their various representations. I’ll leave you with a parade of complex number representations:
In the next post, I’ll dig into what each is and how to go from one form to the other. We’ll also explore the usefulness of complex math. As mentioned above, it involves rotation.
Did this all make sense?
Did I get anything wrong?
Anything you’d like to add? (Or subtract, multiply, or divide?)
Feel free to add a comment!
Until next time…
And the quaternions and octonions
While multiplying two positive numbers also gives you a positive number.
To a dog, each tree is distinct, but humans see 15 trees down the block. We can count them because we see them as belonging to the same class of thing.
The natural numbers begin at zero. Only heathens think otherwise.
For zählen, a German word meaning to count. The usage is attributed to David Hilbert.
The advice to “mind your ps and qs” was for printers. In olde tyme letterset presses, the individual letters used to build text were all backwards (to make the text on the paper right). So, it was very easy to mix up the q and p letters, which are mirror opposites in most fonts.
I love the notion that reality might be rational but not real.
“Unit” (and “unity”) is math-speak for “one”.
We denote such a space ℝ² because of the two real axes.
The comma in the more usual notation for two-dimensional coordinates, (x, y), emphasizes the separateness of the two numbers, and doesn’t imply the math behind complex numbers.
I really like the way you presented this. I have recently been refreshing my math studies since it’s been a while since I’ve had formal training, so I am very interested in your thoughts in other areas, especially statistics and probability. Thanks for sharing!
This is lovely.
Maths was my first love and I made it halfway through a maths degree long ago until work got in the way. I haven't been interested since. Just reading about number sets rekindled a little of that love.
Thank you.