Recently I did a series of posts about how the complex numbers arise from a natural progression of math realizations. I’ve done posts in the past about how the natural numbers lead through the integers and rationals to the real numbers. (And I’ve done posts about how weird the real numbers are, but that’s another topic.)
I recently came across another way a progression of obvious natural questions directly leads to the necessity of a new type of number, and this progression takes us all the way from the naturals to the complex numbers.
All by asking, “What do you get when you…”
I’ll get to that, but first we need to know what a (mathematical) group is. As the name suggests, it does mean “group of things,” but there are a couple of additional requirements.
In math, we have a fundamental idea: a set. The only requirement of a set is a membership function that describes what is in the set. For example:
- All the leaves on the tree outside.
- All the shoes you’ve ever owned.
- All the seas of the Earth.
- All positive numbers.
- All the games ever invented.
- Everything that’s mostly red in color.
- Everyone who loves baseball.
That kind of thing can go on forever. Some sets are finite, some are infinite.
(After enough drinks, naming weird sets becomes kind of fun: All the left-handed red-heads who’ve never been on an airplane.)
The point is, anything can be a set. (Which was great until Bertrand Russell asked: What about the set of sets that don’t include themselves?)
((If that made your head hurt, try it this way: If the town barber shaves every man who doesn’t shave himself, who shaves the town barber? Your head won’t hurt any less, but at least you might see why it hurts.))
(((BTW: The obvious answer is, he isn’t the town barber when he’s at home shaving himself. He’s just a guy shaving himself for free.)))
So a set is any collection of things we can name. A group is a set with one thing extra: operations.
Naturally, there’s a teeny bit more to it, but basically a group is a set of member objects (elements) along with operations that combine any two of those elements to produce a result element.
For example, the set of natural numbers (0, 1, 2,… etc) is a group with addition and multiplication as operations.
If we add any two natural numbers, we get another natural number:
- 2 + 2 = 4
- 5 + 0 = 5
- 33 + 55 = 88
- 100 + 1 = 101
Likewise, if we multiply any two natural numbers, we get another natural number:
- 2 × 2 = 4
- 5 × 1 = 5
- 33 × 55 = 1815
- 100 × 100 = 10000
Therefore, we say that the natural number group is closed under addition and multiplication. An operation always keeps us in the group. That’s the crucial point here.
As an aside, note how the second addition entry, 5+0, gave us 5 back. Likewise, the second multiplication entry, 5×1, also gave us 5 back.
An element of the group that, when used in an operation, returns the other number (5, in this case), is called an identity element. Zero is the additive identity, and one is the multiplicative identity. (Having an identity element is one of those teeny bits I mentioned.)
Now we have all the pieces to bring it home.
The natural numbers seem inevitable and, well, natural. It seems impossible that any intelligence would not discover them. (I’ve argued one wouldn’t even need sensory input to deduce them.)
So let’s consider the natural numbers axiomatic. A starting point.
Now we can ask about what operations are closed for natural numbers. We just saw that addition and multiplication are. But what about subtraction, division, and exponentiation? (For these are the five basic math operations.)
Let’s consider two subtraction examples:
- 5 – 3 = 2 (this works)
- 3 – 5 = ?? (this doesn’t!)
If I have five apples, it’s easy to remove three, but if I have three apples, how do I remove five?
It’s not natural!
But hang on,… what if we invent a new kind of number?
If we think of the natural numbers as a number line, with zero on the left, and counting off to the right (towards infinity, a topic all its own), what if we had numbers going to the left?
Now subtraction is closed:
- 3 – 5 = -2
- 0 – 5 = -5
- 5 – 0 = +5
Note that 0-5 is not the same as 5-0. This is because addition and multiplication both commute — the order of the operands doesn’t matter. With subtraction (and division) the order does matter.
(If we reverse the order, we get the additive or multiplicative inverse, respectively. That is, -5 is the additive inverse of +5, because if we add them we get zero, the additive identity. I’ll mention the division case below, but see how nicely this all holds together?)
So the integers are closed under subtraction. What about division?
- 10 ÷ 2 = 5 (so far so good)
- 2 ÷ 10 = ?? (aw, crap!)
I can break ten things into two groups of five, but how can I break two of something into ten parts?
It’s not wholesome!
We got out of a jam last time by inventing a new kind of number, so let’s try that again. This time we’ll invent the rational numbers.
They just take the form p/q (mind your Ps and Qs), so we can express any fraction as the ratio between two integers. (“Ratio” hence rational numbers.)
Cool. Division solved! In fact, the rational numbers kind of embody the notion of division.
Note that, if we reverse the order, so it’s q/p, we get the multiplicative inverse, because p/q × q/p = 1, the multiplicative identity (the Ps and Qs both cancel).
Speaking of one, we’ve got one more operation left, exponentiation, and by now I’m betting you know what’s coming. Raising one number to the power of another seems okay, but…
- 22 = 4 (no problem)
- 41/2 = 2 (still okay)
- 21/2 = ?? (not okay!)
In N e, if e is a positive whole-number specifying how many times to multiply N, then 1/e is the inverse operation. It asks, what value, multiplied e times, gives us N.
In other words, it asks for the nth root. In the case of 1/2, that’s the square root. (The cube root would be 1/3, and so on.)
It’s not rational!
This new type of number thing is starting to become habit. Does it ever end?
In fact, yes, there’s one more step, and this is the neat one (the point of this post). How we get there makes a cool kind of geometrical sense.
Let’s start with the geometry of multiplication. Let’s say we want to multiply three by two:
We start with +3 out on the number line. That gives us our jump distance. We want to multiply it by +2, which is positive, so we make two jumps in the same direction. The answer is +6.
Now suppose we want to multiply three by minus two:
As before, we have +3 to the right as our jump distance, but this time we’re multiplying by -2, which is negative, so we have to jump the opposite way.
This is essentially a math axiom. If multiplying by a positive (natural) number jumps X-many times in the same direction, then multiplying by a negative number must jump in the opposite direction.
This is necessary for the symmetry of the geometry. We can also look at the previous example as multiplying -2 by +3:
We get the same answer (of course) because this time we start with a negative number and then, because +3 is positive, jump in the same direction three times.
The kicker comes when we multiply a negative times a negative. This time we’ll multiply minus three by minus two:
We start in the negative direction with -3, but the -2 makes us jump in the opposite direction, so the answer is +6. The same thing happens if we start with -2 and then, because minus 3, jump three times in the opposite direction: +6.
This is why multiplying a negative number by another negative number gives us a positive number.
Crucially here, it’s why squaring a negative number gives us the same (positive) number that squaring a positive number does:
We always get a positive number when we square a number. Any number.
This seems to explain why the square root of any negative number makes no sense. All squared numbers have to be positive.
It’s not natural, wholesome, or rational. It’s not even real!
Except we’ve been here before, and we know what comes next. A new type of number. But what type?
Mathematicians had established that the square root of minus one, if accepted as a weird given, seemed to complete math. At first it was taken only as a kind of magic sauce that didn’t map to anything real. (Hence the awful name, imaginary number.)
The German mathematician Carl Friedrich Gauss wanted to name them lateral numbers, and, as you’ll see, that would have been a much better name, because it turns out these new numbers do have a physical reality.
(Math and science have many examples of ‘imaginary conveniences’ that turned out to be physical necessities.)
To understand, let’s take a closer look at the geometry of multiplication:
Multiplying 1×1 is trivial, but note how we can keep multiplying by one forever, and we’ll always get the same answer: +1. We’d end up with a series of answers: +1, +1, +1,…
On the other hand, if we multiply +1 by -1, we jump the opposite way:
We can see this as a rotation. Multiplying by -1 rotates our direction of movement by 180°. (Multiplying by a larger negative number rotates and increases, while multiplying by a negative number less than one rotates and decreases.)
If we multiply by -1 again, we rotate another 180° back to +1. If we continue to multiply by -1, we get the series: -1, +1, -1, +1,… (flipping 180° each time).
I’ll repeat this important point: The geometry of multiplying by a negative number is to rotate on the number line by 180°.
Now let’s consider imaginary i. We know that i×i = -1 (by definition).
If multiplying by -1 is a rotation of 180°, and if i×i equals -1, then multiplying just by one i (half of i×i) is a rotation of 90° (half of 180°).
Consider what we get if we multiply +1 by i repeatedly:
- i × +1 = +i
- i × +i = -1
- i × -1 = –i
- i × –i = +1
And then the cycle repeats. Each multiplication rotates 90° and four such rotations brings us a full 360° back to where we started. (See diagram at the top of the post.)
Which means the imaginary numbers have to be 90° from the real number line. Numbers are inherently two-dimensional.
Far from being something made up, it very much appears (once again) this is something any intelligence must discover as it explores numbers.
The complex numbers have a physical embodiment.
The irony is that, while it’s possible no numbers are real in the mathematical sense, complex numbers do seem to have a real (as in ontological) geometric nature.
(If reality is mathematically rational rather than real, complex numbers would just use the rational numbers, no problem.)
This geometrical view of numbers may explain why complex numbers keep turning up in physics. Reality is very much about geometry.
Stay geometrical, my friends!