I saw a post on reddit asking whether imaginary numbers were actually any use at all.
My response was getting so long, I decided to post it here:
Imagine Ug, the first ever mathematician. She’s at her cave, rocking her baby (on a rock, of course) and thinking about numbers. “Ug knows numbers. There’s 1, 2 and 3”. Simple, and nice.
However, a troubling thought occurs to Ug. While she can count the three current members of her family, she realises, patting her large rounded belly, that her number system has fundamental limitations in its power and expressiveness.
Her mathematics can be used to count her family now, but she runs into trouble if she tries to add one more. The simple equation x = 3 + 1 has no solution. A year ago, x = 2 + 1 presented no difficulty.
She decides to invent a new number, and call it “four”. Her tribesmates look puzzled as she explains. “It’s just a concept”, they say. “What’s the use of it?” They have trouble accepting that four might be a “real” number. She points out that even if it’s not real, the concept of “four” helps solve some advanced problems in demographics and agriculture. “Fine,” they shrug.
In her diary, she writes up the idea of “four” in detail, and also discusses the possibility of numbers such as five or six.
We can see that the weakness of her system, its inability to solve equations like x = 3+1, is evidence that there are numbers missing from her system. The number four – indeed five and six as well – should be included, not just because they are useful, but because make the system more complete, more expressive… and simpler in ways (despie having more numbers), because we can now claim “we can add 1 to any number, we don’t have to have complicated rules to describe which numbers can have 1 added to them”
Seen this way, if a number system can solve some equations, but fails to solve other apparently similar equations, then the number system is not just less than useful, it’s actually not the whole story at all. It’s just a chapter of a book, and we need to add the missing pages.
Fast forward to the dawn of civilization. Mathematicians can solve equations like 2x = 18 and 2x = 16, but not 2x=17. So, they invent fractions. There are objections – numbers like “17/2 of a horse” make no useful sense. Nonetheless, the idea of fractions catches on, and now we all accept these as truly-really numbers.
Likewise, irrational numbers were discovered in ancient greek times. The previous number system could be used to solve, say x^2=9, but not x^2=10, so new numbers were added to the previously incomplete number system. Note, though, these “new” number have been given a rather rude name, a pattern that continues.
Now we’re in the middle ages. We can solve x+4=8 but not x+8=4. Let’s invent some new numbers, and call them negative to indicate how much we dislike them. We’re biased. We hates thems, these negatives and irrational numbers, don’t we, preciousss? Let’s fast forward again.
We can solve x^2 – 1 = 0 but not x^2 + 1 = 0? Let’s invent still more new numbers. This time, we’ll give them not one, but two dirty names. Imaginary. Complex.
It seems like this could go on forever, finding unsolveable equations and adding new numbers, but it doesn’t. With the introduction of complex numbers, something magical happens: algebra becomes simple. A quadratic always factors into two factors. A cubic polynomial always factors into 3 factors, and so on. We hunt and hunt for equations that look like we should be ablke to solve them but can’t, and find none. It’s as if the complex numbers are all the numbers there are, with no new numbers left to add.
We turn back to some calculus problems that were horribly difficult before, and find that complex numbers make them ridiculously easy. We discover that trig functions and exponential functions are almost the same thing. We discover that differentiable functions are all infinitely differentiable, so we can dispense with concepts like ‘twice differentiable” and so on. Radii of convergence of series suddenly make sense. When we turn to physics, we find that complex numbers are phenomenally useful for describing certain phenomena – alternating currents, hydrogen atoms, sound waves and much, much more.
Then, armed with these amazing new numbers, we do something totally stupid. We get bunches and bunches of stressed out calculus students, already struggling with limits, antiderivatives and vectors, and say “Hi guys! How’d you like to learn about something complex and imaginary!?”
They groan, not realising that the words ‘imaginary’ and ‘complex’ are just labels for the numbers that were previously missing, that have been kept secret from them for too long. Unfortunate labels for ideas that give radical shortcuts for some problems and make some parts of math far simpler than they seemed before.
Perhaps calculus profs should say, instead, ‘Hi guys! How’d you like to learn about some useful and beautiful numbers?’ (flashes a slide of Seahorse Valley) ‘in a week or two, you’ll be able to make pictures like this! A few weeks more, and Q3b on yesterday’s quiz will be a five-minute walk in the park!”