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← Mathematics: stupid and clever questions for people who understand

Jos Gibbons's Avatar Jump to comment 6 by Jos Gibbons

These examples have reminded me that quite a lot of common maths confusions, both by modern laypeople and by historical mathematicians, stem from wanting to know “what is” the entity with a notable set of properties rather than what it does. For example, consider this question:

What is the squareroot of minus one?

One way to answer this is to say, “there’s a quantity we call i, or alternatively j, characterised by its square being -1 and its having ordinary interactions with real numbers (those on the familiar number line) in their arithmetic. Obviously, -i also squares to -1, so there are two square roots of -1, just as 5 and -5 both square to 25. Indeed, one convention is that j=-i, so i and j are the two roots of -1.” Note this definition tells you how it all works, e.g. you infer that (1+2i)(3-4i) = 3+2i-8i*i = 11+2i. But what “is” i in a metaphysical sense? Tough luck with that one; mathematical objects are defined “implicitly” by how they interact rather than “explicitly” in the way you’d say “cats are these things over here”. Indeed, if you replace i with –i everywhere in true statements, they remain true, e.g. (1-2i)(3+4i) = 11-2i. The point is not that we’ve “found” an i whose square is -1; the point is that you can claim there is at least one such i, and from that many other things readily follow, such as the examples above.

Several times in the history of mathematics, problems we couldn’t solve were solved by saying “here are some new numbers with the right properties; let’s see what that implies”. So mathematicians said, “but what are they?” Only about 100 years ago did they reach a consensus that that’s the wrong question to ask. It seemed plausible; positive real numbers are lengths; negative numbers help in accounting, and statements including them can be carefully restated to avoid them; rational numbers can be defined with things like slices of cake, and so on. But with complex numbers it gets a lot weirder. Sure, they have applications, e.g. in quantum mechanics; but you can’t seem to trip over i times something in the real world, in the way you can 3.2 kg of objects on the floor.

i is an interesting example because the first reason people disliked square roots of negative numbers was because, at the time, they weren’t even sold on negative numbers! In those days, there were four kinds of monic quadratics. You didn’t say, x squared + ax + b = 0 was the general case; you said x squared = ax + b and x squared + ax = b were 2 different kinds of quadratic. Only positive roots (or a root of 0) would “count”. If a quadratic equation with real coefficients has complex non-real roots, nowadays we easily find them with the quadratic formula, but in those days they’d say “those roots don’t exist”. They didn’t mind saying, “the reason you can’t find 2 roots to every quadratic is because not every quadratic has 2 roots”.

But then something weird happened. Cardano invented a way to solve any cubic equation. But it turned out that even when all three roots were positive real numbers, complex numbers had to be temporarily worked with to get the roots; their role was irreducible. This was called the casus irreducibilis, and ultimately motivated the concession that complex numbers “count”.

Another problem with phrases like “root(-1)” is other systems exist with such roots obeying different rules. For examples, quaternions have infinitely many numbers whose square is -1, because they don’t commute, i.e. when you multiply quaternions the order matters. Wherever you find “square roots of -1”, the important thing is, what are the rules that are obeyed and what can you infer from that?

One last potential problem if you introduce new numbers is it might make the enlarged theory inconsistent. We can show that, if the theory of real numbers is consistent, so is the theory of complex numbers. But now look at this question:

What is the result of any number divided by zero?

Division is an inversion of multiplication, so “a over b = c” means “the unique root of bx = a is x = c”. (If commutativity fails then care is needed concerning whether we prefer to instead define division with the equation xb = a.) So if I say “a over 0 = c” that means “the unique root of 0x = a is x = c”. But if a = 0 any x is a root, while if a isn’t 0 no x is a root, so it’s never unique. But let’s suppose I introduced “z = 1/0”. I can’t keep all the old rules. Look at this argument: “0z = 1, 2 = 2(0z) = (20)z = 0z = (30)z = 3(0z) = 3, a contradiction”. Here it is assumed “z = 1/0 implies 0z = 1” and “a(bc) = (ab)c” (associativity of multiplication). You can invent z if you want, but you’ll have to drop at least one of those rules. One example of an area where we invent division by 0 is wheel theory.

Quine playfully answered “what is root(-1)” with:

The first one is easy, it's ln(cos(π))/π.

Let me explain why that’s right (in a way). cos pi = -1. (Quine, please tell me how you did an actual pi!) To define ln(-1), we use the fact that ln is the inverse of exp, rather like the division-multiplication relationship. But for complex numbers exp doesn’t have a unique inverse, so ln(-1) could be ipi(2n+1) for any integer n. One common convention implies, among other things, that we take n = 0. The argument is now complete; dividing by pi gives i. I hope that makes sense to Neodarwinian. S/he could reply, “ln(-1) is ambiguous, so maybe you have to divide by 3*pi, for example”. Well, true. But Quine’s convention defines ln(z) as the so-called principal natural logarithm of z.

What is the first derivative of X cubed?

3 X squared, if you differentiate with respect to X. An old joke:

Full of himself, e to the x meets a scared 7 running round the corner. “A derivative is after me! He’ll destroy me by turning me into 0.” “Well, he can’t change me however often he mist me,” sniggers e to the x. “Hello,” he says, introducing himself to the derivative as he arrives. “I’m e to the x.” “Hello,” replies the derivative. “I’m d/dy.”

Mon, 07 May 2012 21:06:35 UTC | #940414