You Get What You Order

I’ve heard many times that there is no ordering for the complex numbers. I don’t know why people insist on telling me this so much. They could use that time to tell me to follow my heart, or to tell me not to eat Tropical Fruit Tums like they’re candy, but instead they tell me that the complex numbers are not ordered.

There are some similar facts, all of which I take on faith, that I also hear frequently. Pi is transcendental. The universe is about 3 degrees kelvin, and very nearly a black body. Computers work because of these little tiny logical gates and some stuff to store data and some stuff to organize all the other stuff, and various other things I’m too dumb to understand. Uruguay exists. But there’s a difference between the items on this list and that fact about complex numbers – the ones on the list make sense.

Not ordered? No possible ordering? Yeah, well screw you. I’ll make one up right now. Okay, take two complex numbers and write them in polar form. The one that’s longer (greater magnitude) is bigger. If they’re the same length, the one with the smaller angle (lesser phase, with phase restricted to 0\leq\theta<2\pi) is bigger. The end.

I've been through this seemingly-bulletproof argument in my head every time in the last few years someone's told me the complex numbers aren't ordered. (Frequently, this claim about our imaginary friends is a parenthetical remark made by someone as they write a book or article, so it's a bit of a stretch to say they were telling me about this fact.) It happened again today. I know that it’s not true that everyone in the world is too dumb to come up with an ordering like my brilliant one above, so I finally decided to figure out what they were talking about.

I had seen “partial ordering” and “total ordering” defined in math contexts before. A total ordering is like ranking manliness by penis length. There’s a longest, a shortest, and everyone else falls somewhere in between. (I suppose about half the people in the world would be in a dead tie for last place, but let’s exclude them for didactic purposes. Also ignore the problems with making measurements accurately, and small fluctuations in penis length, etc.)

A partial ordering is like ranking manliness by how many of your favorite alcoholic drink you can consume before upchucking or passing out. Everyone drinking the same drink can be compared, but people drinking different drinks can’t. (I suppose they could, but we set the rules of the manliness competition so it doesn’t work that way. Also, everyone gets to participate in this competition! Unless you’re in a coma.)

But when mathy people say the complex numbers can’t be totally ordered, that’s not what they mean. They mean they can’t be totally ordered in a particularly useful way.

So take a look at my ordering. For one thing, it says that -2 > 1 > -1 > 0, which is certainly counterintuitive. But the real problem is you can’t do normal stuff with it. For example, look at the following seemingly-logical argument:

\begin{array}{ccc} 1 & > & -1 \\ 1 - 1 & > &-1 - 1 \\ 0 & > & -2\end{array}

We begin with a true inequality, subtract one from each side, and end up with an untrue one (remember from above that -2 > 0). So in my ordering of the complex numbers, we can’t do normal arithmetic with inequalities. The mathematical statement, then, is actually quite an interesting one: there is no total ordering of the complex numbers that lets you do the ordinary, sensible things with inequalities. No matter how clever you are, you can’t invent one. I don’t know why that’s true, but there are more true things out there than I can understand, and it’s at least something that makes a bit of sense.

The interesting thing about this was how easy it was to figure out what was meant by the whole no-ordering business when I just went looking for it explicitly. I got it from wikipedia, of course (article). But somehow, many, many sources (perhaps not so many as my exaggerations would lead you to believe, but enough to lead me to believe these exaggerations justified) simply cited the result without bothering to explain what it means. Reciprocally, I never really called anybody out on it. Until today, I guess.

That’s the nice thing about math – it makes sense. When it doesn’t make sense, you or somebody else is doing it wrong. Figure out why.

PS – an interesting fact related to one above is that Pi exists – that is, that in euclidean geometry you can prove that the ratio of the circumference of a circle to its diameter is constant. This is intuitively obvious, but I wouldn’t know how to prove it. No matter, somebody else did it for me. The point is that if nobody had mentioned it to me, I probably wouldn’t even have realized that it needed to be proven.

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2 Responses to “You Get What You Order”

  1. kiwi Says:

    Here’s why the complex numbers cannot be ordered in a nice way. Call a number positive if it is greater than 0, negative if its opposite is positive. Then any square is positive (or zero) because it’s either a positive times a positive or a negative times a negative. But that means both 1 = 1*1 and -1 = i*i are positive.

  2. Paul Murray Says:

    ANd there’s only a partial ordering on space/time coordinates with respect to time.

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