The oddness of numbers

I can divide by zero

Leopold Kronecker is supposed to have said, “God created the integers; all the rest is the work of man.” That work has advanced by extending the natural numbers to include negative numbers, zero, fractions, square roots, pi, e, i, and more. Every time we expand the system we gain and we lose. Things become more abstract, and we have to work harder to convince ourselves what we have now is a number. Not quite in historical order, it might go like this:

The natural numbers are fine: 1,2,3,4…

The rational numbers aren’t bad. Two halves make one whole. And now Johnny can have a pie.

Zero is weird but useful, though it does introduce an exception to remember. “Zero: can’t divide by that.”

Okay, it makes sense to associate any length with a number; and so the square root of two must be a number, even though we can’t say what number exactly. We can give it its own symbol, like zero.

And as long as we’re giving out symbols, here’s pi. And here’s e. And now the numbers aren’t countable. We’ve made a system of “numbers” with which it is in principle impossible to count. That’s bad. No, wait. That’s counter-intuitive; surprising! Yeah, surprising. Surprise is good. Besides, who but us is going to even notice? We may need to redefine “counting.”

And it makes sense that all equations like this should have roots, even if they don’t really. Better to say some roots just aren’t really numbers; or not Real numbers.

These complex numbers are handy. They aren’t so strange after all – just ordered pairs of real numbers. And C is a field, just like R. But we can’t put them in order.

Hmm, what if we consider ordered pairs of complex numbers?

And we press on, eventually getting to the point where we have ordered 16-tuples of real numbers. These we can divide by zero. Of course at that point, the hypercomplex numbers we’re dividing by zero are not commutative or associative or countable or ordered. No reasonable man would say they were numbers at all.

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5 Replies to “The oddness of numbers”

  1. Zero is weird but useful, though it does introduce an exception to remember. “Zero: can’t divide by that.”

    Interesting; we see math in utterly different ways.

    If you put 16 apples into zero groups (16/0), how many apples are there? Same way that saying nonsense in English doesn’t make “exceptions” in my mind, being able to do it in math-notation doesn’t make an exception….

    It’s been a LONG time since my last math class, and even then it wasn’t very advanced, but aren’t all the symbols just shorthand? Pi means “that great big long number that lets you calculate a circle,” 3*3 means 3+3+3 or “a group of three, a group of three and a group of three,” 3^2 means all of those…. The further in you get, the more is packed into the symbol, right?

    1. Sure, that’s a valid point. We’re embedding things in other things. The problem comes up when we call increasingly abstract things numbers. The individual abstractions and expansions are not huge hurdles. All the other numbers you can divide by, but not zero. Okay, maybe “something we can divide by” isn’t an essential property of a number. The numbers like pi and e are uncountable – they can’t be put in one-to-one correspondence with the natural numbers. That seems to me a pretty weird thing for numbers to be, but it does put them in one-to-one correspondence with the points on a line, and “line” is intuitively valid. A Complex number is just a pair of Real numbers. Very nice and handy, but now we can’t put them in order. Is order and essential property of numbers? Maybe not. Quaternions are pairs of complex numbers. Useful, but now a*b is not b*a. Are quaternions numbers? And the process goes on. At some point the abstractions we’re manipulating lack so many intuitive properties of numbers that they seem to me not to be numbers anymore.

      1. I don’t think I said it very well…

        Basically, I don’t see a number when I look at pi, any more than I see “a number” when I look at 3*5, or 7/2, or “split these cakes up so it’s fair, ‘k?” You can get a number from them, sort of, but they’re just shorthand for doing something. (pi: “how many times can the string you used to draw a circle go around the outside of the circle;” 7/2 = 3.5 = “if we’re going to make two even groups, we’ll have to cut one of these in half.”)

        A Complex number is just a pair of Real numbers. Very nice and handy, but now we can’t put them in order.

        If it’s a pair of numbers, then it’s not a number — you can get something out of it, but it’s more a process.

        Heck, 3.5 is rather a “process,” since it means three whole things and one cut in half… it kinda gets to be philosophy, really; do numbers exist, or are they a construct for understanding things that are?

  2. I can’t see the motivation for calling these things numbers. With all of the other additions that you mentioned, the reason that the set exists was to expand the set of numbers so that the set would be closed under common operations. In this case, they are doing something different: first they are interpreting complex numbers as ordered pairs (which are not numbers), then they expand the notion of an ordered pair.

    1. I’m okay with it all through the complex numbers, but maybe that’s only because I’m used to them. When we count something, we get an integer. When we measure, a rational number. The others we get by logical inference: this length must have some numerical value; this equation must have a solution; if this sum approaches a limit, that limit must itself be a number.

      Complex numbers remind me of the wave/particle duality of light. They’re numbers (2+3i; re^i\theta) when we need them to be, and ordered pairs when that makes more sense. On beyond C, quaternions can be written as numbers: a+bi+cj+dk, the others similarly, but they’re not commutative, and to me that makes them no longer numerical enough to be numbers. We use the complex numbers, but the quaternions et al. today are only curiosities, as far as I know. We use vectors and matrices instead.

      Some students want to answer every math question with a number: Q: “What is the equation of a line passing through…” Ans: “3.” He’s in math class, so the answer must be a number. Maybe the same mental process leads mathematicians to call the elements of the sets they work with “numbers.”

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