Love minus zero, plus something

Somewhere down the fanfiction road, I’m tempted to talk Twilight Sparkle into dividing by zero, just to see what happens. In the meantime, Mark Alger, no slouch of a storyteller himself, is playing with the concept, starting, sensibly enough, with Dolly telling him you can’t do that:

Well, properly speaking, you can, but the answer is out of the normal bounds of our concepts of numbers. And, of course, computers lose it when you try to make them calculate it. But, really, it makes logical sense. Zero zeroths is a whole zero, right? I mean, it’s nothing, but it’s ONE nothing. A slippery concept, I’ll admit, but not as weird as n dimensions.

And this also requires admitting that dividing zero by itself to get one is a special case. And what if that means that 0/0=1 is also 0/0=∞? Talk about your special cases. And what does that imply about the question raised in the linked article as to whether infinity actually exists in the real world, or is just a mental construct? See how that blows your dress up.

The reason we have mental constructs in the first place, I suspect, is as placeholders for things we actually haven’t found yet. (Think “Higgs boson”; it explains much, even in its “well, we think we saw one” status.) If you push me, I’d say that infinite anything probably violates at least one law of physics — and that a hundred years from now, those laws will have probably been updated somewhat.

That said, there are transfinite numbers, which I understand barely if at all, and hyperreal numbers, which are at least easy to explain:

The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form

1 + 1 + … + 1.

Such a number is infinite, and its reciprocal is infinitesimal.

I never expect to see a number that is truly infinite, though its reciprocal I see every month on my bank statement: it’s the interest rate they pay me.







8 comments

  1. McGehee »

    6 June 2013 · 2:28 pm

    Zero zeroths is a whole zero, right? I mean, it’s nothing, but it’s ONE nothing.

    Rendered in exponential notation as 0 to the power of 1. Which is still zero, just as 14 to the power of 1 is still 14.

    The one way I can think that you can use zeroes to get 1 is 0 to the power of 0 — any number to the power of zero, IIRC, is 1.

    But that’s not division.

    And really, the only numbers that are not mental constructs are positive integers. Of which zero isn’t one (heh), and neither is infinity.

  2. fillyjonk »

    6 June 2013 · 5:29 pm

    I know (a little) about the “mental constructs” argument, but I tend to prefer this image as being emblematic of what happens when you divide by zero.

    (Also, it was grabbed off of a mathematician’s post about dividing by zero….)

  3. CGHill »

    6 June 2013 · 5:42 pm

    Grayer than the usual black hole, but yes, definitely.

  4. Dick Stanley »

    7 June 2013 · 7:10 am

    The interest rate is so low it’s a wonder anyone uses banks anymore, other than as convenient places to store money—until they lose it all for you, since they’re loaning it out all the time and might not have it when everyone suddenly wants it. Or, as in the Cyprus deal, when the government does.

  5. McGehee »

    7 June 2013 · 8:42 am

    That’s one benefit to not having a whole lot of money, Dick — when I want my money, I’d get it all, even if my banker was the squeegee guy on the corner.

  6. McGehee »

    7 June 2013 · 8:45 am

    Dividing by “almost zero” is not dividing by zero. There is a qualitative difference, because when you’re dividing by “almost zero” you’re dividing by a presence, not an absence.

  7. Dick Stanley »

    7 June 2013 · 10:45 am

    I doubt the rich use banks at all, more likely brokerage houses. It’s the little guys will be screwed by bank collapses.

  8. Mark Alger »

    8 June 2013 · 7:42 am

    Wow! A whole lot of serious for what was meant at the time (albeit admittedly not labeled as such) as a throwaway joke.

    M

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