Sunday, March 30, 2014

Betty and John's Bogus Journey

[From: The Book of John and Betty]

"Wait, what?" Betty said. "Just make another number up? Seriously?"

"Just make another number up! Make another number up!" John was hopping in place now, giggling while he chanted.

"What does that even mean?" Betty said. "I mean, I guess you can do whatever you want in math, sure, but why would we make up an answer to an impossible question?"

"Curiosity!" John said.

"Curiosity? What sort of curiosity are we talking about here? You tried squaring 1 and -1 and you didn't get -1. Great. That's proves it: my question had no answer. It's a great argument, simple and beautiful, so let's move on."

"Sometimes in math we just need to take a chance, Betty. Maybe for once in your life could you learn to take a chance?" John stared at Betty. "We're having fun now, aren't we?"

"Fine then, let's make up another number," Betty said, pacing away from John. "But while we're at it, I've got some other numbers to invent. I mean, you can't divide by zero, right? Let's make up a number for that. Let's invent a number that stays the same when you add five to it, and another number that's bigger than itself. Let's just throw new numbers at every impossibility in mathematics and call it a day."

"Oh, come off it, you big stickler," John said moving his beautiful golden hair off his ears. "You're completely ruining our afternoon, just like you ruin everything."

"Well as long as you're at it why don't you invent a number to fix our ruined afternoon you ridiculous elf."

And suddenly trumpets sounded and Heaven opened and Lord Math Teacher descended upon the White Room where John and Betty fought. And lo a chalkboard appeared as well as chalk and John and Betty were verily impressed.

And the Lord Math Teacher spake, saying: "John, Betty, you are both loyal servants, and I wish to praise you for your great attitudes and overall grit. But let Heaven and Earth bear witness: there indeed is a difference between these numbers you wish us to invent, oh Betty, and that which John has invented. Know it: John's number will lead to no further contradictions, while your numbers lead to contradictions, which is basically a no-go in mathematics, for contradictions are impure in the eyes of the Lord."

"Lord Math Teacher? Lord?" Betty waved in the Lord's direction. "Excuse me? I have a question?"

"Umm, yes Betty."

"How do we know that inventing a square root of -1 won't lead to contradictions?"

"An excellent question, my daughter!" Lord Math Teacher replied. "Yes, and unfortunately proofs of non-contradiction are sort of dicey but I swear upon the heavens that Model Theory has some of the answers. I am Lord Math Teacher, and I hope that answered your question."

And then from the skies a bell rang and the Lord departed.

And this answered all of Betty's questions and she was entirely satisfied with John and complex numbers for the rest of her days, verily.


  1. "The way of mathematics is to make stuff up and see what happens." Vi Hart.

    1. I don't want to pretend to have anything like a wide-ranging knowledge of the history of mathematics and its development. But where exactly does this sort of whimsy have its place in our subject's history?

      To be clear, I'm not denying that a certain amount of whimsy is necessary in constructing and inventing proofs and solutions. That seems just part of the game of creative work -- trying on ideas, seeing what works and then progressing.

      But we're talking about something entirely different with complex numbers. The claim is that you can motivate their existence just by pure invention. What else in the history of math has been created through such what-if-itude?

  2. I don't know... but I tell people that we can't really have less than nothing, either. I explain it as taking the same "opposite" concept that means -1 times -1 is + 1, we can extend the numberline in two dimensions -- voila! a graph! and find an opposite of -1 that's even more opposite... so that it takes i * i to make -1.

    I'm just glad nobody's making 'em work with complicated complex numbers so that foofoo squared = i. Just for fun, you know.

    1. But foofoo is just sqrt2 +sqrt2*i.

    2. The square roots of i are +/-(1 + i)/sqrt(2).

  3. This story makes me uncomfortable because it misrepresents the history of complex numbers. Complex numbers arose from the solution of the cubic equation. If a cubic equation with integer coefficients has three distinct real roots, and it is solved using the cubic formula, then square roots of negative numbers will occur at intermediate steps of the calculation. Although the imaginary parts cancel out in the end, the complex number system was needed to make sense of these calculations.

    Of course, mathematicians are free to ask what-if questions, and this is sometimes fruitful, but important ideas usually come from attempting to solve specific problems.

    1. Rather curious here... how did complex numbers not arise is solving the quadratic before solving the cubic? Was the cubic solved first, or was it due to the knowledge that in the cubic the "imaginary" parts cancelled out and thus had to be meaningful that inspired mathematicians to look at them more closely?

    2. Don't remember the details from my long-ago History of Math course...but from what I recall there were certain cubic equations that mathematicians knew had real roots, but that they couldn't solve using regular strategies without getting this square-roots of negative numbers thing cropping up. So, they went with it, the terms cancelled out and the solutions were confirmed.

      I think this was the only "legitimate" use for imaginary numbers for a while--looking at cases in which they *didn't* cancel was another big step in mathematical thinking.

  4. Touche'. (Now I'll have my students read John and Betty, and then this. Then we can discuss.)