Upside-Down Tooth Numbers
Alex and Eli know that 15 ÷ 3 = 5, that 63 ÷ 7 = 9, and, given a little time, could figure out that 104 ÷ 8 = 13. That’s not bad for five-year-olds.
As we were discussing sharing a treat the other day, we naturally happened upon a situation in which it would be good to know what 1 ÷ 2 is.
Alex suggested, “Two.”
I explained that while the order of the numbers in multiplication doesn’t matter — for example, 2 × 3 = 3 × 2 — order does matter with division. I used the word commutative, but I also tried to explain it with plainer language, too.
I took a rectangular piece of paper and ripped it in half. “How big is each piece?” I asked. The both knew it was one-half. “So there you have it: 1 ÷ 2 = 1/2.”
But why stop there? I divided one of the halves in half again, and I asked, “How big is this piece?” They both knew it was one-quarter. “So that shows that 1/2 ÷ 2 = 1/4.”
They saw that if we continued in this manner, we would get 1/8, 1/16, 1/32, and so on, a pattern they called the upside-down tooth numbers, because the numbers in the sequence are the reciprocals of the powers of two. (For them, tooth = 2th.)
Looking at the pieces of paper on the table, I asked a more advance question. “What do you think we’d get if we added 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …, and we kept adding half as much each time?”
Eli thought for a few seconds. “I think it would be one whole,” he offered.
I was surprised. “Why?” I asked.
“Well,” he said, “I’m thinking about a circle.”
I looked at the rectangular pieces of paper on the table, which had nothing to do with a circle. Sure, he got the right answer, but his reasoning is way off base, I thought.
He explained further. “If you fill half a circle, then a quarter more, then an eighth, and keep going, you’ll eventually fill the whole thing. That’s why I think it’s one.”
Visually, Eli’s argument would look something like this…
Holy sh*t, I thought. That’s pretty good for a five-year-old.
I don’t even care that he doesn’t understand this joke:
An infinite number of mathematicians walk into a bar. The first asks for half a beer. The second asks for a quarter of a beer. The third asks for an eighth of a beer. The bartender interrupts, pours one beer for them, and says, “You guys don’t know your limits.”
I think I was so impressed with Eli’s solution because infinite series can be difficult, even for mathy professionals:
A mathematician will call an infinite series convergent if its terms go to zero. An engineer will call it convergent if the first term is finite.
And let us not forget the Eilenburg swindle, which proves that 1 = 0 since:
1 − 1 + 1 − 1 + … = 1 + (−1 + 1) + (−1 + 1) + … = 1
1 − 1 + 1 − 1 + … = (1 − 1) + (1 − 1) + … = 0
Related to all this, there is a perhaps apocyphal story in which it is rumored that a student once posed the following problem to John von Neumann:
Two trains begin a mile apart and head towards each other at 60 miles an hour. A fly on one train flies at 120 mph to the other train, and when it touches the other train, it immediately turns around and flies back to the first train, and so on, flying back and forth between the two trains until it gets squashed in the middle. How far does the fly travel?
von Neumann thought about it a moment and said, “One mile.” The student said to von Neumann that most people don’t realize that the problem can be figured out easily: the trains meet in 30 seconds, and the fly can travel one mile in half a minute; yet most people think they have to add up the infinite series to figure out how far the fly travels. As the story goes, von Neumann replied, “But that’s how I did it.”