One night over ice cream, we found ourselves talking about repeating decimals. After we filled a napkin with the proof that 0.999… = 1, Jeff had a brainstorm, and wrote the following:
“What the heck is that?” I asked.
“It’s a number where the nines go on forever,” he said. “And afterwards, there’s a four.”
“Yes,” he said. “Exactly.”
“I see. So it’s bigger than this…”
“But smaller,” I continued, “than this.”
“Well, clearly,” he said. “All three numbers are identical up until the infinity-th place past the decimal. They differ at the infinity-and-1st place.”
A woman ordering her ice cream eyed us warily.
“What about this?” I said, and drew the following:
“Naturally,” Jeff said. “The 9’s go on forever. Then there’s a 4. Then the 8’s go on forever .Then there’s a 1.”
“That’s all logical enough,” I said (perhaps bending the meaning of “logical”). “But what if there’s something like this?”
“Is that different,” I continued, “from this?”
“That’s tricky,” Jeff said. “The first one is nines forever, then nines forever again. But the second one is just double-nines forever – which is the same as nines forever.”
“Let me get this straight,” I said. “You’re saying this…”
“Exactly,” he said.
By this point we’d filled most of our napkin. I unfolded another so that the treatise could continue to evolve. “I’m not convinced that these are well-ordered,” I said.
“Well-ordered” is a math term I’d learned in college. A set of items is “well-ordered” if you can put them into a nice, logical order, with no contradictions. For example, tennis players’ heights are well-ordered. But their skills are not – Anita beats Brianna, and Brianna beats Carla – so you’d expect Anita to beat Carla. But instead (in this real-life example from a friend’s high school tennis team), the reverse happens. Carla beats Anita. That’s not well-ordered.
“Well, let’s figure it out,” said Jeff. “If we can find a case where A > B and also B > A, then you’re right. If not, then everybody everywhere will have to start using these numbers, because they’re more fun.”
“Good luck with that,” I said.
At this point, my sister joined us for an ice cream. Holding her cone with one hand, her free hand contributed the following to our napkin:
“Oh!” Jeff exclaimed. “Cool.”
“I see,” I said. “The nines go on forever. And then the forevers go on forever. And then, after all that, there’s a 1.”
This is when things got out of hand (if they hadn’t already).
(the nines go on forever, and then there’s a one, and then that nine-forever-then-a-one pattern goes on forever, and then, at long last, there’s a two)
(as above, except that repeats forever, and then at the very end, there’s a 7)
(as above, except that repeats forever, and then there’s a 3, and then… well, hopefully you get the idea)
“Are these well-ordered?” I asked, as we began to unfold a fourth napkin.
“I don’t know,” Jeff said. “To be honest, I’m not entirely sure what they are.”
By this point, it was 11pm, closing time at J.P. Licks. The last ones there, we stuffed the napkins into our pockets, grabbed our bags, and made for the door.
“What were you guys working on?” one of the employees asked us.
“The Kaufman decimals,” I told him. “On napkins today, in textbooks tomorrow.”
He politely rolled his eyes.
Note: As usual, I’ve used imagination to plug the holes in memory – though Jeff is every bit as weird and wonderful as described. Mathematical minds out there: Are the Kaufman decimals well-ordered? I’m curious mostly about first-order Kaufman decimals. My hunch is that the higher-order ones must run afoul of logic somewhere or other, although I’d love to see a demonstration of that, too.