Successors – More Arithmetic

“What comes after seven?” “Eight”

“What comes after sixty-three?” “Sixty-four”

The Pitch

“What comes after one thousand twenty-six?” “One thousand twenty-seven.”

I’m not sure why the room full of freshmen was playing along with me. Maybe just because their teacher was letting them. Maybe it was a break from solving equations with polynomials over denominators. Maybe they were curious why the strange teacher was asking them such simple things.

“One thousand twenty-seven comes after one thousand twenty-six? Where did you learn that?”

“Kindergarten!” “First grade!” “Really, your first grade teacher taught you about a thousand twenty-six? My first grade teacher didn’t teach me about a thousand twenty-six.”

I addressed the rest of the class “Who here studied a thousand twenty-six in grade school?” “Well, not exactly…” “No, who studied precisely a thousand twenty-six?”

I paused for a beat. For another. “So how do you know what comes next?”

“There’s a rule.” “What rule?”

“Add one!” “Add one? You went to a school that taught you to add before they taught you to count? Who else went to a school where you learned to add before you could count?” The pause was shorter this time.

“Change the six to a seven” It was the first thing this student had said, quiet, maybe a little shaky. “OK,” I got quieter, too “now we are getting somewhere. What comes after three hundred ninety-one?” “Three hundred ninety-two. Change the one to a two.”

And the rules came pouring out. Until I got to five hundred thirty-nine.

And that was my recruitment pitch. Word spread to the other freshmen classes, and I got a few intrepid mathsters, signed up for a once a week arithmetic seminar.

The Seminar(s)

In the first post I explained what was coming. I would make arithmetic strange, and really think about it, and sneak some abstract concepts in. We would reexamine most of it with a new set of symbols: {}. I would teach them a bit of history – 19th century ideas about the axiomatization of arithmetic. I’d also get a discussion going of what the students learned about arithmetic, and how and when they learned it. That’s always fun. Maybe some reflection on pedagogy. I’ll add some history of Sputnik and New Math.

The more advanced group will review proof by contradiction, learn a little proof by induction. Then it is off to examine the Peano Postulates. And, for the older kids, use them to construct the natural numbers, proving every single step.

This work I have assembled from memory, from a wonderful class I took with David Rothchild a quarter of a century ago.

Successors

The first lesson in both seminars is “successors” – what number comes next.

Here’s my set of rules (I pulled it out of the students):

The first numbers succeed each other in this order: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. After that, for numbers past 9 that end in 0, change the 0 to a 1. Numbers that end in 1, change the 1 to a 2. And so on. Numbers that end in 8, change the 8 to a 9. But numbers that end in 9 require two steps. Change the 9 to a 0, and then follow these rules for all the digits to the left of the (new) 0.

And then, in response to my blog post, a later student of Rothchild’s shared the actual handout – I handed seen it in years. I am not far off, but for numbers that end in 9, he treats the rest of the number as a string. It probably works better. He doesn’t have the language for our number system, but if he did it would look like:

If A is any whole number except 0, the successor of A0 is A1, of A1 is A2,… of A8 is A9, and of A9 is B0, where B is the successor of A.

We both move next to the “Abnormal Number System.” As far as I was concerned, Rothchild invented this. David insists he learned it from someone else. Consider the numbers /, ∆, ☐, /❍, //, /∆, /☐, ∆❍, ∆/, ∆∆, ∆☐,  ☐❍,…

Now, my version of the class asks students to come up with a “successor algorithm” for the abnormal number system. Here is what I found from Rothchild:

Here is my expected answer:

Notice his deft use of A and B. Also notice, he is treating ❍ as his first number. I start at /.

Where do we go next? Counting, predecessors, and addition (where I believe I diverge from my teacher). And we must talk about ❍ vs /. Stay tuned.