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When the Digits Fill they must Spill

Another conversation No. 1, our 6 year-old, and I had about number bases.  I'm not sure where I'm headed with all this.  No. 1 and I tend to talk as we ride through San Francisco on its various buses, trains, and cable cars... a lot.  It would be more concise to explain what we're doing math-wise by writing down a short description of the concepts.  It's not what we're actually doing though, so I'm not sure how much help that would be.  I'll just say for now, that I've discovered more about the math No. 1 and talk about by talking than I did by 'learning' it in school, so for now, I'll carry on.




OK, so No. 1 and I had covered the basics of number bases.  You choose you base, you get that many numbers to place in a digit, and you have to include zero as a number.  Choose base 10, and you get our finger-counting system with ten different numbers represented by a single digit: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.  Choose base 3, and you get 0, 1, and 2.  Choose to work in base 2 like a computer, and you get 0 and 1.  The next thing to cover was what happens when you run out of room in a single digit.  So, you’re counting along, and you arrive at 9, (or 2), (or 1).  What do you do when you want to count one more thing?  No. 1 already knew how to count higher than 10, in base 10, so it was as good a place as any to start.

“OK, so I’m counting along in my single digit, and I hit 9.  What do I use for the next number?”

“Ten?”

“Yup.  How do you write it down?”

No. 1 made a 1, and a 0 in the air.

“Awesome.  So, you had to use a second digit.”

“Now, what if we’re in base 2, when do we run out of numbers?

“What?”

“When you’re counting in base 2, what’s the largest you can count with a single digit?”

“One?”

“Yeah, one.  So, when you need to count more than one thing, what will you need to do?”

“What?”

“If I’m counting things, and have two of them, and I want to write down how many I have, how do I do it?”

“Write down a  2?”

“What’s the biggest number a digit in base 2 can be?”

“One.”

“So, can I write down a two then?”

“No.”

“So, what do I do?  What if I write down another digit like you did for 10?”

“ OK.”

By now we’d made it home, so I reached for a piece of paper, and wrote down 10.  “What number is that in base 2?”

“Ten?”

“Nope, what’s the biggest number you can put here before you run out of room?” I asked pointing at the digit on the right.

“One.”

“So, is we had to add a new digit because we ran out of room at one, what’s the new digit for?”

“Two?”

“Yup!  OK, so now, we’ve got two, and we keep counting so now we have a third thing.  There’s room left in the ones place, so I’ going to put a one there,” I said, writing 11 on the sheet of paper.

“OK, so what number is that?”

“I don’t know.”

“Well, what’s in the digit where we can write up to one?”

“One.”

“What’s in the digit for twos?”

“One.”

“What’s a two plus a one?”

Thinking for just a second, No. 1 replied, “Three.”

"So, what’s our new number?"

“Three!”

“Do we have any more room?  Can we make either of the digits any bigger in base 2?”

“No.”

“So, what if we want to count up to four?”
“We’d add another digit?”

“Yup!  We’d write one, zero, zero.”

And then came the exercises.



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