Skip to main content

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?”


“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?


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


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


“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?”


“So, can I write down a two then?”


“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?”


“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.


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


“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?”


“What’s in the digit for twos?”


“What’s a two plus a one?”

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

"So, what’s our new number?"


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


“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.


Popular posts from this blog

Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

Now available as a Kindle ebook for 99 cents! Get a spiffy ebook, and fund more physics
The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differe…

Division: Distributing the Work

Our unschooling math comes in bits and pieces.  The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her.  Here’s how she arrived at the distributive property recently.  Tldr; it came about only because she needed it.
“Give me a math problem!” No. 1 asked Mom-person.

“OK, what’s 18 divided by 2?  But, you’re going to have to do it as you walk.  You and Dad need to head out.”

And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss.

One looked at the ceiling of the library lost in thought as we walked.  She glanced down at her fingers for a moment.  “Is it six?”

“I don’t know, let’s see,” I hedged.  “What’s two times six?  Is it eighteen?”

One looked at me hopefully heading back into her mental math.

I needed to visit the restroom before we left, so I hurried her calculation along.  “What’s two times five?”

I got a grin, and another look indicating she was thinking about that one.

I flashed eac…

The Javascript Google URL Shortener Client API

I was working with the Google API Javascript Client this week to shorten the URLs of Google static maps generated by my ham radio QSL mapper. The client interface provided by Google is very useful. It took me a while to work through some of the less clear documentation, so I thought I'd add a few notes that would have helped me here. First, you only need to authenticate your application to the url shortener application if you want to track statistics on your shortened urls. If you just want the shortened URL, you don't need to worry about this. The worst part for me was that the smaple code only showed how to get a long url from an already shortened rul. If you follow the doucmentaiotn on the insert method, (the method for getting a shortened url from a long one), there is a reference to a rather nebulous Url resource required argument. It's not at all clear how to create one of these in Javascript. The following example code shows how:
var request = gapi.clie…