### 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?”

“Yup!  We’d write one, zero, zero.”

And then came the exercises.

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

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 differential operator in terms of one variable into a series of differential operators in terms of othe…

### The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1.  While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero.  This is typically explained in the following way.  While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare.  That doesn't stop us from looking for them though!

Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2].  Cabrera was utilizing a loop type of magnetic monopole detector.  Its operation is in concept very simpl…

### Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the so…