Binary Math Lessons: The Secret Origin

Unschooling?  How did my last post have anything to do with unschooling?  As soon as I saw the title on the screen, I cringed.  The benefits of binary math, check, anything to do with unschooling?  Nada.

As it turns out, I’d started in the middle of the story.  Our six year-old, No. 1, and I started heading towards binary math—in more proper unschooling form—because she wandered into the room one day and said, “Dad, I want to learn what you do at work.”

All I do at work is test machines whose sole job it is to move ones and zeroes around: microprocessors and other digital devices also known as computer chips in the vernacular.  So, since one and zero are pretty simple concepts, and as it turns out, the logic gate building blocks of digital devices are also really simple, off we went!

The first thing we need to nail down were the handful of logic gates I encounter.  What’s a logic gate you ask?  It’s just an electrical embodiment of a logical construct, (you know like the one’s you had in philosophy 101).  Take the ‘and’ gate we started out with for example.  The device takes two inputs that can be either a one or a zero, and outputs a single number in return, again either one or zero.  If both the inputs are 1, (known as logical true in the vernacular), then the device outputs a one, if not, then the device outputs a 0,(a logical false value).  Hence, if one output AND the other are both true, the ‘and’ gate gives a true output aka 1.  Otherwise its output is 0, aka false .

No. 1 and I made up some homework sheets for her to play around with.  Her homework was to fill in the logic gates on the page with any sets of inputs and outputs from the table.

Did I mention we talk about this stuff all the time too?
"Hey, what's 1 AND 1?"
"1"
"What's 1 AND 0?"
"0 Dad," and so on.

I personally think our constant conversations drive things home more than the homework, but who knows?  In any event, there are a lot of MUNI riders who know more about logic gates than they used to or maybe wanted to.

In case you wanted to play along:

And as a picture:

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…