Faced with the specter of having to memorize addition tables, and with the reward of building a calculator from scratch, our six year old—aka No. 1—and I have been working on math from a slightly different tack.  We switched to base 2 numbers.  Base 2 numbers, also known as binary, are the numbers all computers use.  For those unfamiliar with binary numbers, the binary system, (technically referred to as a ‘base 2’), only gives you two numbers to work with: 0 and 1.  Consequently, the binary addition table is far easier to memorize:

 Addition Table + 0 1 0 0 1 1 1 10

In contrast, the number system we’re all familiar with, (known as ‘base 10’), gives us 10 numbers to work with: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.  Given a single digit in our ‘normal’ base 10 system, we can represent up to 9 things.  If we have ten things, we have to add a new digit—known as the ten’s place—hence 10 uses two digits.  In base 2, given one digit, we can represent at most one thing.  So, when we want to represent two things, we have to add a new digit—known as the two’s place—so in the table above, when we add one with one, the result—two—is written as 10 in base 2.

The concept of ‘carrying’ was easier for us because we didn’t have to use such large numbers to practice.  You might not think adding 4 to 6 is a big deal, but you also might have memorized your addition tables more than a decade ago.

No. 1 and I discovered that we when needed to carry what had really happened was that we’d run out of room adding two digits together.  In other words, when we add two one digit numbers together, and need to write the answer as a two digit number, we’ve carried.  For normal base 10 numbers, you ‘run out of room’ when you add two numbers and wind up with an answer larger than 9.  In base 2, you carry when you add two numbers and come up with an answer larger than one.  Consequently, we wind up carrying in almost every math problem.  No. 1’s getting all the benefit of practicing carrying without having to memorize a 100 entry addition table first.

### 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…

### Kids R Kapable

Just a little note to concerned ‘grownups’ everywhere.  If you look at a kid—and I mean really look—I don’t mean notice a person shorter than you, I mean make eye contact, notice their facial expression and observe their body language—If you look at a kid, don’t assume they need your help unless they’re obviously distressed, or ask for it.  You might think this is difficult call to make.  You might think, not having kids of your own, that you’re unable to make this determination.  You are.  You do in fact, already have the skills even if you’ve never been around kids  It’s a remarkably simple call to make, just use the exact same criteria you would for determining if an adult was in distress.  Because, guess what, kids and adults are in fact the same species of animal and communicate in the same way.  Honest.  If someone—adult or child—doesn’t need your help, feel free to say hello, give a wave, give a smile, but don’t—do not—try to force help on anyone that doesn’t want or need it.

Y…