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…

### Lost Phone

We were incredibly lucky to have both been in university settings when our kids were born.  When No. 1 arrived, we were both still grad students.  Not long after No. 2 arrived, (about 10 days to be exact), mom-person defended her dissertation and gained the appellation prependage Dr.

While there are lots of perks attendant to grad school, not the least of them phenomenal health insurance, that’s not the one that’s come to mind for me just now.  The one I’m most grateful for at the moment with respect to our kids was the opportunities for sheer independence.  Most days, we’d meet for lunch on the quad of whatever university we were hanging out at at the time, (physics research requires a bit of travel), to eat lunch.  During those lunches, the kids could crawl, toddle, or jog off into the distance.  There were no roads, and therefore no cars.  And, I realize now with a certain wistful bliss I had no knowledge of at the time, there were also very few people at hand that new what a baby…

### Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…