January 2013 thus far

James Grime

A lot of things going on in math. We spent a good part of January looking at positional mathematics and various bases, from the standard base 10 (decimal) mathematics to important other bases for computer science: base 2 (binary), base 8 (octal), and base 16 (hexadecimal). Students had a chance to gain a better appreciation of how decimal mathematics works, why positional systems have huge advantages over non-positional systems like Roman numerals (doing arithmetic, even addition, in such systems is excruciatingly slow and painful), and why there is no such thing as THE best base or natural base that we're obligated to use.

Part of this latter issue came up when we watched a video by English mathematician James Grime on dozenal (base 12) mathematics.

I recommend you watch this video and then discuss it with your child. We actually saw it in December, thanks to Noah, but it was clear that we needed to do background work before we could appreciate what Grime discusses. We did so and then rewatched it with more knowledgable eyes and minds.

Before we left bases, we talked about what such exotic beasts as negative and fractional (rational) bases might look like. As one student said, "My mind is blown." And as I said, "I like having my mind blown every once in a while." I'm confident that there was no permanent loss of minds, and that this exploration is already paying dividends.

Starting last week, we changed gears, though only slightly. We began working on a number of problems that emerge from investigating something called "Number Bracelets." Take a look at the basic rules of the game and some elementary questions associated with it here.

We have moved further to start thinking about what happens when you play this game with the same rules but with moduli other than 10. Please take a look at this page.

Today (Tuesday, 1/29/13), we backtracked a bit to look more closely at the definition of modulus arithmetic ideas related to it (in particular, congruences). We began by looking at clock arithmetic, where our friend base 12 makes another appearance, unsurprisingly. Clock math is one of those places where a powerful idea in mathematics is introduced in an elementary context and then (generally) left to lie fallow for most students forever. Only if they take a course in, say, abstract algebra as college junior or senior mathematics majors does this arise again, only in a more formal (and often forbidding) context.

One problem we looked at today from the perspective of modulus arithmetic and congruences appeared (in slightly different form) on an SAT math section in the late 1970s: A machine creates colored banners in a certain order - red, blue, green, yellow, purple, orange, red, blue, green, yellow, purple, orange, etc. If this pattern is repeated indefinitely, what color with the 379th banner be?

Stuck? Ask your child to give you a hint.

After that one, we looked that the following: without using a calculator or actually computing the value of 37^15 (37 raised to the 15th power), figure out what the units (ones) digit will be.

See if you can figure this out by hand. It is utterly unnecessary to know what 37^15 equals. Using modulus arithmetic, it is trivially easy to figure this out by hand with almost no calculations (and certainly nothing difficult or advanced). Again, if you get stuck, speak with your student.

Tomorrow we will return to the number bracelets game to explore some deep questions about how things work when we play the game using various other moduli, from 1 on up. And perhaps we'll explore later a question raised this morning by Jianmarco about the possibility of fractional moduli. :^)