Saturday, March 03, 2007

Add, Multiply or...?

abstract algebra
(a "gnu math" approach)

I'm somewhat proud of this little storyboard I published to Math Forum this morning, which is mostly straight group theory, but then I mention a third operation, neither addition nor multi- plication, using the symbol for the artist formerly known as Prince.

I hope he's OK with that.

When two of these Zoombini-like cartoon characters collide, they do whatever's the currently displayed operation.

Like if the name of the game is "to multiply" and the partners just happen to be reciprocals, like 2 and 1/2, then you get that special one particle.

But if the op is "to add" and the characters are like -2 and +2, then their collision begets a zero, these being the multiplicative and additive identities respectively (1 and 0).

The prince op (shortened name) would have its own notion of inverse, and its own identity element. I haven't shown it in the above "logic gates" view of balls combining (members of Q, the rationals), thinking to leave this op to the students' imaginations.

Note that we don't really consider subtraction and/or division to be ops in their own right, as in abstract algebra we define these as "adding the additive inverse of" and "multiplying by the multiplicative inverse of" respectively.

All this fantasizing about colliding sprites in a Scratch-like playspace got me thinking about Luxor 2 I think, as that's where I ended up, further advancing my career as some Egyptian glass bead gamer.