06 October 2007

Dual numbers

You know that trick where you invent some number ε such that ε2 = 0 and use it to, basically, take derivatives?

For example, (x+ε)2 = x2 + 2xε, so if we change x by some small amount ε then we change x2 by 2x times that amount. Thus the derivative of x2 must be 2x.

It turns out that trick has a name; it's calculation in the algebra of dual numbers, which I discovered by randomly poking around Wikipedia, and it's apparently used in at least some computer algebra systems to do differentiation. I didn't know that.

Edit (Monday, 4:36 pm): Charles of Rigorous Trivialities has pointed out that dual numbers are used extensively in deformation theory.


Pseudonym said...

For a nice practical use of dual arithmetic, check out this paper. Also some background theory.

Micheal cassidy said...

Sorry about your Phillies.

I have four friends at work not talking to me because I keep telling them the Yankees don't deserve to be in the playoffs.

"ε" has always bothered me; its use always seemed like engineering or physics. You are solving to whatever degree of precision you need; BUT its still not absolute; you can never get to infinity so you never quite close the gap.