Wednesday, June 24, 2009

1-x-x^2

The inverse of 1-x2 can be calcuated two ways.

First way: We know 1/(1-y)=(1+y+y2+y3+...) so substitute y=2x, and it comes out.

Second way: Do it directly.

  1. The constant coefficent must be 1.
  2. The x coefficient must 2.
  3. The x2 coefficient must 4.
  4. The x3 coefficient must 8.

We did a couple more examples of computing an inverse. Then we went to an interesting one.

What is the multiplicative inverse of 1-x-x2.

It turns out to be 1+x+2x2+3x3+5x4+8x5+13x6...

The coefficients are the fibonacci series!

Tuesday, June 23, 2009

1/(1-x)

Treating polynomials as abstract symbolics entities (rather than as a formula into which you substitute a real number for x. Q: What do you get when you multiply (1-x) by (1+x+x2+x3+...) ? A: You get 1. Thus 1/(1-x)=(1+x+x2+x3+...).

5-minute math for a 3-year old

My 3-year old child wanted to do five-minute math. I said, OK, I grabbed the marker, and she said "write a 3". So I wrote a 3. Then she said "write a 2". So I wrote a 2. We spent five minutes with her dictating numbers and I filled up the white board.

About Me

I'm research faculty at MIT, and Chief Architect at Tokutek.