Gotham Girls 2008 Championship: Bronx vs. Queens

The final home bout for the Gotham Girls this year did not disappoint.

Both sides brought some serious skills to the track. The Bronx had a ton of raw horsepower. Bonnie Thunders and Luna Impact in particular were wickedly fast—I don’t think I ever saw a Queens jammer beat them off the line. Beyonslay, as always, put in a stellar blocking performance. I’m sure most folks are familiar with her active blocking skills, but where she really shone Saturday was her positional blocking. Once you get behind Beyonslay, you stay behind Beyonslay.

The Queens weren’t exactly slacking in the individual performance department either. Cheapskate was the Queen to watch, with mad blocker-evading skills all night long. Suzy Hotrod proved to be fearsome as well: where Cheapskate would bob and weave to cut through the pack, on more than one occasion, Suzy would smash right through it. (Remind me never to get in her way.)

What tipped the game, as always, was the teams’ ability to pull together. The Queens managed to discombobulate the Bronx more often than not. About two jams out of every three saw the Bronx’s jammer getting tangled up in the pack, while Cheapskate zipped around or Suzy Hotrod crashed through. By the half, the score was 100–37 Queens, and they held on to that lead. Despite an exciting 20-point jam by Bonnie Thunders, the game ended 161–113.

Congratulations to the Queens of Pain, the 2008 GGRD champions!

What *isn’t* the square root of -1?

I stumbled across an article trying to explain what the square root of -1 (j) is. It’s pretty cool that the author’s trying to provide motivation for a not-immediately-intuitive mathematical construct. Unfortunately, the author made a goof that makes his example fall apart.

The place where he goes off the rails is:

In other words, the eigen values, are all values such that:

Ax = ex

Where A is the matrix whose eigenvalues we are trying to find, and e is the scalar eigen-value we are trying to solve for. x is any non-null (non-zero) vector.

What he should have said is that x is some non-zero vector. If x can be any vector, then to find values of e such that Ax = ex, we would need to solve (AeI)x = 0 ⇒ (AeI) = 0 ⇒ A = eI for e. No such e exists.

The author appears to be missing the fact that every eigenvalue of a linear transformation has an associated set of eigenvectors. In this case, the set of eigenvectors associated with the eigenvalue j is all multiples of [j, 1]. For these, and only these vectors does multiplying by j give the same result as rotating 90°. It doesn’t work for, for example, [1,1].

There is a sense in which mutliplying by j is a rotation, but it doesn’t do much to motivate a need for j in the first place. If you represent a complex number as a 2-vector, multiplying by j will rotate the number 90° in the complex plane. I.e, if z = x + yj = rejθ, then jz = (by Euler’s identity) ejπ/2 z = ejπ/2 rejθ = rej(θ + π/2).

If I were trying to motivate the creation of j as the square root of -1, the first example that comes to mind is AC circuit analysis. Euler’s identity makes it easy to view complex numbers as a point in a wave with a given phase and amplitude. You can generalize the notion of resistance into impedance, with capacitors and inductors having imaginary values, and solve for the steady-state solution much the same as you would with just resistors in a DC system. Sure, you can do it without using complex numbers, but it gets clumsy fast.