Get This Man a Doctor

A man, a plan, a canal. Panama.

Crank of the day

Thursday, December 7, 2006

I just got done watching a video from Slashdot. The title of the article was Crank Doesn’t Understand Division. Oh wait, it might of had a more deceptive Slashdotesque title, but I don’t really want to give it pagerank-credence by linking directly to it.

Anyways, it links to a video of some dude claiming to have solved the “two thousand year old problem” (verbally. really.) of 0^0. His innovative idea is that 0/0 is an algebraically consistent number, and he labels it \Phi, pronounced “nullity”. Oh, and 1/0 is also a number, and it’s infinity.

Now, there’s nothing crazy about claiming that infinity is a number, or that there are positive numbers that are smaller than every positive real number. But then he goes to the board and starts writing out an equation that treats 0/0 and 1/0 just like fractions.

Okay, let \infty = \frac{1}{0}. That is, \infty is the multiplicative inverse of 0. Then by definition, it must multiply with 0 to obtain 1. In the video, Herr Doctor Professor claims that multiplying the numerator and denominator of fractions holds with infinity and nullity. So,

1 = \infty \cdot 0 = \frac{1}{0} \cdot \frac{0}{1} = \frac{1\cdot 0}{0\cdot 1} = \frac{0}{0} = \Phi

So the magical “nullity” number in the video is actually equal to one. Hmm….

On the first day of abstract algebra, you have to prove that you can’t divide by zero without either redefining multiplication or proving a contradiction. If there was a multiplicative inverse of zero, say a, then we would have,

1 = a \cdot 0 = a \cdot (1-1) = a - a = 0

Which is generally not a good result.

To see an examination of infinitesimally small and infinitely large numbers that isn’t crazy, read about surreal or hyperreal numbers.

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