Monday, March 26, 2012

Mess With Your Mind Monday - Volume 001

Welcome to the first of a new series of posts that I call "Mess With Your Mind Monday" (MWYMM). The intent of these posts is to get you thinking and asking questions, not necessarily to give you answers. I plan to post something new on this series every Monday, but I may occasionally skip a week. The topics will vary, just about anything is fair game if it will mess with your mind and make you think. Here's a permanent (or at least semi-permanent) link to the MWYMM posts. Now, on to the first post.

Today's topic is math (it's a good topic for messing with your mind). Specifically, I'm going to look at some oddities involving imaginary numbers, complex numbers, and transcendental numbers.

You probably know that the square root of any negative real number is an imaginary number. The base unit of imaginary numbers is the square root of -1, an imaginary number we call "i". i² = -1.

You might even know that ℯ^(π*i) = -1 (Euler's identity). This one still blows my mind because ℯ (Euler's number) and π (pi) are both transcendental numbers. Yet somehow, ℯ raised to the power of pi*i = -1. How does a transcendental number raised to an imaginary transcendental power yield a negative integer? But that's not really one of today's MWYMM questions.

What is the square root of i ?
There is an answer. In fact, like other square roots, there are two answers, one with a positive real component, and one with a negative real component (e.g. sqrt(4) = 2 or -2, with the positive root being the implied answer in most cases). So that you can ponder it for a while, I'm not going to give the answer here. Instead, here's a link to the answer along with an explanation.

In case I haven't messed with your mind enough, here's one more.

What is i to the Power of i (i.e. i^i) ?
Now, we're raising an imaginary number to an imaginary power. Any guesses about the answer? Let me warn you now, this one may completely blow your mind. Again, I'm only giving a link to the answer here, but here are a couple clues; there is more than one answer, and the answers are real numbers.

I hope you've enjoyed pondering these questions as much as I've enjoyed messing with your mind. Check back for next week's installment of MWYMM.