Why is 0⁰ = 1?
Because that’s the trend.
This is a journey into a place called the Mandelbrot Set using a tool in Linux (and DOS) called Xaos. The Mandelbrot Set is a mindscape, it’s a place anyone in the universe can explore. Aliens in another galaxy can visit the same exotic places in the Mandelbrot Set that we do. The red arrow indicates where I’m going to zoom in.
To understand what the Mandelbrot Set is, I have to explain complex numbers. That name doesn’t mean the numbers are complicated, it just means they are numbers which have two parts, and when the two parts are multiplied, they have to be handled seperately just like a binomial. For instance, 2 + 3i is a complex number. If I multiply it by itself (square it), I end up with 4+ 6i + 6i + 9i^2. Now to continue I have to explain what i is.
i is defined as the square root of -1. The normal operations of algebra don’t give us the square roots of negative numbers, because the squares of negative numbers are always positive. But in Algebra Deluxe we have defined i as the square root of -1 by fiat so we don’t end up quadratic equations with only one solution, or even none. In an unfortunate choice of math jargon i is called an imaginary number but that doesn’t mean it’s pure fantasy. For instance, i^i = e^(-π/2)
So now that we know i is the square root of -1, it follows that i^2 is simply -1. In my squared complex number above, the solution becomes 4 + 12i – 9, or -5 + 12i. Now what good are complex numbers? They extend the one-dimensional number line of real numbers to the two-dimensional Argand Plane. The real part of my complex numbers (+2 in the first case, -5 in the second case) represent the horizontal position of a point. The “imaginary” part of my complex number (+3 in the first case, +12 in the second case) represent the vertical position of a point.
So here’s what Xaos does to generate the Mandelbrot set. Each point on the screen is a complex number which is multiplied by itself repeatedly. Each solution is a new point. If the solutions remain on the screen after, say, 100 iterations, that point is assigned a dark color. If the solutions fly off the screen, they are assigned lighter colors based on how soon they depart. And the result is a wonderful fractal universe that you can dive into real time with your mouse, as it smoothly zooms in (or out if you hold down the right-clicker instead). This universe is like Mount Everest. We didn’t make it. God didn’t even make it. It’s just “there”, and it has always been there, like the distribution of primes. You can go down and down, forever and ever, and the funny thing is, sometimes you find little baby copies of the whole Mandelbrot Set down there, but they are never exact copies, only variations of the theme. It’s really a mind blower.