Transcendental Infinity

I just read the most awesome, goosebumpish thing I’ve come across in a long while.

Well, at least since I re-read the proof that 1 = 0.999… last month.

I wish I could explain it better, but I can’t. So bear with me a moment as I fuddle through this.

Envision a number line in your head, like the one you had on your desk in second grade. Zero smack in the middle. Positive integers to the right, negative integers to the left. Fractions between the integer slashes on that number line.

Now, there are these things called transcendental numbers. They are constants which have nonrepeating decimals stretching out to infinity. The only ones I know of are pi, e, and some things called Liouville numbers.

Anyway, the awesome part comes when considering infinity. Think of all the integers, stretching out to infinity. Now think of all the fractions, an infinity of which reside between each pair of integers. Turns out, and I can’t rightly explain it, that the set of all integers and the set of all fractions, both infinite, are equal in size.

But even more amazing is that both these sets pale in comparison to the set of all transcendental numbers. There are an infinite infinite amount of transcendental numbers as opposed to rational numbers such as integers and fractions, so much so that the two really can’t be compared. In fact, if you put every number out of infinity into a great big bag, reached in, and pulled out a number, the odds that you’d pick a transcendental number out is, well, a hundred percent.

I sit before you with a “whoa” upon my lips.