We know that e and π are both irrational numbers. Neither can be written as a ratio between two integers.
The high-school math class mantra is this:
“Will we ever actually use this in the Real World?”
I found the following thumbnail in my YouTube feed recently. Try out the problem first. Then watch the video here. I’ll wait.
The solution is a cinch. Hahahaha. Okay, sorry.
Here’s a fun curve-sketching challenge for you to try. My advice is to work your way through it yourself first. Then read my solution below. Or watch the walk-through in the video.
We want to sketch the curve, sinh(ln x) over the Real Numbers. Because of the logarithm, we’ll impose the restriction, x>0. We’ll need to remember that restriction later.
Attack the function of a function from the inside out, with this substitution:
The irrationality of e can be demonstrated by a surprisingly accessible piece of mathematics. This proof, developed by Joseph Fourier, is perhaps the most well-known.
We need to begin with this infinite sum expression for e:
What is the last digit of π? We may as well ask what colour infinity is. It isn’t that the answer is unknowable. The question is meaningless. Right?
The digits π — 3.14 something something something — have a definite beginning which we all know. Yet there is nothing natural about representing π as a string of digits.
Consider this alternative representation:
Adam Hrankowski is a maths/physics tutor in BC. You can find his stuff on YouTube: youtube/c/mathadam