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.

MATH CHALLENGE

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?

Sort of.

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:

The Cosine Law. It looks like the Pythagorean Theorem. But what is that extra term?

Here’s an exercise you might find in a child’s maths class. See how well you do. To help you out, we’ll even make it multiple choice.

Find the next number in the series.