This I Know I Think
Life on the verge of understanding
August 15, 2018 Categories: maths.

YouTube now routinely recommends videos to watch based, it says, on my browsing history. I’m not convinced that the algorithm is all that accurate, although so far, at least, I haven’t been offered anything that is completely unconnected with what I might be interested in.

Yesterday I was pushed towards a video discussing Euler’s equation. I may have posted about this before, but it’s worth saying again: it is one of the most beautiful expressions in the world, almost mystical in its simplicity and wonderment. The equation, named after Leonhard Euler, a Swiss mathematician, links five of the most fundamental elements in mathematics:

• $e$ which is the base of natural logarithms, an irrational number (goes on without end) with the (very) approximate value 2.718281828459045…. It is found in many areas, including the study of compound interest.
• $i$, one of the square roots of -1. (The other one is $-i$). I know, you cannot square a number and get -1, but mathematicians found it convenient to do that and so to expand the domain of Real numbers into the domain of Complex numbers. (“With one bound he was free”).
• $\pi$, another constant, and originally defined as the ratio of the circumference of a circle to its diameter. The fact that that is a constant is wonderful in itself. It’s also an irrational (aka transcendental) number, approximately 3.1415926535….
• $1$ and $0$, two of the fundamental units in the Natural or counting numbers that form the basis of all the Natural numbers.

Euler’s equation links all of these fundamentals in the most elegant way:

$$e^{i\pi}+1=0$$
The mystical part comes when you start to wonder:

• how can I raise something to a the value of a complex number?
• how do I multiply a Complex number and an irrational number?
• How come when I do that I get “back” to $-1$ again?
• What is the meaning of life?

A constant source of wonder to me.