Euler's Identity

A little off the track of software but not entirely disconnected...

I've recently taken to mentioning Euler's Identity when we talk in Algorithms class about Euler and the Bridges of Königsberg.

I point out that it includes five of the most important numbers in mathematics: 0, 1, π, i (the square root of -1), and e, the base of natural logarithms (Euler's number); it also involves four of the most important operators: +, =, * and exponentiation.

The only numbers or operators that could be reasonably considered to complete the set would be the number 2 and division.

Have you ever wondered about the definition of π as the ratio of the circumference of a circle to its diameter? Why the diameter? Why not the radius? There are so many situations where we have to talk about 2π, for example the number of radians in a complete circle, or the "reduced" Planck constant (h/2π) as used in Schrödinger's equation).

So, what would be the effect of redefining π as the ratio of the circumference of a circle to its radius? To avoid the most appalling confusion, we would of course have to give it a different symbol. The greek letter tau has been proposed. Employing 𝝉 = 2π, the Euler identity would appear thus:

Now, we would have six numerical quantities and five operators. I have to admit though that it doesn't look quite so elegant this way.

For a more complete discussion of this use of 𝝉, please see Turn (geometry): section Tau Proposals.

OK, back to work!