Euler’s Identity

In my previous blog I mentioned that “if there’s one mathematical constant that reigns supreme in the heady world of statistics, it’s Pi”.  But what about that other mathematical constant, Euler’s Number, e?  I don’t think statistics would have advanced very far without e.  Well in fact Pi, e and the square root of -1, denoted by i, share a special bond, known as Euler’s Identity:

e^{i \pi} = -1

Many mathematicians consider Euler’s Identity to be the most beautiful equation of all time.  And I have to admit it sends tingles down my spine too.  I also get the same sensation when I see fractals, or Einstein’s famous proof of the equivalence of mass and energy, E=mc2.

I must be some nerd.

Approximating Pi with a fraction

π = 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679…
A million digits of Pi

A little while ago my blogging thoughts turned to (Monte Carlo) biscuits.  Now I’m thinking about pie.  Or, to be more precise, Pi.  If there’s one mathematical constant that reigns supreme in the heady world of statistics, it’s Pi.  And of all the approximations of Pi my favourite is the Milü, derived by the Chinese mathematician and astronomer Zu Chongzhi.

π ≈ \frac{355}{113} = 3.14159292035…

What I love about the Milü is that from just 3 digits in the numerator and denominator, you get an approximation of Pi that is accurate to six decimal places (after rounding).  And I’d argue that this is close enough for most practical purposes.  As an added bonus the denominator is a prime.

The common π ≈ 22/7 = 3.14285714286… isn’t bad, but only accurate to two decimal places.  The next best approximation is π ≈ 52163/16604 = 3.14159238738… But this is unwieldy and doesn’t offer any practical improvement in accuracy.

Neither of these approximations is as useful, or as elegant, as the Milü.

\frac{355}{113}.  A good idea for a tattoo perhaps?