Why the world needs pi

Today, written as 3/14 the American way, is $\pi $ day! This special day happens every year, but today is extra special! It’s 3/14/15, giving us the first five digits of that lovely number, rather than just three! But why should we care about all these digits?

If you had a sloppy maths teacher at school you might have grown up with the idea that the number $\pi $ is equal to $22/7.$ Now that is completely wrong. Writing those numbers out in decimal gives $22/7 = 3.142...$ while $\pi = 3.141...$. There’s a difference in the third decimal place after the decimal point!

Pi

How accurately do we need to know the value of π?

Surely this small inaccuracy doesn't matter? Well, as the following extract from a longer article by Chris Budd shows, it really does.

The point is that $\pi $ is not any number. It lies at the heart of any technology that involves rotation or waves, and that is much of mechanical and electrical engineering. If rotating parts in, say, a typical jet engine are not manufactured to high tolerance, then the parts simply won’t rotate. This typically involves measurements correct to one part in 10,000 and, as these measurements involve $\pi $, we require a value of $\pi $ to at least this order of accuracy to prevent errors. In medical imaging using CAT or MRI scanners, the scanning devices move on a ring which has to be manufactured to a tolerance of one part in 1,000,000, requiring an even more precise value of $\pi $.

However, even this level of accuracy pales into insignificance when we look at modern electrical devices. In high frequency electronics, with frequencies in the order of 1GHz (typical for mobile phones or GPS applications), electrical engineers require a precision in the value used for $\pi $ in the order of one part in 1,000,000,000,000,000.

So, the modern world needs $\pi $ and it needs it accurately!