Five Fascinating Facts about Pi

The knowledge that a constant of nature exists in the geometry of circles has been around for more than 4000 years. The ancient Babylonians used to estimate the area of a circle by squaring its radius and multiplying their answer by three. Thus, the very first estimate of Pi was 3. There is evidence, though, that the Babylonians refined this value over time, with one tablet dating from around 1700 BC stating the value of Pi as 3.125. Not long after that the Egyptians had suggested a value of 3.1605. Not bad for ancient peoples.

As far as we know, it was Archimedes who proposed the first formal method for estimating Pi. He proposed a method where a regular polygon is inscribed inside a circle and then a larger regular polygon is constructed in which the circle is circumscribed. The areas of these two polygons could be calculated using the Pythagorean Theorem (which was relatively newly discovered at the time), and Archimedes proposed that the area of the circle lay somewhere in between these two areas. He used this method to show that Pi’s value was between 22/7 and 223/71.

Despite the fact that the ancient Greeks were thought leaders on the topic, they never proposed the Greek letter Pi (π) to represent this elusive value. That did not appear until the 18th century when the Welsh mathematician William Jones started using it, and it was then picked up by the Swiss mathematician Leonhard Euler. As influencers go, Euler was the Kim Kardashian of the 18th century math community, and so it was at this point that the notation we use today went viral.

Here are five fascinating facts about the wonderful natural constant that we know as π, but which we can never, ever write down in numbers.

1. Pi was estimated independently in China around 400 AD

Zu Chongzhi (429–501) was a Chinese philosopher and astronomer who would have had no idea of the works of the ancient Babylonians, Egyptians or Greeks, but who estimated the value of Pi to be 355/113.

His workings have been lost to history, but if Archimedes’ method was used, then to obtain this level of accuracy he would have had to construct regular polygons with 24,576 sides, and work with hundreds of square roots calculated to at least 9 decimal places.

2. Pi is irrational

At school many of us learned to use 22/7 as an approximation for Pi, but in reality Pi cannot be expressed accurately in the form a/b where a and b are integers. This was proven in 1761 by the Swiss-French mathematician Johann Heinrich Lambert. Lambert showed that the trigonometric ratio tan(x) can be expressed as a continued fraction as follows:

He also proved that if x is a rational number, then this expression will be irrational. Now since we know that tan(π/4) = 1, which is rational, we have to conclude that π/4 is not a rational number. and hence π is not a rational number.

One consequence of this is that there is no way to write Pi in finite decimal notation. But that hasn’t stopped people from trying. At a filmed event in a public hall in Tokyo in 2006, Akira Haraguchi recited Pi to 100,000 decimal places from memory, starting at 9am, finishing at 1.28am the next day, and sustaining himself with five minute breaks every two hours to eat onigiri.

3. Pi is transcendental

I feel like, whenever I say this, I need to have long hair, put my fingers up in a peace sign and finish my sentence with ‘man’, ‘dude’, or ‘brother’. A transcendental number is a number which can never be the root of a polynomial with rational coefficients.

Think about that for a minute. It’s astounding. Basically, we could spend the rest of time writing out polynomials of any finite degree with any coefficients that are integers or fractions, and Pi would never be the root of any of these. The proof that Pi is transcendental is a trivial consequence of the Lindemann-Weierstrass Theorem in the field of Algebraic Number Theory, proved in 1885.

This is a stronger result than proving that Pi is irrational, because any transcendental number is necessarily irrational. The fact that Pi is transcendental also allows us to prove that ‘squaring a circle’ is impossible using a straight edge and compass, something the Ancient Greeks tortured themselves trying to do.

4. Any numerical expression for Pi is infinite

There are many ways of expressing Pi using numbers, but all of them are either infinitely long or involve infinite sums or products. The legendary Indian mathematician Ramanujan came up with a smorgasboard of infinite sums to determine Pi, many of them looking completely off-the-wall, like this one:

It’s also possible to use double angle trigonometric identities to create infinitely nested surd expressions for Pi like this one:

Another beautiful expression came from Leibniz, which uses the fact that π = 4arctan(1), and a known series for arctan to give this simple series:

5. Pi can be estimated arbitrarily closely using toothpicks

Using a probabilistic experiment often called a ‘Pi toss’, where you throw toothpicks at random on a piece of paper, you can progressively get closer and closer estimates for Pi. It’s an incredibly simple experiment and great to use with kids. All you need to do is draw two lines on the paper that are exactly the length of two toothpicks apart. Then throw toothpicks randomly on the paper one by one, and divide the total number thrown by the number that touch one of the lines. The more you throw, the closer your answer will be to Pi.

Here’s a quick and fun video in case you want to give it a go:

Did you enjoy these facts about Pi? Do you have any others to add? Feel free to comment!