It’s nearly the 14th of March (3/14 using the US dating convention, representing pi’s first 3 digits, 3.14) and the world’s greatest nerds will demonstrate their prowess at memorising many digits of a number that many people are familiar with, although it may seem to be confusing and irrelevant.

Most people know that it has something to do with circles or trigonometry and that it goes on forever and forever without repeating itself (an irrational number), but far fewer people know where it came from, what it does and how we know what we do about it.

And what about tau? That’s even less well-known … and we’ll get to that too!

#### What is it? And where did it come from?

Did you realise that we’ve been using the number pi (π) for longer than recorded history? That means we don’t know who first came up with it or what they used it for.

But most simply, it is the ratio of a circle’s circumference (C) to it’s diameter (D), so that π = C/D.

You can easily try to measure this by selecting a circular shape, measuring the circumference around the outside and then the diameter across the circle. Then simply divide the circumference by the diameter. If you measure accurately, your value should be slightly larger than 3.

The bigger your circle is, the more accurate your estimates should be and should start approaching 3.14, or perhaps 3.14159, but never the actual value. The more digits are measured, the more accurate calculations using pi become. For example, if you know ten digits of pi (3.1415926535), you could calculate the circumference of the Earth with only 1cm of error!

However wonderful that seems, ten digits isn’t very many for mathematicians. Knowing that pi is the ratio of circumference to diameter, mathematicians have devoted themselves to understanding what pi actually is and how far its digits extend.

This feat began back in Ancient Greece when Archimedes developed a method to estimate the circumference of circles by drawing polygons (shapes like squares (4 sides), pentagons (5 sides), hexagons (6 sides) and so on) inside and outside circles so that all the vertices of the polygon (the points where sides meet) all touch the circle. He developed an algorithm (a process) for calculating the perimeter (this is the total outside length of the shape) of polygons with *n* sides, where *n* is any number you want. (You can read more about this method here)

Archimedes took this process up to 96-side polygons and came up with a value of pi to 3 decimal places. Theoretically, you could keep repeating his process with larger and larger polygons and come up with more and more digits of pi. The problem is that this takes a lot of time (or in modern terms, computing power).

So today we have different algorithms. They’re called quadratically converging algorithms and can give over 45 million digits of pi after being applied only 25 times (compared to getting only 3 digits for 5 times in Archimedes algorithm). However, this one’s also computer intensive and a little bit pointless.

#### How do we know that pi goes on forever and ever and never repeats?

So it’s all well and good that we can calculate lots and lots of digits of pi and check that none of the digits we’ve found contain any repeating patterns, but going on from there, how can we know if pi might just stop a few digits later, or repeat itself?

This brings us to the exciting world of mathematical proofs, where things can be definitively shown (rather than just a good guess, which is the case in most of science).

The proof that pi is an irrational number (with decimal digits going on forever without repeating) first arose in 1761. Unfortunately, all of the different proofs rely on at least some calculus to demonstrate pi’s irrationality. They’re also really complicated, but they boil down to demonstrating that π = 4 * (1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11 + 1/13 – 1/15 + …..) continuing with every single odd number (and the odd numbers continue infinitely), getting closer to the actual value of pi with every step. If you are really on top of your mathematical skills, you can read one of the proofs with an explanation on the brilliant site, Ask A Mathematician.

#### What about tau? What is that?

Tau is a number that equals 2 pi (~6.28), or the ratio of the circumference to the radius of a circle. We’ve almost always used pi because, historically, it was easier to measure diameter than radius of a circle … and the number stuck around. But today, many argue that tau would make sense to replace pi, for several reasons:

- The formula for circumference of a circle is C = 2π * r. So if we use tau, C = τ * r, a simpler formula.
- In fact, a lot of complicated equations (covering concepts such as Fourier transforms, Riemann zeta functions, Gaussian distributions and far more) use a 2π value that could be easily and simply replaced by τ.
- When you calculate area or volume of circular or round objects, you use the radius, so it would make sense to express our circle constant in terms of the radius.
- In trigonometry, angles are often expressed in radians (instead of degrees), where a full rotation of a unit circle is 2π radians. If tau was the commonly used unit, we’d instead have circles with a convenient τ radians. Then, an angle that was a third of a circle would simply be 1/3 τ radians. An early tau convert described this idiocy as defining an hour as 30 minutes but leaving the clocks exactly the same … a system that just isn’t logical or sensible!

To allow for this possible change, proponents of tau have also created a tau day …. ready for us to celebrate on the 28th of June (to match 6.28 using the American dating conventions).

But whatever your opinions, I hope you’ve learnt something from this. Now go prepare yourself to eat pie this Thursday!

Using Pi longer than recorded history… really?

Dad

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It has been used in different scenarios and building templates at historical sites without any record of where it came from or arose.

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Excellent, informative post!

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