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How many places of pi do we need?

Tuesday 12th April 2016 10:00 am

Pi is a very long and a very important number, but how many decimal places of it do we really need to know? Dr Karl investigates.

In the land of food, a pie is something from an oven that is usually delicious. But in the land of mathematics, pi (‘Π’) is a very special number — and also quite tasty. With regard to a circle, it’s the ratio of the circle’s circumference to its diameter — it’s the magic number for a circle.

People have memorised Π to tens of thousands of decimal places, but we need only a handful of digits to know our place in the universe. In most cases, you can get away with Π as being equal to 3.1416 — that’s just four decimal places. So one roll of a wheel will cover just over three times its diameter — 3.1416.

Pi underlies much of what we know about the universe — because many objects and their movements are pretty close to spherical or circular. The shapes of stars, the orbits of planets, the movement of radiation and energy — all roughly circular or spherical.

So where did the name ‘pi’ come from? Well, in the Greek language, ‘peri’ means around, while ‘meter’ means measure. So ‘perimetros’ referred to the circumference of a circle. The first letter in that Greek word is the letter “pi”, or “Π”.

Π was first used to mean the circumference/diameter ratio in the year 1647 by the mathematician William Oughtred. But Leonard Euler, possibly the most brilliant mathematician of all time, made Π popular after he used it in one of his books about a century later, in 1748.

The idea that there was a fixed ratio between a circle’s diameter and its circumference was known way back in ancient Egypt and Babylon. By around 1800 BC (that’s nearly four thousand years ago), they each had values for Π that were accurate to within 1 per cent.

This brings me to a major fact about Π. Π is not equal to any simple fraction of one number divided by another number. You’re fairly close with 22/7, and even closer with 333/106, but one of the major properties of Π is that it is not a simple fraction.

So, how do you work out Π?

Sure, you can actually measure (with a tape) the circumference and diameter of a circle — but there are always errors in measurement.

Archimedes, around 250 BC, realised that a polygon with 96 sides was very close to a circle. So he did the maths for two such polygons — one just inside a circle, the other just outside. He came up with Π being somewhere between 223/71 (for the outside polygon) and 22/7 (for the inside polygon). He showed that Π was somewhere between 3.1408 and 3.1429 — a pretty good result for scribbling with a stick in the sand.

Of course, if you went for more than 96 sides, you could get better estimates. By the late 1500s, other mathematicians had achieved accuracies of 15 and 20 decimal places in the value of Π. The best accuracy for working out Π they ever reached by using polygons (with huge numbers of sides) was 39 digits.

No, the best way to get lots and lots of places of Π lay in the development of infinite series in the 1500s and 1600s. Now infinite series can be really weird. If you choose your terms, the sum of an infinite number of terms is NOT infinite — it can be finite. Sure, the infinite series 1 + 2 + 3 + 4 … etc will eventually add up to infinity. But what about the series 1 – 1/3 + 1/5 – 1/7 + 1/9 … etc? Well, it adds up to Π/4.

So in 1699, the English mathematician, Abraham Sharp, used this particular infinite series to push Π up to 71 digits. But this job took him a huge amount of time.

Since then, the mathematicians have devised better infinite series, and our scientists and engineers have given us increasingly faster computers. So we worked out Π to 10,000 digits in 1958, a million in 1973, and now we’re up in the trillions. And humans have been able to memorise up to about 70,000 places of Π.

So how many places of Π do we need in our modern society? (I guess some of you might be quietly wondering if we actually need any at all.)

Well, consider the Voyager 1 spacecraft — the most distant object we humans have ever sent into space. It’s about 20 billion kilometres away – about four times the distance of Pluto. To know its position accurate to one centimetre, we need to know Π only to 15 places.

And consider the biggest thing we know — the entire known universe. It has a radius of about 46 billion light years. If we want to find out the circumference of the entire known universe to the diameter of a tiny hydrogen atom, we need only 39 places of Π.

So when it comes to measuring the size of the universe, it’s just all pi in the sky.

 

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This blog first appeared on Dr Karl's Great Moments in Science

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