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3. Everything / Maths, Science & Technology / Mathematics


No, this isn't an entry about apple pie, or cream pie, or any other sort of pie you can think of. It's an entry about pi1, the irrational number which is a vital part of mathematics. Pi is defined as the circumference of any circle divided by its diameter.

Irrational Number?

An irrational number is one that cannot be written in the form p/q, where both p and q are whole numbers. For example, 1/2, 3/4, and 10971/182936 are all rational numbers, whereas pi, otherwise known as π is not. All irrational numbers have an infinite number of digits after the decimal point, and these digits never form a repeating pattern2.

How do I Calculate Pi?

From the definition, π can be found by taking a circle (any circle at all) and dividing the length of its circumference3 by the length of its diameter4. This will give you a value close to the number stated below, depending on how accurate your measurements are.

What Exactly is the Value of π?

π, to 2,000 decimal places, is:


Of course, that's not where it ends. π continues for an infinite number of decimal places, and there is no pattern to the order of the digits. Any pattern that you may think is emerging is just an illusion, and will quickly disappear to be replaced by another 'pattern'. For ease of mathematical calculations, π is usually reduced to 3.142, or 3.14159 for a more accurate result.

The value of π has been calculated to 1.24 trillion decimal places so far, by Professor Yasumasa Kanada, his team, and some amazing computing. However, the digits of π still seem to occur randomly, meaning that we are no nearer predicting what the next digit will be than were the Ancient Greeks.

The History of Pi

One well-known reference to the value of π is in the Bible:

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about.
- I Kings 7:23

This refers to a list of specifications for the Temple of Solomon, which was built around 950 BC, and gives the value of π = 3. Even for its day, this was quite an inaccurate figure, as the ancient Egyptians and Mesopotamians are believed to have calculated π as 25/8 = 3.125 before this date.

The first attempt to actually calculate, rather than measure, π seems to have been by Archimedes in around 295 BC. He came up with the inequality 223/71 < π < 22/7, by working with many-sided polygons rather than circles. Estimating the average of these two boundaries, we can come up with the value of π = 3.1418.

After Archimedes' success, others used the same method to calculate π to more and more decimal places. By 1600 AD, Van Ceulen had calculated the first 36 digits of π. Through this time, no changes to the method were made, just more and more calculations carried out.

During the Renaissance in Europe, more algorithms5 were proposed for calculating the value of π, the most well-known of these being π/4 = 1 - 1/3 + 1/5 - 1/7 + ... This is attributed variously to Leibniz or Gregory, and was proposed at some point during the late 17th Century, and is an accurate method of calculating π to as many decimal places as you can be bothered to work out; however it is hugely labour-intensive as you need about 10,000 terms of the series to work out π to just four decimal places.

Gregory later came up with a more useful series, using properties of tan to calculate π, and using this result the number of known decimal places of π shot up. It was proved in 1761 that π was irrational, and in 1873 π was calculated to 707 decimal places. It was later discovered, however, that the last 180 were incorrect6.

In 1949, π was calculated to 2,000 decimal places with the help of one of the first computers, and since then computers have been used to increase the known digits of π into the trillions.

What Use is Pi?

π, to a few decimal places, is highly useful in mathematics, and in construction, and anywhere else that accurate measurements of circles are needed. Increasing numbers of decimal places are used depending on the accuracy needed, but really accurate values of π are not needed for any real-world applications, and more than 10 decimal places would be unlikely to be necessary. Working out the value of π is more useful for developing computer systems capable of the task, which could then be used for other purposes.

π is also used when measuring angles in radians7. There are 360/(2π) degrees in one radian, and so measurements in degrees can be converted to radians and vice versa. Radians are more commonly used than degrees for advanced mathematics, with 2π radians being a full circle, and π/2 radians being a right-angle.

π has applications in cryptography, as it is useful for generating random keys for ciphers such as the Vigenère cipher.

A game that can be played with π is finding a book or website that gives π to as many decimal places as possible, and then trying to find your phone number, birth date, and interesting combinations of numbers (like 123456) within the digits. There is even a website called PiSearch which will do this for you.

Memorising π to as many decimal places as possible has become a hobby for some people, and the current official record for memorising π is 42,195 digits, set by Hiroyuki Goto in 1995.

The Reciprocal of Pi

The reciprocal of π is the number y, such that yπ = 1, or to put it another way, y = 1/π. This is useful in calculations that include π as the denominator of a fraction - rather than dividing by π we just multiply by the reciprocal. This reciprocal is 0.318310 to 6 decimal places.

So How Do I Memorise Pi?

There are many different memory techniques (otherwise known as mnemonics) used, most of which can be used to remember any sequence at all, from your shopping list to phone numbers to π. These include taking an imaginary walk around your house, and seeing various objects that have a connection with the thing you're trying to remember. For example, you could relate each digit with something that rhymes with it, like a nun for one, a shoe for two, a tree for three, etc. This can be a very effective method for remembering sequences.

There are other methods that relate strictly to π. One of these is that the first 13 digits seem to rhyme (three-point-one-four-one-five-nine, two-six-five-three-five-eight-nine) but there is only so far you can get with that technique. Another method is using any of the following sayings:

How I wish I could determine
In circle round
The exact relation
Lindemann8 found.
How I wish I
Could calculate Pi
How I want a drink,
alcoholic of course,
after all those chapters
involving quantum mechanics.

in which the number of letters in each word indicates the digit, but again this method is limited.

Another phrase can be used to memorise the reciprocal of π:

Can I remember the reciprocal?

where again, the number of letters in each word relates to the digit.

In the end, if you really want to memorise π, it's up to you to find the technique which suits you and allows you to remember the most digits. Good luck!

1 Pronounced 'pie' in English, and 'pee' in most other languages.
2 Although irrational numbers with non-repeating patterns can be constructed, such as 0.12345678910111213...
3 All the way around the circle.
4 From one side of the circle to the other, passing through the center.
5 An algorithm is a set of rules or processes to follow in order to solve a problem.
6 It is worth pointing out at this stage that all calculations were done by hand, without the aid of calculators and computers, which hadn't yet been invented.
7 A radian is defined by the Merriam-Webster dictionary as being a unit of plane angular measurement that is equal to the angle at the center of a circle subtended by an arc equal in length to the radius.
8 Lindemann proved in 1882 that π was transcendental, meaning that it isn't a root of any polynomial equation with integer (whole number) coefficients. A consequence of this is that it is impossible to construct a square equal in area to a given circle using only compass and straight-edge, one of the most sought-after constructions in history.

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Entry Data
Entry ID: A211500 (Edited)

Written and Researched by:

Edited by:

Date: 02   December   1999

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Referenced Guide Entries
The Bible - a Perspective
Rational Numbers
Irrational Numbers
Pie Recipes
Mnemonics and Other Learning Devices
A Beginner's Guide to Mean, Median and Mode
Early Electronic Computers
Modern Cryptography - Methods and Uses

Referenced Sites
Merriam-Webster dictionary

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