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3. Everything / Maths, Science & Technology / Mathematics

Pi

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:

3.1415926535897932384626433832795028841971693993751058209749445
923078164062862089986280348253421170679821480865132823066470938
446095505822317253594081284811174502841027019385211055596446229
489549303819644288109756659334461284756482337867831652712019091
456485669234603486104543266482133936072602491412737245870066063
155881748815209209628292540917153643678925903600113305305488204
665213841469519415116094330572703657595919530921861173819326117
931051185480744623799627495673518857527248912279381830119491298
336733624406566430860213949463952247371907021798609437027705392
171762931767523846748184676694051320005681271452635608277857713
427577896091736371787214684409012249534301465495853710507922796
892589235420199561121290219608640344181598136297747713099605187
072113499999983729780499510597317328160963185950244594553469083
026425223082533446850352619311881710100031378387528865875332083
814206171776691473035982534904287554687311595628638823537875937
519577818577805321712268066130019278766111959092164201989380952
572010654858632788659361533818279682303019520353018529689957736
225994138912497217752834791315155748572424541506959508295331168
617278558890750983817546374649393192550604009277016711390098488
240128583616035637076601047101819429555961989467678374494482553
797747268471040475346462080466842590694912933136770289891521047
521620569660240580381501935112533824300355876402474964732639141
992726042699227967823547816360093417216412199245863150302861829
745557067498385054945885869269956909272107975093029553211653449
872027559602364806654991198818347977535663698074265425278625518
184175746728909777727938000816470600161452491921732172147723501
414419735685481613611573525521334757418494684385233239073941433
345477624168625189835694855620992192221842725502542568876717904
946016534668049886272327917860857843838279679766814541009538837
863609506800642251252051173929848960841284886269456042419652850
222106611863067442786220391949450471237137869609563643719172874
677646575739624138908658326459958133904780275901

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:
Cyzaki

Edited by:
U284


Date: 02   December   1999


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


Referenced Sites
Merriam-Webster dictionary
PiSearch

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