The Transcendental Number Pi

Jai Ganesh · 2021-12-13 14:37:34

1. Introduction

The number π (/paɪ/; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159. It is defined in Euclidean geometry as the ratio of a circle's circumference to its diameter, and also has various equivalent definitions. The number appears in many formulas in all areas of mathematics and physics. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in 1706. It is also referred to as Archimedes' constant.

Being an irrational number, π cannot be expressed as a common fraction, although fractions such as 22/7 are commonly used to approximate it. Equivalently, its decimal representation never ends and never settles into a permanently repeating pattern. Its decimal (or other base) digits appear to be randomly distributed, and are conjectured to satisfy a specific kind of statistical randomness.

It is known that π is a transcendental number: it is not the root of any polynomial with rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.

Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of π for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, Chinese mathematics approximated π to seven digits, while Indian mathematics made a five-digit approximation, both using geometrical techniques. The first computational formula for π, based on infinite series, was discovered a millennium later, when the Madhava–Leibniz series was discovered by the Kerala school of astronomy and mathematics, documented in the Yuktibhāṣā, written around 1530.

The invention of calculus soon led to the calculation of hundreds of digits of π, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits. The primary motivation for these computations is as a test case to develop efficient algorithms to calculate numeric series, as well as the quest to break records. The extensive calculations involved have also been used to test supercomputers and high-precision multiplication algorithms.

Because its most elementary definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. In more modern mathematical analysis, the number is instead defined using the spectral properties of the real number system, as an eigenvalue or a period, without any reference to geometry. It appears therefore in areas of mathematics and sciences having little to do with the geometry of circles, such as number theory and statistics, as well as in almost all areas of physics. The ubiquity of π makes it one of the most widely known mathematical constants—both inside and outside the scientific community. Several books devoted to π have been published, and record-setting calculations of the digits of π often result in news headlines.

What Is Pi, and How Did It Originate?

Succinctly, pi—which is written as the Greek letter for p, or π—is the ratio of the circumference of any circle to the diameter of that circle. Regardless of the circle's size, this ratio will always equal pi. In decimal form, the value of pi is approximately 3.14. But pi is an irrational number, meaning that its decimal form neither ends (like 1/4 = 0.25) nor becomes repetitive (like 1/6 = 0.166666...). (To only 18 decimal places, pi is 3.141592653589793238.) Hence, it is useful to have shorthand for this ratio of circumference to diameter. According to Petr Beckmann's A History of Pi, the Greek letter π was first used for this purpose by William Jones in 1706, probably as an abbreviation of periphery, and became standard mathematical notation roughly 30 years later.

Try a brief experiment: Using a compass, draw a circle. Take one piece of string and place it on top of the circle, exactly once around. Now straighten out the string; its length is called the circumference of the circle. Measure the circumference with a ruler. Next, measure the diameter of the circle, which is the length from any point on the circle straight through its center to another point on the opposite side. (The diameter is twice the radius, the length from any point on the circle to its center.) If you divide the circumference of the circle by the diameter, you will get approximately 3.14—no matter what size circle you drew! A larger circle will have a larger circumference and a larger radius, but the ratio will always be the same. If you could measure and divide perfectly, you would get 3.141592653589793238..., or pi.

Otherwise said, if you cut several pieces of string equal in length to the diameter, you will need a little more than three of them to cover the circumference of the circle.

Pi is most commonly used in certain computations regarding circles. Pi not only relates circumference and diameter. Amazingly, it also connects the diameter or radius of a circle with the area of that circle by the formula: the area is equal to pi times the radius squared. Additionally, pi shows up often unexpectedly in many mathematical situations. For example, the sum of the infinite series

The importance of pi has been recognized for at least 4,000 years. A History of Pi notes that by 2000 B.C., "the Babylonians and the Egyptians (at least) were aware of the existence and significance of the constant π," recognizing that every circle has the same ratio of circumference to diameter. Both the Babylonians and Egyptians had rough numerical approximations to the value of pi, and later mathematicians in ancient Greece, particularly Archimedes, improved on those approximations. By the start of the 20th century, about 500 digits of pi were known. With computation advances, thanks to computers, we now know more than the first six billion digits of pi.

The first 100 digits of pi are 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679. The value of pi starts with a 3 followed by a decimal point.

Jai Ganesh · 2022-05-12 02:03:36

2. Addenda

The number

(spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulas across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as 22/7 are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an equation involving only sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found.

For thousands of years, mathematicians have attempted to extend their understanding of

, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated π to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula forπ, based on infinite series, was discovered a millennium later. The earliest known use of the Greek letter to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706.

The invention of calculus soon led to the calculation of hundreds of digits of

, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of to many trillions of digits. These computations are motivated by the development of efficient algorithms to calculate numeric series, as well as the human quest to break records. The extensive computations involved have also been used to test supercomputers.

Because its definition relates to the circle,

is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such as cosmology, fractals, thermodynamics, mechanics, and electromagnetism. In modern mathematical analysis, it is often instead defined without any reference to geometry; therefore, it also appears in areas having little to do with geometry, such as number theory and statistics. The ubiquity of π makes it one of the most widely known mathematical constants inside and outside of science. Several books devoted to π have been published, and record-setting calculations of the digits of π often result in news headlines.

Jeremy Desmond · 2022-08-28 20:38:25

So how do we know that pi has an infinite number of decimal places that never repeat? I mean, if the fastest modern computers have not yet calculated the last decimal digit of pi, how do we know that the end is not just one or two decimal places further on?

Sorry if that’s a silly question!

Jai Ganesh · 2022-08-28 23:08:04

Hi,

Calculating

:

In some ways Pi (π) is a really straightforward number – calculating Pi simply involves taking any circle and dividing its circumference by its diameter.

On the other hand Pi (π) is the first number we learn about at school where we can’t write it as an exact decimal – it is a mysterious number which has digits which go on forever and has fascinated people for thousands of years.

We learn that we can start to write down Pi (π) = 3.141592653589….. but that we can never finish it. Pi (π) goes on forever and has no repeating pattern to its digits – it is what is called an irrational number. In fact if you search long enough within the digits of Pi (π) you can find any number, including your birthday.

Pi (π) is also a really useful number. It appears everywhere in mathematics and also has countless uses in Engineering and Science. Lots of things are round, and whenever something is round, Pi (π) usually becomes important. For example if an engineer wants to calculate the volume of a water pipe they will use the following formula for a cylinder:

(Where r is the radius of the pipe and h is the height of the pipe.)

Calculating Pi (π)

Because Pi (π) has so many important uses, then we need to be able to start to calculate it, at least to several decimal places accuracy. Someone had to come up with the approximate value for Pi (

) which appears on your calculator – it didn’t get there by magic!

Measuring circles

The first and most obvious way to calculate Pi (π) is to take the most perfect circle you can, and then measure its circumference and diameter to work out Pi (π). This is what ancient civilisations would have done and it is how they would have first realised that there is a constant ratio hidden within every circle. The problem with this method is accuracy – can you trust your tape measure to deliver Pi (π) correct to 10 or more decimal places?

Hexagons Using Polygons to approximate Pi (π)

The Ancient Greek mathematician Archimedes came up with an ingenious method for calculating an approximation of Pi (π). Archimedes began by inscribing a regular hexagon inside a circle and then circumscribing another regular hexagon outside the same circle. He was then able to calculate the exact circumferences and diameters of the hexagons and could therefore obtain a rough approximation of Pi (π) by dividing the circumference by the diameter.

Archimedes then found a way to double the number of sides of his hexagons. He could then find a more accurate approximation of Pi (π) by using polygons with more sides, which were closer to the circle. He did this four times until he was using 96 sided polygons. Archimedes calculated the circumference and diameter exactly and therefore could approximate Pi (π) to being between

and The fraction has remained as one of the most popular and memorable approximations of Pi (π) ever since.

Around 600 years after Archimedes, the Chinese mathematician Zu Chongzhi used a similar method to inscribe a regular polygon with 12,288 sides. This produced an approximation of Pi (π) as

which is correct to six decimal places. It was nearly 600 more years until a totally new method was devised that improved upon this approximation.

Calculating Pi (π) using infinite series

Mathematicians eventually discovered that there are in fact exact formulas for calculating Pi (π). The only catch is that each formula requires you to do something an infinite number of times. (Which makes sense given that the digits of Pi (π) go on forever.) One of the amazing things which interests people about Pi (π) is that there isn’t just one formula, but a large number of different ones for people to study.

One of the most well known and beautiful ways to calculate Pi (π) is to use the Gregory-Leibniz Series:

If you continued this pattern forever you would be able to calculate

exactly and then just multiply it by 4 in order to get . If however you start to add up the first few terms, you will begin to get an approximation for Pi (π). The problem with the series above is that you need to add up a lot of terms in order to get an accurate approximation of Pi (π). You need to add up more than 300 terms in order to produce Pi (π) accurate to two decimal places!

Another series which converges more quickly is the Nilakantha Series which was developed in the 15th century. Converges more quickly means that you need to work out fewer terms for your answer to become closer to Pi (π) .

Nilakantha Series:

Mathematicians have also found other more efficient series for calculating Pi (π). Computer programs can add up more and more terms, calculating Pi (π) to extraordinary degrees of accuracy. In 2014 the world record was that a computer has calculated Pi (π) correct to 13,300,000,000,000 decimal places.

Before the advent of computers it was much harder to calculate Pi (π). In the 19th Century William Shanks took 15 years to calculate Pi (π) correct to 707 decimal places. Unfortunately it was later found that he had made a mistake and was only right to 527 decimal places! The nine or 10 digits of Pi (π) which you see on your calculator have been known about probably since 1400.

Now that you know how to calculate Pi (π), you could always try your hand at memorising the decimal places of Pi (π). The most recent record was created on Pi Day in 2019 by Google, who calculated Pi to 31.4 trillion decimal places!. On the other hand, you could simply use the following mnemonic for learning the first six decimal places of Pi (π): “How I wish I could calculate Pi”

Jeremy Desmond · 2022-08-29 00:05:38

Thank you Ganesh for that extra interesting information about pi. But you still haven’t explained how we know pi has an infinite number of decimal places containing non-repeating digits. You stated that computers have calculated pi correct to 1.33 x 10^13 decimal places but how do we know the end is not just round the corner? Is it possible that one day a computer will calculate the exact value of pi to a finite number of decimal places?

Jai Ganesh · 2022-08-29 00:37:30

Hi Jeremy Desmond,

Jeremy Desmond wrote:

Is it possible that one day a computer will calculate the exact value of pi to a finite number of decimal places?

By definition, a Transcendental Number:

Transcendental number is a number that is not algebraic, in the sense that it is not the solution of an algebraic equation with rational-number coefficients. Transcendental numbers are irrational, but not all irrational numbers are transcendental. For example,

has the solutions ; thus, Square root of 2, an irrational number, is an algebraic number and not transcendental. Nearly all real and complex numbers are transcendental, but very few numbers have been proven to be transcendental. The numbers and are transcendental numbers.

For the last question, I would say 'No'.

zetafunc · 2022-08-29 02:02:32

Jeremy Desmond wrote:

Thank you Ganesh for that extra interesting information about pi. But you still haven’t explained how we know pi has an infinite number of decimal places containing non-repeating digits. You stated that computers have calculated pi correct to 1.33 x 10^13 decimal places but how do we know the end is not just round the corner? Is it possible that one day a computer will calculate the exact value of pi to a finite number of decimal places?

Hi Jeremy,

Welcome to the forum.

The reason that pi doesn't have a final digit is because it is an irrational number, and irrational numbers have decimal expansions which continue forever. In other words, to answer your question it suffices to:

(1) Prove that pi is irrational, and then
(2) Prove that all irrational numbers have an infinite, non-recurring decimal expansion.

There are quite a few proofs that pi is irrational, some of which are more complex than others. One of the most common proofs is Lambert's, where he essentially (a) shows that all infinite continued fractions are irrational and then (b) finds an infinite continued fraction for pi, which automatically implies that it must be irrational from part (a). If you're interested in continued fractions I've got some videos about it on my YouTube channel (although it doesn't discuss the continued fraction of pi). This takes care of (1).

Now for (2). Suppose instead that you could find an irrational number which didn't have an infinite, non-recurring decimal expansion. Let's say for example, the number 0.12345123451234512345... This decimal expansion has an infinitely recurrent pattern (the '12345' bit). But we see that if we let x = 0.12345123451234512345... then:

Multiplying both sides by 100000:

Subtracting x from both sides:

And finally, dividing both sides by 99999 gives us:

But hang on -- if we can express it as a fraction of two integers, it must be a rational number! (This is pretty much the definition of what it means for a number to be rational.) Similarly, if pi had a 'final digit' we could also express it as a fraction -- for example, if pi terminated after 5 decimal places, i.e. 3.14159, then we could write that as 314159/100000, which is a rational number.

Let me know if that makes sense -- happy to clarify anything if needed.

Jeremy Desmond · 2022-08-29 21:20:18

Thanks very much, zetafunc. Your explanation is helpful. I watched your introductory video about continued fractions on your YouTube channel and that has stimulated my interest in that topic. I plan to watch more of your videos. Thank you for spreading your expertise and enthusiasm.
Jeremy

Jai Ganesh · 2022-09-05 22:02:54

Chronology of computation of

20th and 21st centuries

In 1910, the Indian mathematician Srinivasa Ramanujan found several rapidly converging infinite series of

, including

which computes a further eight decimal places of

with each term in the series. His series are now the basis for the fastest algorithms currently used to calculate . Even using just the first term gives

From the mid-20th century onwards, all calculations of

have been done with the help of calculators or computers.

In 1944, D. F. Ferguson, with the aid of a mechanical desk calculator, found that William Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were incorrect.

In the early years of the computer, an expansion of π to 100000 decimal places was computed by Maryland mathematician Daniel Shanks (no relation to the aforementioned William Shanks) and his team at the United States Naval Research Laboratory in Washington, D.C. In 1961, Shanks and his team used two different power series for calculating the digits of

. For one, it was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 100,265 digits of π were published in 1962. The authors outlined what would be needed to calculate to 1 million decimal places and concluded that the task was beyond that day's technology, but would be possible in five to seven years.

In 1989, the Chudnovsky brothers computed π to over 1 billion decimal places on the supercomputer IBM 3090 using the following variation of Ramanujan's infinite series of

:

Records since then have all been accomplished using the Chudnovsky algorithm. In 1999, Yasumasa Kanada and his team at the University of Tokyo computed

to over 200 billion decimal places on the supercomputer HITACHI SR8000/MPP (128 nodes) using another variation of Ramanujan's infinite series of . In November 2002, Yasumasa Kanada and a team of 9 others used the Hitachi SR8000, a 64-node supercomputer with 1 terabyte of main memory, to calculate to roughly 1.24 trillion digits in around 600 hours (25 days).

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The Transcendental Number Pi

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