Pi (mathematical constant)

Pi, denoted by the lower-case Greek letter $\scriptstyle \pi$, is a mathematical constant that is approximately equal to 3.14159. In Euclidean geometry, $\pi$ represents the ratio between the circumference and the diameter of any circle, or equivalently, the ratio between a circle's area and the square of its radius. This ratio is also found in many other geometrical objects, such as spheres and cones. Pi has further uses in many other areas of mathematics.

Pi is an irrational number, which means that it cannot be expressed exactly as a ratio of integers, and transcendental, meaning that it does not satisfy any algebraic equation with rational coefficients. The latter property implies that $\pi$ is not constructible: in particular, the problem of squaring the circle in a finite number of steps using a compass and straightedge is impossible. Since $\pi$ is irrational, it cannot be written as a finite or periodically repeating decimal. The first 50 digits after the decimal point are

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 ...

Early geometrical study

The fact that the ratio of the circumference to the diameter of a circle is the same for all circles, and that it is slightly more than 3, was known to ancient Egyptian, Babylonian, Indian and Greek geometers. Preserved documents do not refer to the number explicitly, but contain approximations for $\scriptstyle \pi$ as part of methods for calculating circumference or area. A Babylonian stone tablet from 1900 BC uses the proportion 25/8 = 3.125, and an Egyptian document from the same time gives the ratio $\scriptstyle (16/9)^{2}\,\approx \,3.16$. The ancient Indian text Shatapatha Brahmana gives $\scriptstyle \pi$ as $\scriptstyle 339/108\,\approx \,3.139$. All of these values differ from the true value by less than one percent. The Books of Kings (600 BC) appears to suggest $\scriptstyle \pi \,=\,3$, which is notably worse than other estimates available at the time, although it is disputed whether the text suggests an exact value.

Archimedes (287–212 BC), in whose honor $\scriptstyle \pi$ is sometimes called Archimedes' constant, was the first to treat the number with mathematical rigor. He knew how to calculate the circumference of a regular polygon, and realized that a polygon inscribed in a circle has a smaller circumference than the circle whereas a circumscribed polygon has a greater circumference. By calculating the circumferences of two 96-sided polygons, one slightly smaller than a unit circle and the other slightly larger, he rigorously proved the bounds $\scriptstyle 223/71\,<\,\pi \,<\,22/7$. Taking the average of the bounds yields the approximation $\scriptstyle \pi \,\approx \,3.1419$. The method of Archimedes is the first known algorithm for calculating $\scriptstyle \pi$ with arbitrary accuracy.

In the following centuries, most significant mathematical development took place in India and China. Around 480, the Chinese mathematician Zu Chongzhi gave the approximation $\scriptstyle \pi \,\approx \,355/113$ and showed that $\scriptstyle 3.1415926\,<\,\pi \,<\,3.1415927$, which would stand as the most accurate value for $\scriptstyle \pi$ over the next 900 years.

Computer calculations

The advent of digital computers in the 20th century led to an increased rate of new $\pi$ calculation records. John von Neumann used ENIAC to compute 2037 digits of $\pi$ in 1949, a calculation that took 70 hours. Additional thousands of decimal places were obtained in the following decades, with the million-digit milestone passed in 1973. Progress was not only due to faster computer hardware, but also the discovery of new algorithms for doing arithmetic. The discovery of the Karatsuba algorithm and the fast Fourier transform (FFT) in the 1960s facilitated multiplication and division of numbers with millions of digits. The ordinary "school book" method of multiplication requires $n^{2}$ operations to find the product of two $n$-digit numbers, whereas the FFT is essentially linearithmic, using $n\cdot \log(n)\cdot \log(\log(n))$ steps. A 1000-digit multiplication thus requires a few thousand calculations rather than a million.

Computer calculations of $\pi$ have also been aided by new formulas for $\pi$ found in the 20th century. In his study of modular functions, the Indian mathematician Srinivasa Ramanujan found many remarkable new formulas, such as the series

${\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}$

which gives eight decimals of π per term. Due to scarce distribution of Ramanujan's published work, it was however not widely known until around 1985 when William Gosper used it to calculate 17 million decimals of $\pi$. Based on Ramanujan's ideas, Gregory and David Chudnovsky found a formula that delivers 14 digits per term. The Chudnovsky brothers used this formula to set several $\pi$ computing records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691) decimals in 1989.

Series typically give a fixed number of digits for each term. In 1975, Richard Brent and Eugene Salamin independently discovered the Brent-Salamin algorithm in which the number of correct digits doubles with each step. The algorithm consists of setting

$a_{0}=1\quad \quad \quad b_{0}={\frac {1}{\sqrt {2}}}\quad \quad \quad t_{0}={\frac {1}{4}}\quad \quad \quad p_{0}=1$

and iterating

$a_{n+1}={\frac {a_{n}+b_{n}}{2}}\quad \quad \quad b_{n+1}={\sqrt {a_{n}b_{n}}}$

$t_{n+1}=t_{n}-p_{n}(a_{n}-a_{n+1})^{2}\quad \quad \quad p_{n+1}=2p_{n}$

until $a_{n}$ and $b_{n}$ are close enough. Then the estimate for $\pi$ is given by

$\pi \approx {\frac {(a_{n}+b_{n})^{2}}{4t_{n}}}.$

A mere 25 iterations gives 45 million correct decimals of $\pi$. A similar algorithm that quadruples the accuracy in each step has been found by Jonathan and Peter Borwein. The methods have been used by Yasumasa Kanada and team to set most of the $\pi$ calculation records since 1980, up to a calculation of 206,158,430,000 decimals of $\pi$ in 1999. The current record is 1,241,100,000,000 decimals, set by Kanada and team in 2002. Although most of Kanada's previous records were set using the Brent-Salamin algorithm, the 2002 calculation made use of two Machin-like formulas that were slower but crucially reduced memory consumption. The calculation was performed on a 64-node Hitachi supercomputer with 1 terabyte of main memory, capable of carrying out 2 trillion operations per seconds.

Aside from prestige, enormous $\pi$ calculations are useful for testing the integrity and performance of supercomputers. Pi calculation is a popular benchmark for personal computer hardware and arbitrary-precision arithmetic software as well. With optimized $\pi$ calculating software, a modern (2006) home computer can calculate a million digits of $\pi$ in a few seconds and a billion digits in a few hours.

Properties of $\pi$'s digits

It appears that $\pi$'s digits are "random", although it is not entirely easy to define the "randomness" of a number. The most important measure besides irrationality is perhaps normality: if a number is normal, all digits appear equally often in its decimal expansion, and all subsequences of digits appear equally often. Further, this should hold in every base, such as binary (in which $\pi$ begins 11.0010010000111111...). A popular statement is that if $\pi$ is normal, then, for example, every work by Shakespeare can be found encoded somewhere in the digits of $\pi$. This seems likely, but no proof that $\pi$ is normal has been found. It is not even known whether any single digit, such as "1", appears infinitely many times in the decimal expansion of $\pi$. The value of $\pi$ calculated by Kanada in 2002 shows that the first trillion digits of $\pi$ are statistically random, but a finite calculation says nothing about the infinitely many digits that follow.

Another definition of randomness is that there should be no "simple formula" for the $n$-th digit of $\pi$. Although there are simple formulas for $\pi$, the calculations required to produce explicit digits are not so simple. Computational complexity theory allows the difficulty to be expressed quantitatively: the time needed to calculate $n$ digits of $\pi$ on a Turing machine using the best known algorithm is proportional to $n\cdot (\log(n))^{2}\cdot \log(\log(n))$. It is conjectured that computing the $n$-th digit of $\pi$ cannot be done significantly faster than in a time proportional to $n$: such a method could be said to be a "simple formula" for the digits of $\pi$.

The closest thing to a simple formula for the digits of $\pi$ is the BBP formula, found in 1995 by Simon Plouffe together with David H. Bailey and Peter Borwein. The formula,

$\pi =\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right),$

allows the $n$-th individual hexadecimal or binary digit of $\pi$ to be extracted in $n\cdot \log(n)$ time, which is significantly faster than calculating all the first $n$ digits and extracting the last one in the result. Between 1998 and 2000, the distributed computing project PiHex used a modification of the BBP formula due to Fabrice Bellard to compute the quadrillionth (1,000,000,000,000,000-th) binary digit of $\pi$, which turned out to be 0. The BBP formula has led to theoretical progress as well: it has been used to show that a plausible conjecture from chaos theory would imply that $\pi$ is normal in binary.

Unfortunately, the BBP formula does not allow one to extract decimal digits of $\pi$. Although decimal BBP-type formulas are known for some other mathematical constants, it has been proved that no BBP-type formula exists for $\pi$. There could still exist some completely different scheme for quickly extracting isolated decimals of $\pi$, however. Decimal digit-extraction algorithms have in fact been constructed, but they are slower than calculating all the first $n$ digits of $\pi$ and throwing away the first $n-1$ ones. A spigot algorithm that generates the decimal digits of $\pi$ one by one, rather than computing a fixed number of digits in one large calculation, was found by Stanley Rabinowitz and Stan Wagon in 1995. Again, this algorithm is however slower than fast methods for calculating a fixed amount of decimals at once.

Relation to other constants

Although $\pi$ itself is transcendental, it is not easy to determine whether many of the other well known transcendental numbers (such as $e$) are algebraically independent from $\pi$. It is, for example, not known whether $\pi$ and $e$ are algebraically independent. It is known by the Lindemann–Weierstrass theorem that at least one of $\pi +e$ and $\pi e$ is transcendental, but not which one or whether both are. By contrast, the Gelfond-Schneider theorem implies that $e^{\pi }$ is transcendental, and Yuri Nesterenko proved in 1996 that both $e^{\pi }$ and the gamma function value $\Gamma (1/4)$ are algebraically independent from $\pi$.

Another open problem concerns the value of the so-called zeta constants, which are the Riemann zeta function evaluated at the integers. Euler proved that ζ(2) = π²/6 and that ζ(n) is a rational multiple of πⁿ whenever n is an even number. For example,

$\zeta (4)=1+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}=1.0823\dots$

$\zeta (6)=1+{\frac {1}{2^{6}}}+{\frac {1}{3^{6}}}+\cdots ={\frac {\pi ^{6}}{945}}=1.0173\dots .$

Although it may seem plausible that ζ(n) is a multiple of πⁿ also when n is odd, no such formula has been found. The special case ζ(3) ≈ 1.20205 is called Apéry's constant since it was proved irrational by Roger Apéry in 1977.

References

• Jonathan Borwein & David Bailey (2003). Mathematics by Experiment - Plausible Reasoning in the 21st Century. ISBN 1-56881-211-6
• http://www.cecm.sfu.ca/organics/papers/borwein/paper/html/paper.html