What is π?

The number π is a mathematical constant, the ratio of a circle's circumference to its diameter, commonly approximated as 3.14159.

Any circle in naturally occuring Elucidean space, would have a value of π when its circumference is divided by its diameter.

Being an irrational number, π cannot be expressed exactly as a fraction (equivalently, its decimal representation never ends and never settles into a permanent repeating pattern). Still, fractions such as 227${\displaystyle \frac{22}{7}}$ and other rational numbers are commonly used to approximate π. The digits appear to be randomly distributed. In particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a transcendental number – a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.

Ancient civilizations needed the value of π to be computed accurately for practical reasons. It was calculated to seven digits, using geometrical techniques, in Chinese mathematics and to about five in Indian mathematics in the 5th century AD. The historically first exact formula for π, based on infinite series, was not available until a millennium later, when in the 14th century the Madhava–Leibniz series was discovered in Indian mathematics.

In the 20th and 21st centuries, mathematicians and computer scientists discovered new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits after the decimal point. Practically all scientific applications require no more than a few hundred digits of π, and many substantially fewer, so the primary motivation for these computations is the human desire to break records. However, the extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms.

π is commonly defined as the ratio of a circle's circumference C to its diameter d:

π=Cd${\displaystyle \pi =\frac{C}{d}}$

The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d.

This definition of π implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curved (non-Euclidean) geometry, these new circles will no longer satisfy the formula π=C/d$\pi =C/d$.

Like all irrational numbers, π cannot be represented as a common fraction, as it is an irrational number. But every irrational number, including π, can be represented by an infinite series of nested fractions, called a continued fraction:

Truncating the continued fraction at any point yields a rational approximation for π; the first four of these are 3, 22/7, 333/106, and 355/113. These numbers are among the most well-known and widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to π than any other fraction with the same or a smaller denominator. Due to π being known to be transcendental, it is by definition not algebraic and so cannot be a quadratic irrational. Therefore, π cannot have a periodic continued fraction.

The development of computers in the mid-20th century again revolutionized the hunt for digits of π. Algorithms were developed to compute more and more digits of π. A popular algorithm made for this purpose was the Gauss-Legendre algorithm, which states:

1. Initial value setting:
   a0=1b0=12–√t0=14p0=1.${\displaystyle a_0=1b_0=\frac{1}{\sqrt{2}}{t}_0=\frac{1}{4}{p}_0=1.}$
2. Repeat the following instructions until the difference of an${\displaystyle a_n}$ and bn${\displaystyle b_n}$ is within the desired accuracy:
   an+1bn+1tn+1pn+1=an+bn2,=anbn−−−−√,=tn−pn(an−an+1)2,=2pn.${\displaystyle \begin{array}{rl}{a}_{n+1}& =\frac{a_n+b_n}{2},\\ b_{n+1}& =\sqrt{a_n{b}_n},\\ t_{n+1}& =t_n-p_n(a_n-a_{n+1}{)}^2,\\ p_{n+1}& =2p_n.\end{array}}$
3. π is then approximated as:
   π≈(an+1+bn+1)24tn+1.${\displaystyle \pi \approx \frac{(a_{n+1}+b_{n+1}{)}^2}{4t_{n+1}}.}$

The first three iterations give (approximations given up to and including the first incorrect digit):

3.140…${\displaystyle 3.140\dots }$

3.14159264…${\displaystyle 3.14159264\dots }$

3.1415926535897932382…${\displaystyle 3.1415926535897932382\dots }$

The algorithm has quadratic convergence, which essentially means that the number of correct digits doubles with each iteration of the algorithm.

Due to π being closely related to the circle, it is found in many formulae from the fields of geometry and trigonometry, particularly those concerning circles, spheres, or ellipses. Other branches of science, such as statistics, physics, Fourier analysis, and number theory, also include π in some of their important formulae. appears in formulae for areas and volumes of geometrical shapes based on circles, such as ellipses, spheres, cones, and tori. Below are some of the more common formulae that involve π.

• The circumference of a circle with radius r is 2πr.
• The area of a circle with radius r is πr2.
• The volume of a sphere with radius r is 4/3πr3.
• The surface area of a sphere with radius r is 4πr2.

The formulae above are special cases of the volume of the n-dimensional ball and the surface area of its boundary, the (n−1)-dimensional sphere, given below.

Definite integrals that describe circumference, area, or volume of shapes generated by circles typically have values that involve π. For example, an integral that specifies half the area of a circle of radius one is given by:

∫1−11−x2−−−−−√dx=π2.${\displaystyle {\int }_{-1}^1\sqrt{1-x^2}dx=\frac{\pi }{2}.}$

In that integral the function √1 − x2 represents the top half of a circle (the square root is a consequence of the Pythagorean theorem), and the integral ∫1
−1 computes the area between that half of a circle and the x axis.

Sine and cosine functions repeat with period 2π.

The trigonometric functions rely on angles, and mathematicians generally use radians as units of measurement. π plays an important role in angles measured in radians, which are defined so that a complete circle spans an angle of 2π radians.

The angle measure of 180° is equal to π radians, and 1° = π/180 radians.

Common trigonometric functions have periods that are multiples of π; for example, sine and cosine have period 2π, so for any angle θ and any integer k,

sinθ=sin(θ+2πk) and cosθ=cos(θ+2πk).${\displaystyle \sin \theta =\sin \left(\theta +2\pi k\right)\text{ and }\cos \theta =\cos \left(\theta +2\pi k\right).}$

π≈3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778960917363717872146844090122495343014654958537105079227968925892420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989

I love π.