The ratio C/d is constant, regardless of the circle's size.
This definition of Pi is not universal, because it is valid only in flat (Euclidean) geometry; it is not valid in curved (non-Euclidean) geometries. For this reason, some mathematicians prefer definitions of pi based on calculus or trigonometry that do not rely on the circle. One such definition is: Pi is twice the smallest positive x for which cos(x)=0.
Pi is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate Pi); consequently, its decimal representation never ends and never settles into a permanent repeating pattern. The digits appear to be randomly distributed, although no proof of this has yet been discovered. Pi is a transcendental number – a number that is not the root of any nonzero polynomial having rational coefficients. The transcendence of Pi implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge.
Like all irrational numbers, Pi cannot be represented as a simple fraction. But every irrational number, including Pi, can be represented by an infinite series of nested fractions, called a continued fraction.
For thousands of years, mathematicians have attempted to extend their understanding of Pi, sometimes by computing its value to a high degree of accuracy. Before the 15th century, mathematicians such as Archimedes and Liu Hui used geometrical techniques, based on polygons, to estimate the value of Pi. Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of Pi, and were used by mathematicians including Madhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.
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 Pi to, as of late 2011, over 10 trillion (1013) digits. Scientific applications generally require no more than 40 digits of pi, so the primary motivation for these computations is the human desire to break records, but the extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms.
Because its definition relates to the circle, Pi is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, or spheres. It is also found in formulae from other branches of science, such as cosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism. The ubiquitous nature of Pi makes it one of the most widely known mathematical constants, both inside and outside the scientific community: Several books devoted to it have been published; the number is celebrated on Pi Day; and news headlines often contain reports about record-setting calculations of the digits of Pi. Several people have endeavored to memorize the value of Pi with increasing precision, leading to records of over 67,000 digits.
Some approximations of Pi include:
The first 100 decimal digits are 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 ....
The base 16 approximation to 20 digits is 3.243F6A8885A308D31319 ....
A base 60 approximation is 3:8:29:44: