Claim: A fraction cannot exactly express the number pi.

The number is a mathematical constant , the ratio of a circle 's circumference to its diameter , commonly approximated as 3.14159 . It has been represented by the Greek letter '' '' since the mid-18th century , though it is also sometimes spelled out as `` pi '' -LRB- -LSB- paɪ -RSB- -RRB- .   Being an irrational number , can not be expressed exactly as a fraction -LRB- equivalently , its decimal representation never ends and never settles into a permanent repeating pattern -RRB- . Still , fractions such as 22/7 and other rational numbers are commonly used to approximate . The digits appear to be randomly distributed . In particular , the digit sequence of is conjectured to satisfy a specific kind of statistical randomness , but to date no proof of this has been discovered . Also , is a transcendental number ; that is , a number that is not the root of any non-zero polynomial having rational coefficients . This transcendence of implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge .   Ancient civilizations required fairly accurate computed values for for practical reasons . It was calculated to seven digits , using geometrical techniques , in Chinese mathematics , and to about five digits in Indian mathematics in the 5th century AD . The historically first exact formula for , 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 to many trillions of digits after the decimal point . Practically all scientific applications require no more than a few hundred digits of , and many substantially fewer , so the primary motivation for these computations is the quest to find more efficient algorithms for calculating lengthy numeric series , as well as the desire 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 , 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 the 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 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 record-setting calculations of the digits of often result in news headlines . Attempts to memorize the value of with increasing precision have led to records of over 70,000 digits .