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In the 1760s, Johann Heinrich Lambert was the first to prove that the number π is irrational, meaning it cannot be expressed as a fraction /, where and are both integers. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus.
The number π (/ p aɪ / ⓘ; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter.It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.
Pi: 3.14159 26535 89793 23846 [Mw 1] [OEIS 1] Ratio of a circle's circumference to its diameter. ... Continued fractions with more than 20 known terms have been ...
The latter fraction is the best possible rational approximation of ... PiFast 4.4 is available from Stu's Pi page. PiFast 4.3 is available from Gourdon's page.
3.8 Continued fractions. 3.9 Iterative algorithms. ... A History of Pi; In culture; Indiana pi bill; Pi Day; ... [3] () + = (see also Proof that 22/7 ...
An easy mnemonic helps memorize this fraction by writing down each of the first three odd numbers twice: 1 1 3 3 5 5, then dividing the decimal number represented by the last 3 digits by the decimal number given by the first three digits: 1 1 3 分之(fēn zhī) 3 5 5. (In Eastern Asia, fractions are read by stating the denominator first ...
The purpose of the proof is not primarily to convince its readers that 22 / 7 (or 3 + 1 / 7 ) is indeed bigger than π. Systematic methods of computing the value of π exist. If one knows that π is approximately 3.14159, then it trivially follows that π < 22 / 7 , which is approximately 3.142857.
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510... (sequence A000796 in the OEIS). Unusually good approximations are given by the fractions 22/7 and 355/113. Memorizing as well as computing increasingly more digits of π is a world record pursuit.