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where C is the circumference of a circle, d is the diameter, and r is the radius.More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width.
In the special case where the numerator =, there are exactly four solutions having only two terms. [ 7 ] [ 8 ] All four were found by John Machin in 1705–1706, but only one of them became widely known when it was published in William Jones 's book Synopsis Palmariorum Matheseos , so the other three are often attributed to other mathematicians.
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.
[0; 4, 4, 8, 16, 18, 5, 1, 1, 1, 1, 7, 1, 1, 6, 2, 9, 58, 1, 3, 4, …] [OEIS 100] Computed up to 1 011 597 392 terms by E. Weisstein. He also noted that while the Champernowne constant continued fraction contains sporadic large terms, the continued fraction of the Copeland–Erdős Constant do not exhibit this property. [Mw 85] Base 10 ...
Using the P function mentioned above, the simplest known formula for π is for s = 1, but m > 1. Many now-discovered formulae are known for b as an exponent of 2 or 3 and m as an exponent of 2 or it some other factor-rich value, but where several of the terms of sequence A are zero.
Alternatively, the quick approximation 99/70 (≈ 1.41429) for the square root of two was frequently used before the common use of electronic calculators and computers. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000 (approx. 7.2 × 10 −5). Its simple continued fraction is periodic and given by:
Viète obtained his formula by comparing the areas of regular polygons with 2 n and 2 n + 1 sides inscribed in a circle. [ 1 ] [ 2 ] The first term in the product, 2 / 2 {\displaystyle {\sqrt {2}}/2} , is the ratio of areas of a square and an octagon , the second term is the ratio of areas of an octagon and a hexadecagon , etc.
(Pi function) – the gamma function when offset to coincide with the factorial Rectangular function π ( n ) {\displaystyle \pi (n)\,\!} – the Pisano period