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Viète's formula

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From Wikipedia, the free encyclopedia

Infinite product converging to 2/π

This article is about a formula for π. For formulas for symmetric functions of the roots, see Vieta's formulas.

Viète's formula, as printed in Viète's Variorum de rebus mathematicis responsorum, liber VIII (1593)<br>In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant π:

{\displaystyle {\begin{aligned}{\frac {2}{\pi }}&={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots \\[5mu]&={\sqrt {\frac {1}{2}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}}}\cdots \end{aligned}}}

It can also be represented as

cos

{\displaystyle {\frac {2}{\pi }}=\prod _{n=1}^{\infty }\cos {\frac {\pi }{2^{n+1}}}.}

The formula is named after François Viète, who published it in 1593.[1] As the first formula of European mathematics to represent an infinite process,[2] it can be given a rigorous meaning as a limit expression[3] and marks the beginning of mathematical analysis. It has linear convergence and can be used for calculations of π,[4] but other methods before and since have led to greater accuracy. It has also been used in calculations of the behavior of systems of springs and masses[5] and as a motivating example for the concept of statistical independence.

The formula can be derived as a telescoping product of either the areas or perimeters of nested polygons converging to a circle. Alternatively, repeated use of the half-angle formula from trigonometry leads to a generalized formula, discovered by Leonhard Euler, that has Viète's formula as a special case. Many similar formulas involving nested roots or infinite products are now known.

Significance<br>[edit]

François Viète (1540–1603) was a French lawyer, privy councillor and code-breaker to two French kings, and amateur mathematician. He published this formula in 1593 in his work Variorum de rebus mathematicis responsorum, liber VIII. At this time, methods for approximating π to (in principle) arbitrary accuracy had long been known. Viète's own method can be interpreted as a variation of an idea of Archimedes of approximating the circumference of a circle by the perimeter of a many-sided polygon,[1] used by Archimedes to find the approximation[6]

223<br>71

22

{\displaystyle {\frac {223}{71}}

By publishing his method as a mathematical formula, Viète formulated the first instance of an infinite product known in mathematics,[7][8] and the first example of an explicit formula for the exact value of π.[9][10] As the first representation in European mathematics of a number as the result of an infinite process rather than of a finite calculation,[11] Eli Maor highlights Viète's formula as marking the beginning of mathematical analysis[2] and Jonathan Borwein calls its appearance "the dawn of modern mathematics".[12]

Using his formula, Viète calculated π to an accuracy of nine decimal digits.[4] However, this was not the most accurate approximation to π known at the time, as the Persian mathematician Jamshīd al-Kāshī had calculated π to an accuracy of nine sexagesimal digits and 16 decimal digits in 1424.[12] Not long after Viète published his formula, Ludolph van Ceulen used a method closely related to Viète's to calculate 35 digits of π, which were published only after van Ceulen's death in 1610.[12]

Beyond its mathematical and historical significance, Viète's formula can be used to explain the different speeds of waves of different frequencies in an infinite chain of springs and masses, and the appearance of π in the limiting behavior of these speeds.[5] Additionally, a derivation of this formula as a product of integrals involving the Rademacher system, equal to the integral of products of the same functions, provides a motivating example for the concept of statistical independence.[13]

Interpretation and convergence<br>[edit]

Viète's formula may be rewritten and understood as a limit expression[3]

lim

{\displaystyle \lim _{n\rightarrow \infty }\prod _{i=1}^{n}{\frac {a_{i}}{2}}={\frac {2}{\pi }},}

where

{\displaystyle {\begin{aligned}a_{1}&={\sqrt {2}}\\a_{i}&={\sqrt {2+a_{i-1}}}.\end{aligned}}}

For each choice of

{\displaystyle n}

, the expression in the limit is a finite product, and as

{\displaystyle n}

gets arbitrarily large, these finite products have values that approach the value of Viète's formula arbitrarily closely. Viète did his work long before the concepts of...