View source for Viète's formula

{{Short description|Infinite product converging to 2/π}}
{{About|a formula for {{pi}}|formulas for symmetric functions of the roots|Vieta's formulas}}
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[[File:Viète's formula.png|thumb|400px|Viète's formula, as printed in Viète's {{lang|la|Variorum de rebus mathematicis responsorum, liber VIII}} (1593)]]

In [[mathematics]], '''Viète's formula''' is the following [[infinite product]] of [[nested radical]]s representing twice the [[Multiplicative inverse|reciprocal]] of the mathematical constant [[pi|{{pi}}]]:
<math display=block>\frac2\pi = \frac{\sqrt 2}2 \cdot \frac{\sqrt{2+\sqrt 2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt 2}}}2 \cdots</math>
It can also be represented as
<math display=block>\frac2\pi = \prod_{n=1}^{\infty} \cos \frac{\pi}{2^{n+1}}.</math>

The formula is named after [[François Viète]], who published it in 1593.{{r|beckmann}} As the first formula of European mathematics to represent an infinite process,{{r|maor}} it can be given a rigorous meaning as a [[Limit (mathematics)|limit]] expression{{r|eymlaf}} and marks the beginning of [[mathematical analysis]]. It has [[linear convergence]] and can be used for calculations of {{pi}},{{r|kreminski}} 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{{r|cullerne}} 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 [[perimeter]]s of nested [[polygon]]s 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==
François Viète (1540–1603) was a French lawyer, [[privy councillor]] 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 [[Approximations of π|approximating {{pi}}]] 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,{{r|beckmann}} used by Archimedes to find the approximation{{sfn|Beckmann|1971|p=67}}
<math display=block>\frac{223}{71} < \pi < \frac{22}{7}.</math>

By publishing his method as a mathematical formula, Viète formulated the first instance of an [[infinite product]] known in mathematics,{{r|desmith|mgc}} and the first example of an explicit formula for the exact value of {{pi}}.{{r|morrison|oms}} As the first representation in European mathematics of a number as the result of an infinite process rather than of a finite calculation,{{r|plofker}} [[Eli Maor]] highlights Viète's formula as marking the beginning of [[mathematical analysis]]{{r|maor}} and [[Jonathan Borwein]] calls its appearance "the dawn of modern mathematics".{{r|borwein}}

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

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

==Interpretation and convergence==
Viète's formula may be rewritten and understood as a limit expression{{r|eymlaf}}
<math display=block>\lim_{n \rightarrow \infty} \prod_{i=1}^n \frac{a_i}{2} = \frac2\pi,</math>
where
<math display=block>\begin{align} a_1 &= \sqrt{2} \\ a_n &= \sqrt{2+a_{n-1}}. \end{align}</math>

For each choice of <math>n</math>, the expression in the limit is a finite product, and as <math>n</math> 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 limits and rigorous proofs of [[Convergence (math)|convergence]] were developed in mathematics; the first proof that this limit exists was not given until the work of [[Ferdinand Rudio]] in 1891.{{r|beckmann|rudio}}

[[File:comparison_pi_infinite_series.svg|thumb|upright=1.5|Comparison of the convergence of Viète's formula ({{red|'''×'''}}) and several historical infinite series for {{pi}}. {{math|''S<sub>n</sub>''}} is the approximation after taking {{mvar|n}} terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times.]]

The [[rate of convergence]] of a limit governs the number of terms of the expression needed to achieve a given number of digits of accuracy. In Viète's formula, the numbers of terms and digits are proportional to each other: the product of the first {{mvar|n}} terms in the limit gives an expression for {{pi}} that is accurate to approximately {{math|0.6''n''}} digits.{{r|kreminski|osler}} This convergence rate compares very favorably with the [[Wallis product]], a later infinite product formula for {{pi}}. Although Viète himself used his formula to calculate {{pi}} only with nine-digit accuracy, an [[Series acceleration|accelerated]] version of his formula has been used to calculate {{pi}} to hundreds of thousands of digits.{{r|kreminski}}

==Related formulas==
Viète's formula may be obtained as a special case of a formula for the [[sinc function]] that has often been attributed to [[Leonhard Euler]]{{r|euler}}, more than a century later:{{r|beckmann}} 
<math display=block>\frac{\sin x}{x} = \cos\frac{x}{2} \cdot \cos\frac{x}{4} \cdot \cos\frac{x}{8} \cdots</math>

Substituting {{math|''x'' {{=}} {{pi}}/2}} in this formula yields<ref>{{cite book |last1=Wilson |first1=Robin J. |title=Euler's pioneering equation: the most beautiful theorem in mathematics |date=2018 |publisher=Oxford University Press |location=Oxford, United Kingdom |isbn=9780198794929 |pages=57–58 |edition=1st }}</ref>
<math display=block>\frac{2}{\pi} = \cos\frac{\pi}{4} \cdot \cos\frac{\pi}{8} \cdot \cos\frac{\pi}{16} \cdots</math>

Then, expressing each term of the product on the right as a function of earlier terms using the [[half-angle formula]]:
<math display=block>\cos\frac{x}{2} = \sqrt\frac{1+\cos x}{2}</math>
gives Viète's formula.{{r|morrison}}

It is also possible to derive from Viète's formula a related formula for {{pi}} that still involves nested square roots of two, but uses only one multiplication:{{r|servi}}
<math display=block>\pi = \lim_{k\to\infty} 2^{k} \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}}}}_{k\text{ square roots}},</math>
which can be rewritten compactly as
<math display=block>\begin{align}
 \pi &= \lim_{k\to\infty} 2^k\sqrt{2-a_{k}}, \\
 a_1 &= 0, \\
 a_k &= \sqrt{2+a_{k-1}}.
\end{align}</math>

Many formulae for {{pi}} and other constants such as the [[golden ratio]] are now known, similar to Viète's in their use of either nested radicals or infinite products of trigonometric functions.{{r|mgc|servi|nyblom|levin|levin2|osler2|stolarsky|allen}}

==Derivation==
[[File:Viète nested polygons.svg|thumb|A sequence of [[regular polygon]]s with numbers of sides equal to [[power of two|powers of two]], inscribed in a circle. The ratios between areas or perimeters of consecutive polygons in the sequence give the terms of Viète's formula.]]
Viète obtained his formula by comparing the [[area]]s of [[regular polygon]]s with {{math|2<sup>''n''</sup>}} and {{math|2<sup>''n'' + 1</sup>}} sides inscribed in a [[circle]].{{r|beckmann|maor}} The first term in the product, <math>\sqrt{2}/2</math>, 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. Thus, the product [[Telescoping series|telescopes]] to give the ratio of areas of a square (the initial polygon in the sequence) to a circle (the limiting case of a {{math|2<sup>''n''</sup>}}-gon). Alternatively, the terms in the product may be instead interpreted as ratios of perimeters of the same sequence of polygons, starting with the ratio of perimeters of a [[digon]] (the diameter of the circle, counted twice) and a square, the ratio of perimeters of a square and an octagon, etc.{{r|rummler}}

Another derivation is possible based on [[trigonometric identity|trigonometric identities]] and Euler's formula.
Repeatedly applying the [[List of trigonometric identities#Multiple-angle and half-angle formulae|double-angle formula]]
<math display=block>\sin x = 2\sin\frac{x}{2}\cos\frac{x}{2},</math>
leads to a proof by [[mathematical induction]] that, for all positive integers {{mvar|n}},
<math display=block>\sin x = 2^n \sin\frac{x}{2^n}\left(\prod_{i=1}^n \cos\frac{x}{2^i}\right).</math>

The term {{math|2<sup>''n''</sup> sin(''x''/2<sup>''n''</sup>)}} goes to {{mvar|x}} in the limit as {{mvar|n}} goes to infinity, from which Euler's formula follows. Viète's formula may be obtained from this formula by the substitution {{math|''x'' {{=}} {{pi}}/2}}.{{r|morrison|kac}}
{{clear}}

== See also ==

* [[Morrie's law]], same identity taking  <math>x=2^{n}\alpha</math> on Viète's formula

* [[List of trigonometric identities]]

==References==
{{reflist|refs=

<ref name=allen>{{cite journal
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 | doi = 10.2307/3617569
 | issue = 450
 | journal = [[The Mathematical Gazette]]
 | jstor = 3617569
 | pages = 261–263
 | title = Continued radicals
 | volume = 69
 | year = 1985| s2cid = 250441699
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<ref name=beckmann>{{cite book
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 | edition = 2nd
 | isbn = 978-0-88029-418-8
 | mr = 0449960
 | pages = [https://books.google.com/books?id=XqqUUSyz138C&pg=PA94 94–95]
 | publisher = The Golem Press | location = Boulder, Colorado
 | title = A History of {{pi}}
 | title-link = A History of Pi
 | year = 1971}}</ref>

<ref name=euler>{{cite journal
 | last = Euler | first = Leonhard | author-link = Leonhard Euler
 | journal = Commentarii Academiae Scientiarum Petropolitanae
 | language = la
 | pages = 222–236
 | title = De variis modis circuli quadraturam numeris proxime exprimendi
 | trans-title = On various methods for expressing the quadrature of a circle with verging numbers
 | url = https://scholarlycommons.pacific.edu/euler-works/74/
 | volume = 9
 | year = 1738}} Translated into English by Thomas W. Polaski. See final formula. The same formula is also in {{cite journal
 | last = Euler | first = Leonhard | author-link = Leonhard Euler
 | journal = Opuscula Analytica
 | language = la
 | pages = 345–352
 | title = Variae observationes circa angulos in progressione geometrica progredientes
 | trans-title = Various observations about angles proceeding in geometric progression
 | url = https://scholarlycommons.pacific.edu/euler-works/561/
 | volume = 1
 | year = 1783}} Translated into English by Jordan Bell, {{arxiv|1009.1439}}. See the formula in numbered paragraph 3.</ref>

<ref name=borwein>{{cite book
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 | title = From Alexandria, Through Baghdad
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 | contribution = The life of Pi: From Archimedes to ENIAC and beyond
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<ref name=cullerne>{{cite journal
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 | title = Teaching wave propagation and the emergence of Viète's formula
 | volume = 47| s2cid = 122368450
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<ref name=desmith>{{cite book
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<ref name=eymlaf>{{cite book
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 | isbn = 978-0-8218-3246-2
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 | publisher = American Mathematical Society | location = Providence, Rhode Island
 | title = The Number pi
 | year = 2004}}</ref>

<ref name=kac>{{cite book
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 | contribution = Chapter 1: From Vieta to the notion of statistical independence
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 | series = [[Carus Mathematical Monographs]]
 | title = Statistical Independence in Probability, Analysis and Number Theory
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<ref name=kreminski>{{cite journal
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 | title = {{pi}} to thousands of digits from Vieta's formula
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<ref name=levin>{{cite journal
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 | title = A geometric interpretation of an infinite product for the lemniscate constant
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<ref name=levin2>{{cite journal
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<ref name=maor>{{cite book
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<ref name=mgc>{{cite journal
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<ref name=morrison>{{cite journal
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 | title = Cosine products, Fourier transforms, and random sums
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<ref name=nyblom>{{cite journal
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 | title = Some closed-form evaluations of infinite products involving nested radicals
 | volume = 42
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<ref name=oms>{{cite book
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<ref name=osler>{{cite journal
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 | pages = 136–142
 | title = A simple geometric method of estimating the error in using Vieta's product for {{pi}}
 | volume = 38
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<ref name=osler2>{{cite journal
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 | mr = 2437033
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 | title = Vieta-like products of nested radicals with Fibonacci and Lucas numbers
 | volume = 45
 | year = 2007}}</ref>

<ref name=plofker>Very similar infinite trigonometric series for <math>\pi</math> appeared earlier in [[Indian mathematics]], in the work of [[Madhava of Sangamagrama]] (c. 1340 – 1425), but were not known in Europe until much later. See: {{cite book
 | last = Plofker | first = Kim | author-link = Kim Plofker
 | contribution = 7.3.1 Mādhava on the circumference and arcs of the circle
 | contribution-url = https://books.google.com/books?id=DHvThPNp9yMC&pg=PA221
 | isbn = 978-0-691-12067-6
 | pages = 221–234
 | publisher = Princeton University Press | location = Princeton, New Jersey
 | title = Mathematics in India
 | title-link = Mathematics in India (book)
 | year = 2009}}</ref>

<ref name=rudio>{{cite journal
 | last = Rudio | first = F. | author-link = Ferdinand Rudio
 | journal = Historisch-litterarische Abteilung der Zeitschrift für Mathematik und Physik
 | jfm = 23.0263.02
 | language = de
 | pages = 139–140
 | title = {{not a typo|Ueber}} die {{not a typo|Convergenz}} einer von Vieta herrührenden eigentümlichen Produktentwicklung
 | trans-title = On the convergence of a special product expansion due to Vieta
 | url = https://gdz.sub.uni-goettingen.de/id/PPN599415665_0036?tify=%7B%22pages%22%3A%5B557%5D%7D
 | volume = 36
 | year = 1891}}</ref>

<ref name=rummler>{{cite journal
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 | doi = 10.2307/2324662
 | issue = 9
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2324662
 | mr = 1247533
 | pages = 858–860
 | title = Squaring the circle with holes
 | volume = 100
 | year = 1993}}</ref>

<ref name=servi>{{cite journal
 | last = Servi | first = L. D.
 | doi = 10.2307/3647881
 | issue = 4
 | journal = [[The American Mathematical Monthly]]
 | jstor = 3647881
 | mr = 1984573
 | pages = 326–330
 | title = Nested square roots of 2
 | volume = 110
 | year = 2003}}</ref>

<ref name=stolarsky>{{cite journal
 | last = Stolarsky | first = Kenneth B.
 | doi = 10.2140/pjm.1980.89.209
 | issue = 1
 | journal = [[Pacific Journal of Mathematics]]
 | mr = 596932
 | pages = 209–227
 | title = Mapping properties, growth, and uniqueness of Vieta (infinite cosine) products
 | url = https://projecteuclid.org/euclid.pjm/1102779384
 | volume = 89
 | year = 1980| doi-access = free
 }}</ref>

}}

==External links==
{{Portal|Mathematics}}
* Viète's [https://books.google.com/books?id=7_BCAAAAcAAJ ''Variorum de rebus mathematicis responsorum, liber VIII''] (1593) on [[Google Books]]. The formula is on the second half of p.&nbsp;30.

{{DEFAULTSORT:Viete's Formula}}
[[Category:Articles containing proofs]]
[[Category:Infinite products]]
[[Category:Pi algorithms]]