Viète's formula

In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant $π$ : ${\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots$ It can also be represented as ${\frac {2}{\pi }}=\prod _{n=1}^{\infty }\cos {\frac {\pi }{2^{n+1}}}.$

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

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 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] ${\frac {223}{71}}<\pi <{\frac {22}{7}}.$

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

Viète's formula may be rewritten and understood as a limit expression^[3] $\lim _{n\rightarrow \infty }\prod _{i=1}^{n}{\frac {a_{i}}{2}}={\frac {2}{\pi }},$ where ${\begin{aligned}a_{1}&={\sqrt {2}}\\a_{n}&={\sqrt {2+a_{n-1}}}.\end{aligned}}$

For each choice of $n$ , the expression in the limit is a finite product, and as $n$ 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 limits and rigorous proofs of convergence were developed in mathematics; the first proof that this limit exists was not given until the work of Ferdinand Rudio in 1891.^[1]^[14]

Comparison of the convergence of Viète's formula (×) and several historical infinite series for

π

.

S n

is the approximation after taking

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 $n$ terms in the limit gives an expression for $π$ that is accurate to approximately $0.6 n$ digits.^[4]^[15] This convergence rate compares very favorably with the Wallis product, a later infinite product formula for $π$ . Although Viète himself used his formula to calculate $π$ only with nine-digit accuracy, an accelerated version of his formula has been used to calculate $π$ to hundreds of thousands of digits.^[4]

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^[16], more than a century later:^[1] ${\frac {\sin x}{x}}=\cos {\frac {x}{2}}\cdot \cos {\frac {x}{4}}\cdot \cos {\frac {x}{8}}\cdots$

Substituting $x = π /2$ in this formula yields^[17] ${\frac {2}{\pi }}=\cos {\frac {\pi }{4}}\cdot \cos {\frac {\pi }{8}}\cdot \cos {\frac {\pi }{16}}\cdots$

Then, expressing each term of the product on the right as a function of earlier terms using the half-angle formula: $\cos {\frac {x}{2}}={\sqrt {\frac {1+\cos x}{2}}}$ gives Viète's formula.^[9]

It is also possible to derive from Viète's formula a related formula for $π$ that still involves nested square roots of two, but uses only one multiplication:^[18] $\pi =\lim _{k\to \infty }2^{k}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2}}}}}}}}}}}} _{k{\text{ square roots}}},$ which can be rewritten compactly as ${\begin{aligned}\pi &=\lim _{k\to \infty }2^{k}{\sqrt {2-a_{k}}},\\a_{1}&=0,\\a_{k}&={\sqrt {2+a_{k-1}}}.\end{aligned}}$

Many formulae for $π$ 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.^[8]^[18]^[19]^[20]^[21]^[22]^[23]^[24]

Derivation

A sequence of regular polygons with numbers of sides equal to 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 areas of regular polygons with $2 n$ and $2 n + 1$ sides inscribed in a circle.^[1]^[2] The first term in the product, ${\sqrt {2}}/2$ , 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. Thus, the product telescopes to give the ratio of areas of a square (the initial polygon in the sequence) to a circle (the limiting case of a $2 n$ -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.^[25]

Another derivation is possible based on trigonometric identities and Euler's formula. Repeatedly applying the double-angle formula $\sin x=2\sin {\frac {x}{2}}\cos {\frac {x}{2}},$ leads to a proof by mathematical induction that, for all positive integers $n$ , $\sin x=2^{n}\sin {\frac {x}{2^{n}}}\left(\prod _{i=1}^{n}\cos {\frac {x}{2^{i}}}\right).$

The term $2 n sin(x /2 n)$ goes to $x$ in the limit as $n$ goes to infinity, from which Euler's formula follows. Viète's formula may be obtained from this formula by the substitution $x = π /2$ .^[9]^[13]

References

^ ^a ^b ^c ^d ^e Beckmann, Petr (1971). A History of $π$ (2nd ed.). Boulder, Colorado: The Golem Press. pp. 94–95. ISBN 978-0-88029-418-8. MR 0449960.
^ ^a ^b ^c Maor, Eli (2011). Trigonometric Delights. Princeton, New Jersey: Princeton University Press. pp. 50, 140. ISBN 978-1-4008-4282-7.
^ ^a ^b Eymard, Pierre; Lafon, Jean Pierre (2004). "2.1 Viète's infinite product". The Number pi. Translated by Wilson, Stephen S. Providence, Rhode Island: American Mathematical Society. pp. 44–46. ISBN 978-0-8218-3246-2. MR 2036595.
^ ^a ^b ^c ^d Kreminski, Rick (2008). " $π$ to thousands of digits from Vieta's formula". Mathematics Magazine. 81 (3): 201–207. doi:10.1080/0025570X.2008.11953549. JSTOR 27643107. S2CID 125362227.
^ ^a ^b Cullerne, J. P.; Goekjian, M. C. Dunn (December 2011). "Teaching wave propagation and the emergence of Viète's formula". Physics Education. 47 (1): 87–91. doi:10.1088/0031-9120/47/1/87. S2CID 122368450.
^ Beckmann 1971, p. 67.
^ De Smith, Michael J. (2006). Maths for the Mystified: An Exploration of the History of Mathematics and Its Relationship to Modern-day Science and Computing. Leicester: Matador. p. 165. ISBN 978-1905237-81-4.
^ ^a ^b Moreno, Samuel G.; García-Caballero, Esther M. (2013). "On Viète-like formulas". Journal of Approximation Theory. 174: 90–112. doi:10.1016/j.jat.2013.06.006. MR 3090772.
^ ^a ^b ^c Morrison, Kent E. (1995). "Cosine products, Fourier transforms, and random sums". The American Mathematical Monthly. 102 (8): 716–724. arXiv:math/0411380. doi:10.2307/2974641. JSTOR 2974641. MR 1357488.
^ Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2010). An Atlas of Functions: with Equator, the Atlas Function Calculator. New York: Springer. p. 15. doi:10.1007/978-0-387-48807-3. ISBN 978-0-387-48807-3.
^ Very similar infinite trigonometric series for $\pi$ 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: Plofker, Kim (2009). "7.3.1 Mādhava on the circumference and arcs of the circle". Mathematics in India. Princeton, New Jersey: Princeton University Press. pp. 221–234. ISBN 978-0-691-12067-6.
^ ^a ^b ^c Borwein, Jonathan M. (2014). "The life of Pi: From Archimedes to ENIAC and beyond" (PDF). In Sidoli, Nathan; Van Brummelen, Glen (eds.). From Alexandria, Through Baghdad. Berlin & Heidelberg: Springer. pp. 531–561. doi:10.1007/978-3-642-36736-6_24. ISBN 978-3-642-36735-9. Retrieved 2024-08-20.
^ ^a ^b Kac, Mark (1959). "Chapter 1: From Vieta to the notion of statistical independence". Statistical Independence in Probability, Analysis and Number Theory. Carus Mathematical Monographs. Vol. 12. New York: John Wiley & Sons for the Mathematical Association of America. pp. 1–12. MR 0110114.
^ Rudio, F. (1891). "Ueber die Convergenz einer von Vieta herrührenden eigentümlichen Produktentwicklung" [On the convergence of a special product expansion due to Vieta]. Historisch-litterarische Abteilung der Zeitschrift für Mathematik und Physik (in German). 36: 139–140. JFM 23.0263.02.
^ Osler, Thomas J. (2007). "A simple geometric method of estimating the error in using Vieta's product for $π$ ". International Journal of Mathematical Education in Science and Technology. 38 (1): 136–142. doi:10.1080/00207390601002799. S2CID 120145020.
^ Euler, Leonhard (1738). "De variis modis circuli quadraturam numeris proxime exprimendi" [On various methods for expressing the quadrature of a circle with verging numbers]. Commentarii Academiae Scientiarum Petropolitanae (in Latin). 9: 222–236. Translated into English by Thomas W. Polaski. See final formula. The same formula is also in Euler, Leonhard (1783). "Variae observationes circa angulos in progressione geometrica progredientes" [Various observations about angles proceeding in geometric progression]. Opuscula Analytica (in Latin). 1: 345–352. Translated into English by Jordan Bell, arXiv:1009.1439. See the formula in numbered paragraph 3.
^ Wilson, Robin J. (2018). Euler's pioneering equation: the most beautiful theorem in mathematics (1st ed.). Oxford, United Kingdom: Oxford University Press. pp. 57–58. ISBN 9780198794929.
^ ^a ^b Servi, L. D. (2003). "Nested square roots of 2". The American Mathematical Monthly. 110 (4): 326–330. doi:10.2307/3647881. JSTOR 3647881. MR 1984573.
^ Nyblom, M. A. (2012). "Some closed-form evaluations of infinite products involving nested radicals". Rocky Mountain Journal of Mathematics. 42 (2): 751–758. doi:10.1216/RMJ-2012-42-2-751. MR 2915517.
^ Levin, Aaron (2006). "A geometric interpretation of an infinite product for the lemniscate constant". The American Mathematical Monthly. 113 (6): 510–520. doi:10.2307/27641976. JSTOR 27641976. MR 2231136.
^ Levin, Aaron (2005). "A new class of infinite products generalizing Viète's product formula for $π$ ". The Ramanujan Journal. 10 (3): 305–324. doi:10.1007/s11139-005-4852-z. MR 2193382. S2CID 123023282.
^ Osler, Thomas J. (2007). "Vieta-like products of nested radicals with Fibonacci and Lucas numbers". Fibonacci Quarterly. 45 (3): 202–204. MR 2437033.
^ Stolarsky, Kenneth B. (1980). "Mapping properties, growth, and uniqueness of Vieta (infinite cosine) products". Pacific Journal of Mathematics. 89 (1): 209–227. doi:10.2140/pjm.1980.89.209. MR 0596932.
^ Allen, Edward J. (1985). "Continued radicals". The Mathematical Gazette. 69 (450): 261–263. doi:10.2307/3617569. JSTOR 3617569. S2CID 250441699.
^ Rummler, Hansklaus (1993). "Squaring the circle with holes". The American Mathematical Monthly. 100 (9): 858–860. doi:10.2307/2324662. JSTOR 2324662. MR 1247533.

External links

Viète's Variorum de rebus mathematicis responsorum, liber VIII (1593) on Google Books. The formula is on the second half of p. 30.

[beckmann-1] Beckmann, Petr (1971). A History of $π$ (2nd ed.). Boulder, Colorado: The Golem Press. pp. 94–95. ISBN 978-0-88029-418-8. MR 0449960.

[maor-2] Maor, Eli (2011). Trigonometric Delights. Princeton, New Jersey: Princeton University Press. pp. 50, 140. ISBN 978-1-4008-4282-7.

[eymlaf-3] Eymard, Pierre; Lafon, Jean Pierre (2004). "2.1 Viète's infinite product". The Number pi. Translated by Wilson, Stephen S. Providence, Rhode Island: American Mathematical Society. pp. 44–46. ISBN 978-0-8218-3246-2. MR 2036595.

[kreminski-4] Kreminski, Rick (2008). " $π$ to thousands of digits from Vieta's formula". Mathematics Magazine. 81 (3): 201–207. doi:10.1080/0025570X.2008.11953549. JSTOR 27643107. S2CID 125362227.

[cullerne-5] Cullerne, J. P.; Goekjian, M. C. Dunn (December 2011). "Teaching wave propagation and the emergence of Viète's formula". Physics Education. 47 (1): 87–91. doi:10.1088/0031-9120/47/1/87. S2CID 122368450.

[FOOTNOTEBeckmann197167-6] Beckmann 1971, p. 67.

[desmith-7] De Smith, Michael J. (2006). Maths for the Mystified: An Exploration of the History of Mathematics and Its Relationship to Modern-day Science and Computing. Leicester: Matador. p. 165. ISBN 978-1905237-81-4.

[mgc-8] Moreno, Samuel G.; García-Caballero, Esther M. (2013). "On Viète-like formulas". Journal of Approximation Theory. 174: 90–112. doi:10.1016/j.jat.2013.06.006. MR 3090772.

[morrison-9] Morrison, Kent E. (1995). "Cosine products, Fourier transforms, and random sums". The American Mathematical Monthly. 102 (8): 716–724. arXiv:math/0411380. doi:10.2307/2974641. JSTOR 2974641. MR 1357488.

[oms-10] Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2010). An Atlas of Functions: with Equator, the Atlas Function Calculator. New York: Springer. p. 15. doi:10.1007/978-0-387-48807-3. ISBN 978-0-387-48807-3.

[plofker-11] Very similar infinite trigonometric series for $\pi$ 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: Plofker, Kim (2009). "7.3.1 Mādhava on the circumference and arcs of the circle". Mathematics in India. Princeton, New Jersey: Princeton University Press. pp. 221–234. ISBN 978-0-691-12067-6.

[borwein-12] Borwein, Jonathan M. (2014). "The life of Pi: From Archimedes to ENIAC and beyond" (PDF). In Sidoli, Nathan; Van Brummelen, Glen (eds.). From Alexandria, Through Baghdad. Berlin & Heidelberg: Springer. pp. 531–561. doi:10.1007/978-3-642-36736-6_24. ISBN 978-3-642-36735-9. Retrieved 2024-08-20.

[kac-13] Kac, Mark (1959). "Chapter 1: From Vieta to the notion of statistical independence". Statistical Independence in Probability, Analysis and Number Theory. Carus Mathematical Monographs. Vol. 12. New York: John Wiley & Sons for the Mathematical Association of America. pp. 1–12. MR 0110114.

[rudio-14] Rudio, F. (1891). "Ueber die Convergenz einer von Vieta herrührenden eigentümlichen Produktentwicklung" [On the convergence of a special product expansion due to Vieta]. Historisch-litterarische Abteilung der Zeitschrift für Mathematik und Physik (in German). 36: 139–140. JFM 23.0263.02.

[osler-15] Osler, Thomas J. (2007). "A simple geometric method of estimating the error in using Vieta's product for $π$ ". International Journal of Mathematical Education in Science and Technology. 38 (1): 136–142. doi:10.1080/00207390601002799. S2CID 120145020.

[euler-16] Euler, Leonhard (1738). "De variis modis circuli quadraturam numeris proxime exprimendi" [On various methods for expressing the quadrature of a circle with verging numbers]. Commentarii Academiae Scientiarum Petropolitanae (in Latin). 9: 222–236. Translated into English by Thomas W. Polaski. See final formula. The same formula is also in Euler, Leonhard (1783). "Variae observationes circa angulos in progressione geometrica progredientes" [Various observations about angles proceeding in geometric progression]. Opuscula Analytica (in Latin). 1: 345–352. Translated into English by Jordan Bell, arXiv:1009.1439. See the formula in numbered paragraph 3.

[17] Wilson, Robin J. (2018). Euler's pioneering equation: the most beautiful theorem in mathematics (1st ed.). Oxford, United Kingdom: Oxford University Press. pp. 57–58. ISBN 9780198794929.

[servi-18] Servi, L. D. (2003). "Nested square roots of 2". The American Mathematical Monthly. 110 (4): 326–330. doi:10.2307/3647881. JSTOR 3647881. MR 1984573.

[nyblom-19] Nyblom, M. A. (2012). "Some closed-form evaluations of infinite products involving nested radicals". Rocky Mountain Journal of Mathematics. 42 (2): 751–758. doi:10.1216/RMJ-2012-42-2-751. MR 2915517.

[levin-20] Levin, Aaron (2006). "A geometric interpretation of an infinite product for the lemniscate constant". The American Mathematical Monthly. 113 (6): 510–520. doi:10.2307/27641976. JSTOR 27641976. MR 2231136.

[levin2-21] Levin, Aaron (2005). "A new class of infinite products generalizing Viète's product formula for $π$ ". The Ramanujan Journal. 10 (3): 305–324. doi:10.1007/s11139-005-4852-z. MR 2193382. S2CID 123023282.

[osler2-22] Osler, Thomas J. (2007). "Vieta-like products of nested radicals with Fibonacci and Lucas numbers". Fibonacci Quarterly. 45 (3): 202–204. MR 2437033.

[stolarsky-23] Stolarsky, Kenneth B. (1980). "Mapping properties, growth, and uniqueness of Vieta (infinite cosine) products". Pacific Journal of Mathematics. 89 (1): 209–227. doi:10.2140/pjm.1980.89.209. MR 0596932.

[allen-24] Allen, Edward J. (1985). "Continued radicals". The Mathematical Gazette. 69 (450): 261–263. doi:10.2307/3617569. JSTOR 3617569. S2CID 250441699.

[rummler-25] Rummler, Hansklaus (1993). "Squaring the circle with holes". The American Mathematical Monthly. 100 (9): 858–860. doi:10.2307/2324662. JSTOR 2324662. MR 1247533.

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