Viète's formula

In mathematics, Viète's formula is the following infinite product of nested radicals representing 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 is named after François Viète (1540–1603), who published it in 1593 in his work Variorum de rebus mathematicis responsorum, liber VIII.^[1]

Significance

At the time Viète published his formula, 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 area of a circle by that of a many-sided polygon,^[1] used by Archimedes to find the approximation

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

However, by publishing his method as a mathematical formula, Viète formulated the first instance of an infinite product known in mathematics,^[2]^[3] and the first example of an explicit formula for the exact value of $π$ .^[4]^[5] As the first formula representing a number as the result of an infinite process rather than of a finite calculation, Viète's formula has been noted as the beginning of mathematical analysis^[6] and even more broadly as "the dawn of modern mathematics".^[7]

Using his formula, Viète calculated $π$ to an accuracy of nine decimal digits.^[8] 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.^[7] Not long after Viète published his formula, Ludolph van Ceulen used a closely related method to calculate 35 digits of $π$ , which were published only after van Ceulen's death in 1610.^[7]

Interpretation and convergence

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

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

where $a n = \sqrt 2 + a n - 1$ , with initial condition $a 1 = \sqrt 2$ .^[9] 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]^[10]

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 the case of Viète's formula, there is a linear relation between the number of terms and the number of digits: the product of the first $n$ terms in the limit gives an expression for $π$ that is accurate to approximately $0.6 n$ digits.^[8]^[11] This convergence rate compares very favorably with the Wallis product, a later infinite product formula for $π$ . Although Viète himself only used his formula to calculate $π$ with nine-digit accuracy, an accelerated version of his formula has been used to calculate $π$ to hundreds of thousands of digits.^[8]

Related formulas

Viète's formula may be obtained as a special case of a formula given more than a century later by Leonhard Euler. Euler discovered that:

{\frac {\sin x}{x}}=\cos {\frac {x}{2}}\cdot \cos {\frac {x}{4}}\cdot \cos {\frac {x}{8}}\cdots

Substituting $x = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠π/2⁠$ , and expressing each term of the product 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.^[1]

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:^[12]

\pi =\lim _{k\to \infty }2^{k}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2}}}}}}}}}}}} _{k\ \mathrm {square} \ \mathrm {roots} }

By now many formulas similar to Viète's involving either nested radicals or infinite products of trigonometric functions are known for $π$ , as well as for other constants such as the golden ratio.^[12]^[13]^[14]^[15]^[16]^[17]^[18]^[19]

Derivation

Viète obtained his formula by comparing the areas of regular polygons with $2 n$ and $2 n + 1$ sides inscribed in a circle.^[1]^[6] The first term in the product, ⁠√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.^[20]

Another derivation is possible based on trigonometric identities and Euler's formula. By repeatedly applying the double-angle formula

\sin x=2\sin {\frac {x}{2}}\cos {\frac {x}{2}},

one may prove 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 $x = ⁠ π / 2 ⁠$ .^[4]

References

^ ^a ^b ^c ^d ^e Beckmann, Petr (1971). A history of $π$ (2nd ed.). Boulder, CO: The Golem Press. pp. 94–95. ISBN 978-0-88029-418-8. MR 0449960.
^ 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. Troubador Publishing Ltd. p. 165. ISBN 9781905237814.
^ 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 Morrison, Kent E. (1995). "Cosine products, Fourier transforms, and random sums". The American Mathematical Monthly. 102 (8): 716–724. doi:10.2307/2974641. MR 1357488.
^ Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2010). An Atlas of Functions: with Equator, the Atlas Function Calculator. Springer. p. 15. ISBN 9780387488073.
^ ^a ^b Maor, Eli (2011). Trigonometric Delights. Princeton University Press. pp. 50, 140. ISBN 9781400842827.
^ ^a ^b ^c Borwein, Jonathan M. (2013). "The Life of Pi: From Archimedes to ENIAC and Beyond". From Alexandria, Through Baghdad: Surveys and Studies in the Ancient Greek and Medieval Islamic Mathematical Sciences in Honor of J.L. Berggren (PDF). Springer. ISBN 9783642367359.
^ ^a ^b ^c Kreminski, Rick (2008). " $π$ to Thousands of Digits from Vieta's Formula". Mathematics Magazine. 81 (3): 201–207. JSTOR 27643107.
^ Eymard, Pierre; Lafon, Jean Pierre (2004). "2.1 Viète's infinite product". The Number $π$ . American Mathematical Society. pp. 44–46. ISBN 9780821832462.
^ Rudio, F. (1891). "Über die Konvergenz einer von Vieta herrührenden eigentümlichen Produktentwicklung". Z. Math. Phys. 36: 139–140.
^ Osler, T. 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.
^ ^a ^b Servi, L. D. (2003). "Nested square roots of 2". The American Mathematical Monthly. 110 (4): 326–330. doi:10.2307/3647881. MR 1984573.
^ Nyblom, M. A. (2012). "Some closed-form evaluations of infinite products involving nested radicals". The Rocky Mountain Journal of Mathematics. 42 (2): 751–758. doi:10.1216/RMJ-2012-42-2-751. MR 2915517.
^ 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.
^ Levin, Aaron (2006). "A geometric interpretation of an infinite product for the lemniscate constant". American Mathematical Monthly. 113 (6): 510–520. doi:10.2307/27641976. MR 2231136.
^ Levin, Aaron (2005). "A new class of infinite products generalizing Viète's product formula for $π$ ". Ramanujan Journal. 10 (3): 305–324. doi:10.1007/s11139-005-4852-z. MR 2193382.
^ Osler, Thomas J. (2007). "Vieta-like products of nested radicals with Fibonacci and Lucas numbers". The 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". Mathematical Gazette. 69 (450): 261–263. JSTOR 3617569.
^ Rummler, Hansklaus (1993). "Squaring the circle with holes". The American Mathematical Monthly. 100 (9): 858–860. doi:10.2307/2324662. 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, CO: The Golem Press. pp. 94–95. ISBN 978-0-88029-418-8. MR 0449960.

[2] 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. Troubador Publishing Ltd. p. 165. ISBN 9781905237814.

[3] 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-4] Morrison, Kent E. (1995). "Cosine products, Fourier transforms, and random sums". The American Mathematical Monthly. 102 (8): 716–724. doi:10.2307/2974641. MR 1357488.

[5] Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2010). An Atlas of Functions: with Equator, the Atlas Function Calculator. Springer. p. 15. ISBN 9780387488073.

[maor-6] Maor, Eli (2011). Trigonometric Delights. Princeton University Press. pp. 50, 140. ISBN 9781400842827.

[borwein-7] Borwein, Jonathan M. (2013). "The Life of Pi: From Archimedes to ENIAC and Beyond". From Alexandria, Through Baghdad: Surveys and Studies in the Ancient Greek and Medieval Islamic Mathematical Sciences in Honor of J.L. Berggren (PDF). Springer. ISBN 9783642367359.

[kreminski-8] Kreminski, Rick (2008). " $π$ to Thousands of Digits from Vieta's Formula". Mathematics Magazine. 81 (3): 201–207. JSTOR 27643107.

[9] Eymard, Pierre; Lafon, Jean Pierre (2004). "2.1 Viète's infinite product". The Number $π$ . American Mathematical Society. pp. 44–46. ISBN 9780821832462.

[10] Rudio, F. (1891). "Über die Konvergenz einer von Vieta herrührenden eigentümlichen Produktentwicklung". Z. Math. Phys. 36: 139–140.

[11] Osler, T. 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.

[servi-12] Servi, L. D. (2003). "Nested square roots of 2". The American Mathematical Monthly. 110 (4): 326–330. doi:10.2307/3647881. MR 1984573.

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

[14] 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.

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

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

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

[18] 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.

[19] Allen, Edward J. (1985). "Continued radicals". Mathematical Gazette. 69 (450): 261–263. JSTOR 3617569.

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

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