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:

2

π

=

2

2

⋅

2

+

2

2

⋅

2

+

2

+

2

2

⋯

=

1

2

⋅

1

2

+

1

2

1

2

⋅

1

2

+

1

2

1

2

+

1

2

1

2

⋯

{\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

2

π

=

∏

n

=

1

∞

cos

⁡

π

2

n

+

1

.

{\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. As the first formula of European mathematics to represent an infinite process, it can be given a rigorous meaning as a limit expression and marks the beginning of mathematical analysis. It has linear convergence and can be used for calculations of π, 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 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 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.