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The mathematical constant π (Greek pi) is commonly used in mathematics. It is also known as Archimedes' constant.

Pi is an irrational number. Furthermore, it is a transcendental number.

Pi is defined as the ratio of the circumference of any circle to its diameter. As all circles are similar and therefore proportional in dimensions, pi is therefore always the same for all circles and is a constant.

Consequently, pi can also be viewed as the area of a circle whose radius is one.

Value

Its value can be approximated to fourteen significant digits as: 3.14159265358979…, often sufficient for most calculations.

The true value can never be exactly implemented (not with, but only approximated, and therefore π should often remain a factored constant.

Pi to one-hundred significant digits: $\pi =$ 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679$\cdots$.

The current known value has been computed using super-computers to in excess of 2.7 trillion digits.

Computing Pi

Pi can be computed in a variety of techniques and represented in a variety of ways. Even the value, itself, has been represented in base numbering systems other than decimal.

As an infinite summation:

$\pi = \sum_{n=0}^\infty \frac{4{(-1)}^n}{2n+1} = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} \cdots$

As an integral:

$\pi = \int^{\infty}_{-\infty} \frac{dx}{1 + x^2}$

The value can also be viewed in limit form, as the perimeter of a regular polygon, with an apothem of one-half, as the number of sides increases to infinity.

Applications

Pi is used to relate properties of spheres and circles to their radii; however, due to the unique properties and origins of the number, the value has uses throughout mathematics, including outside the realm of strict circular geometry. Most notably, pi is the basis of the angular measurement radians, and therefore has huge implications for trigonometry, complex analysis, and calculus. An especially important application is in Euler's formula.

Pi appears in many integrals; for example:

$\int^{\infty}_{-\infty} \frac{1}{1+x^2} \ dx = \pi$

,

$\int^{\infty}_{-\infty} e^{-x^2} dx = \sqrt{\pi}$

and

$\int\limits_{-1}^{1}\frac{dx}{\sqrt{1-x^2}} = \pi$

The square root of pi frequently appears in gamma functions and Gaussian integrals.