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# What is mathematical Pi?

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Pi is a mathematical ratio symbolized by the Greek letter π.

Pi (symbol π ) is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. (Area of a circle = πr2)

This number is the same for every circle. This means that for any circle, the length of the circumference divided by the length of the diameter equals pi. It's expressed in the formula pi = C/d , usually written as C = pi x d , finding the circumference when the diameter is known. The concept has been handed down for thousands of years, and it developed alongside mathematics for much of that time.

The exact value of pi can never be expressed exactly in numerical form. This is because pi is a transcendental number; it is irrational. But here are its first few digits (for more digits, see the discussion page):

3.1415926535897932384626433832795028841971693993751

Because pi is a non-repeating, non-terminating number, it is often approximated as 22/7, or 3.14, or 3.1416 in order to make calculations a little more manageable. As a fraction, its closest approximations for basic mathematical purposes are 22/7, 333/106 or 355/113.

As a decimal figure, 3.14 or 3.14159 are most generally used.

Pi is the ratio of the circumference of a circle to its diameter.

π is an irrational number, which means that its value cannot be expressed exactly as a fraction, Consequently, its decimal representation never ends or repeats.

Other properties of pi

Pi isn't just about circles. It's not even just about geometric shapes. For instance:

* pi/4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - ...

* pi^2/6 = 1/1 + 1/4 + 1/9 + 1/16 + ... (the denominators are the square numbers)

* Pi appears in complex analysis, in a formula which expresses the nth derivative of a function as an integral.

* It also appears in Fourier analysis - the mathematical theory which allows us to, for instance, take a sound wave and calculate how much of it is at each frequency.

* In probability theory, the probability density function for the (normalized) normal distribution is exp(-x^2/2)/sqrt(2*pi).

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