Why has PI infinite decimal digits?

Welcome to my first English post! A friend did ask me why PI has infinite decimal digits. Behold, my answer.

π (pi, about 3.14159) is the ratio of a circle’s circumference C to its diameter d, or π = C / d.


A diagram of a circle, with the width labeled as diameter, and the perimeter labeled as circumference
By Kjoonlee, based on previous work by w:User:PapeschrOwn work, Public Domain, Link


As those two values are proportional (the circumference of a circle increases with its diameter and vice versa), the number is constant, i.e. the same for every circle [1] (at least in flat Euclidean geometry on which I will spare you the details).

We will explore the reason for the infinit number of π’s decimal digits by the means of an example. However, we will not use C/d, but the simpler ratio 1/3 (i.e. the number fitting into one exactly three times). This is equivalent to 0.3333…, a zero with an infinite number of threes after the decimal point.

3 * 0.333 only equals 0,999. Doubling the number of decimal digits, we arrive at 3 * 0.333333 = 0.999999. The more threes we append, the closer we get to one – reaching it not before doing this for an infinite number of steps, though. Therefore, there is no number with a finite amount of decimal digits fitting into one exactly three times. After every decimal digit we can still express the number even more precise by adding yet another. [2]

π is an irrational number and thus its decimal digits are chaotic, but the principle is the same: there is no number that can represent the ratio of a circle’s circumference to its diameter with a finite amount of decimal digits.


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Sources

[1] https://en.wikipedia.org/wiki/Pi, as of 2020-03-29
[2] https://www.gutefrage.net/frage/warum-liefern-manche-rechnungen-unendlich-nachkommastellen, as of 2020-03-29