We know pi has infinite digits because it is an irrational number.
Understanding Pi and Rational Numbers
Pi (π) is defined as the ratio of a circle's circumference to its diameter. While it's often approximated as 3.14 or 22/7, these are just simplified representations.
A rational number is any number that can be expressed as a fraction p/q, where p and q are integers, and q is not zero. Rational numbers have decimal representations that either terminate (e.g., 0.5 = 1/2) or repeat (e.g., 0.333... = 1/3).
Why Pi is Irrational
In 1761, Johann Heinrich Lambert proved that pi is irrational. This means that pi cannot be expressed as a fraction of two integers.
Because pi is irrational, its decimal representation neither terminates nor repeats. If the decimal representation of pi did terminate or repeat, it would be possible to express it as a fraction, contradicting the established proof of its irrationality.
The Implications of Irrationality
The irrationality of pi directly implies that its decimal representation continues infinitely without any repeating pattern. This is why we say pi has infinite digits. While we can calculate billions of digits of pi, we will never find a point where the digits stop or begin to repeat in a predictable cycle.
