In this article we give an overview of some of the most famous irrational numbers and their origin. Our curriculum does not talk about the origin and calculation of these numbers, and definitely doesn’t talk about how we figured the fact that they are never ending non repeating, or how so many numbers behind the decimal place are calculated.
The golden ratio
The incompleteness of contradiction method
The contradiction method does tell us about why √2 can’t be rational, but not how. If I try to measure the side of that triangle, I would get a rational number. Some will say that that is an approximation, but then how is anything not an approximation? How can you be sure about measuring anything at all?
Can we measure irrational numbers?
We can’t measure anything accurate enough till infinite decimal places. A number is said to be irrational if it can’t be depicted as p/q, where both p and q are integers and q is not 0. It also has to have a never ending non repeating pattern in its decimals.
While you can draw root 2 on the number line, you can’t exactly measure the length of that line segment, or any line segment for that matter. Our measurements are based on approximations to a certain decimal place, and can never reveal the exact value as theorized by mathematics. Therefore we use mathematical theorems and algorithms to provide an exact value.
There are a lot of algorithms to find the exact values of irrational numbers, for example, some of these for finding roots are explained here and here on wikipedia. The algorithm for finding pi is explained below.