How Do We Know the Square Root of 2 Is Irrational?

There are all kinds of numbers. Natural numbers, which are also called counting numbers, are familiar to young children. Whole numbers adds the number zero to the list of natural numbers. Integers go a step further, adding negative numbers to the list of whole numbers.

Then we get to rational numbers. Calling a number rational is a fancy way of saying that its value can be precisely defined with a fraction made up of integers. As such, rational numbers fill in the spaces between the integers.

But not all fractions whose values fall in between the integers are rational. Some of these fractions cannot be written using integers at all. Long ago, mathematicians decided to call these numbers irrational because there is no way to use fractions using integers to express their exact value.

One of the earliest of the list of irrational numbers to be discovered is the square root of 2. While you might be able to approximate the square root of two using fractions with integers, you’ll always come up short of expressing its exact value because it just doesn’t work, no matter what integers you might use.

That scenario leads to an interesting question. How do we know that the square root of 2 is an irrational number? The answer lies is some clever thinking, as illustrated in the following two-minute video that features Taylor Swift proving the square root of 2 is irrational!

And that’s how we know the square root of 2 cannot be exactly expressed using fractions made up of integer values.

On a closing note, although we recognize Taylor Swift as incredibly talented, credit for the discovery probably lies with Hippasus of Metapontum, who had to deal with his day’s own version of irrational fandom.