The square root of 2 is another one of my favorite irrational numbers. It is equal to the hypotenuse of a right triangle whose sides are equal to 1. Unlike π, e and φ, the irrationality of this number is not very apparent, hence it’s not as popular as some of the other irrational numbers. But I feel that it deserves more popularity as there’s something that we use pretty much every day that involves this fascinating number: the standard size paper.
The ratio of the two sides of an A4 size paper – the standard letter format in many countries including US – equals the square root of 2 (~ 1.41421). Because of this aspect ratio, when you cut the paper in half along its longer side, the resulting halves retain the same ratio.
To understand why the square root of 2 is irrational, we can start by asking the following question: ‘Can it be expressed as a ratio of two whole numbers?’ If the answer to this question is ‘yes’ than it is a rational number which can be expressed as n/m, where both n and m are whole numbers. By squaring both sides and multiplying by m2, we get 2m2=n2. This implies that it is possible to have two identical squares with their sides equal to m, whose total area is identical to a bigger square whose sides are equal to n. A visual proof emerges based on this proposition – check out this link (PDF) if you’re interested in understanding how the irrationality of the square root of 2 can be proven based on this hypothesis.
Let me end this post with a nerdy pick-up line: “You must be the square root of two, because I feel irrational around you.”