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 *m*^{2}, we get *2**m*^{2}=n^{2}. 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.”

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Good one.

Thanks abhijeet!

Hectic! One more from the Mathematician… The facts and content apart, I really love your love for irrational numbers. As I mentioned earlier as well, your articles keep taking me back to those solidly meaningful things I’m otherwise ignorant about. Thank you! 🙂

And a neat a finish – “You must be the square root of two, because I feel irrational around you.”

And I love that you love my love for irrational numbers! 🙂