The square root of two
In first year maths, they briefly study the five families of number: the natural numbers N, the integers Z, the rational numbers Q, the real numbers R, and the complex numbers C. In particular, they focus on the distinction between the rational numbers and the real numbers. A classic proof they are given at this time is one that the number √2 is irrational. This blog post is about some alternative proofs.
The following comments were left on the original blog post:
Jim Propp 25 May 2017:
I really like the second proof (though some purists might object that its use of facts about reduction of fractions amounts to a tacit appeal to the fundamental theorem of arithmetic that makes the proof less elementary than the other two). For one thing, the second proof readily generalizes to show that for all positive integers n, not just n=2, the square root of n cannot be rational unless it is an integer. Nice!
David Butler 25 May 2017:
Yes I rather liked that you could use it to show that square roots can only be rational if they’re integers.