r/learnmath

11h ago

u/Some_Brazilian_Guy

Are irrational numbers closed under the operation of taking roots (radication)?

I tried posting this on r/math, but I don't have enough karma. I hope someone here can help clarify this for me.

I was reviewing the basic number sets when I came across the concept of closure. The natural numbers are closed under addition and multiplication, the integers are also closed under subtraction, and so on. It's easy to show that the irrational numbers aren't closed under the four basic arithmetic operations, but what about taking roots? I couldn't find a counterexample.

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shadowyams

10h ago

It's not closed under exponentiation, at least. Let x=sqrt(2)^sqrt(2). Either it is rational or irrational. If it is rational, then the proof is done.…