Inquiry about the properties of square roots

[spherenine]

What is the proof that states that if the square root of a natural number is not another natural number, it must be irrational? In other words, the square root of a natural number must be either natural or irrational.

 

[g_edgar]

What is the proof that states that if the square root of a natural number is not another natural number, it must be irrational? In other words, the square root of a natural number must be either natural or irrational.

Use unique factorization into primes.

 

[Mark44]

And continuing with g_edgar's advice, a perfect square will always have pairs of equal factors. E.g., 81 = 9*9, 100 = 10*10 and so on.

 

[Elucidus]

The critical component is that the square root of a prime number is irrational and by extension so is the square root of any square-free number. The proofs of these facts are not necessarily straight forward.

--Elucidus