Sorry if this has been asked before, but can't find what I'm looking for on here.
I'm currently preparing an early lesson for my new A-Level class about surds. I'd like to be quite formal and thorough with them and I'm struggling with a good way of communicating surds.
So for my own curiosity and for a good definition for my pupils I've got a couple of questions I'm hoping someone could answer.
So
Question 1: Are all square roots of prime numbers irrational, or square roots of non-perfect powers for that matter i.e. all $ n \in \mathbb{N} , \text{where} \ n \ne a^b$ for $ a, b \in \mathbb{N} $ ?
Question 2: If we define a surd as: $ \sqrt{p \cdot q …..} $
where $p,q…$ unique primes. Then is there anyway to guarantee that this will be irrational or is it a brute force method for each individual number taking the form of the $ \sqrt{2} $ proof.
Final question: What is a good definition of a surd? For me it's the square root of a number that has a prime decomposition in which no prime is raised to any power greater than 1.
Best Answer
The answer to question 1 is yes for primes, and extremely well known. My favorite of many many proofs is that if for prime $p \ne 1$ we have that if $\sqrt{p} = a/b$ (i.e., were rational), then $p b^2 = a^2$. The left side has an odd number of prime factors, while the right side has an even number of prime factors. By the Fundamental Theorem of Arithmetic, this is a contradiction.
The answer to question 2 is no for arbitrary reals (since your definition would not apply to $\sqrt{\pi}$ or an infinite number of other real numbers), but yes for any natural number, all of which of course have a unique prime factorization.