What does the notation (p,q) = 1 mean for a rational number p/q?

Question

I'm beginning to work through Murty and Rath's Transcendentals and came across this notation which I've never seen before.

The theorem states:

Given a real algebraic number $\alpha$ of degree $n>1$, there is a positive constant $c=c(\alpha)$ such that for all rational numbers $p/q$ with $(p,q)=1, q>0$, we have $\mid \alpha - \frac{p}{q} \mid > \frac{c(\alpha )}{q^n}$.

So I was wondering what the notation $(p,q)=1$ meant.

Thanks.

Answer 1

Basically, it means $p$ and $q$ are coprime (do not have any common factor other than $1$) and, therefore, the fraction $\frac{p}{q}$ is not reducible anymore.

Answer 2

$(p,q)$ a common notation for $\gcd(p,q)$ and so $(p,q)=1$ means that the fraction is irreducible.

Answer 3

This is probably the notation for the greatest common divisor. Many authors, like Apostol, prefer to use the notation $(a,b)$ rather than $\gcd(a,b)$.The notation $(p,q)=1$ means that $p$ and $q$ are relatively prime.