We know that in every PID, for every two elements, there is one GCD (up to being associate). Now, since the set of real numbers is a field and consequently PID, this theorem holds for them.
Then, what's the GCD of a rational (for instance $2$) and an irrational number (for instance $\sqrt2)$)?
Suppose that $D$ is an integral domain (so associates are unit multiples).
If $\,\forall i\!: a\mid a_i\,$ then $\,(a,a_1,a_2,\ldots) = (a)\ $ so $\,\gcd(a,a_1,a_2,\ldots) \approx a,\,$ where $\approx$ means associate.
In particular a unit $a$ divides all elements $\in D$ so any ideal containing it $=(a)=(1)$ and any gcd containing it is $\approx a\approx 1\,$ (usually unit ideals and gcds are unit normalized to $\,1)$
In particular the above applies to every $\,a\neq 0\,$ in a field, since then $a$ is a unit.