Field extension $\mathbb{Q} (\sqrt2)$; why is adding $\sqrt2$ not enough?

Question

This an extract from Visual Group theory book section 10.5.1 page-234 ,which says that:

What I can't understand from this extract is for what purpose do we need to add other elements except $\sqrt{2}$ only?

Answer 1

The purpose is to have a field, a "number system capable of arithmetic", meaning a set of numbers closed under the arithmetic operations $+,-,\times,\div$.

Edited to address some comments: I think the writer's phrasing is legitimately confusing. The first sentence makes it sound like the writer's only goal is to solve $x^2-2=0$; if that were the only goal, then considering the entire field $\Bbb Q(\sqrt2)$ wouldn't be necessary. But a second priority (and, I submit, the more important priority) is to have "a number system capable of arithmetic, a field". Once we recognize that as a goal, we agree that "It is not enough to add the single number $\sqrt2$". But the writer should have rearranged the order of these sentences to make this clearer.