Proof of equality of complex numbers needed?

Question

Is $a+bi=c+di\iff a=c, b=d$, where $a,b\in\mathbb{R}$ something that requires proof? My instinct is telling me that proof is required to demonstrate that playing around with real numbers (using field operations) cannot make them 'escape' into the realm of non-real numbers. Is there a term for this, perhaps involving metric spaces?

Answer 1

As Dave keenly pointed out, one way of framing this question is whether the notations $(a, b)$ and $a + bi$ are compatible, or more precisely, whether the space of pairs $(a, b)$ and the space of complex numbers $a + bi$, each endowed respectively with their usual operations, are isomorphic (say, as rings) via the identification $\Phi: (a, b) \mapsto a + bi$, which is obviously a bijection.

In particular, we must check that:

• $\Phi((a, b) + (a', b')) = \Phi((a, b)) + \Phi((a', b'))$ and
• $\Phi((a, b) \cdot (a', b')) = \Phi((a, b)) \Phi((a', b'))$.

But the first identity is almost immediate, and the multiplication rule $$(a, b) \cdot (a', b') := (aa' - bb', ab' + a'b)$$ is chosen precisely so that the second identity holds too. (If this isn't clear, evaluate both sides of the identity to check for yourself.)