From what set of numbers can $a$ and $b$ be from in the complex number $a+bi\in\mathbb{C}$

Question

I've done extremely little with complex numbers to date, so my understanding of them is very low. But from what I've seen, and what I read online, a complex number is of the form $a+bi$. What I haven't been explicitly told, however, and can't find it said explicitly online, is what set of numbers do $a$ and $b$ come from? Every example I've been shown has them coming from $\mathbb{Z}$ only... but can they come from $\mathbb{Q}$ or $\mathbb{R}$, even?

Answer 1

From what set of numbers can a and b be from in the complex number $a+bi \in \mathbb{C}$? $a$ and $b$ are members of the set of real numbers, $\mathbb{R}$.

By definition, $\mathbb{C}=\{a+bi: a,b \in \mathbb{R}\}$, where $i$ is an imaginary unit satisfying $i^2=-1$.

See this for more info on the matter.

Every example I've been shown has them coming from $\mathbb{Z}$ only...

That's because it's easier to calculate the modulus and argument of a complex number with integer real and imaginary parts.

$\pi^\pi+e^{-200}i \in \mathbb{C}$ is a valid complex number, but to calculate its argument and modulus requires you to push a few more buttons on your calculator.

On the other hand, (with a bit of practice), one can spot almost immediately that $|1+i|=\sqrt{2}$ and that $\rm{Arg}$$(1+i)=\pi/4$.

Answer 2

They are almost always assumed to be from $\mathbb{R}$

Answer 3

They are from $\Bbb R$. So you can have, for instance, $(\sqrt{5} + \ln(7)) - (e^{3.78}-\pi)i$, but textbook examples and exercises usually use $\Bbb Z$ or something close to it (square roots of $2$ and $5$ frequently appear, due to the Pythagorean theorem) so that you can actually calculate anything.