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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?

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  • $\begingroup$ The numbers $a$ and $b$ are real. That is, the set of complex numbers is $\mathbb{C} = \{ a + bi \colon a, b \in \mathbb{R} \}$. $\endgroup$ – msteve Aug 4 '14 at 0:08
  • $\begingroup$ They could come from $\mathbb{C}$, as well, but that would just be silly. $\endgroup$ – Andrew Maurer Aug 4 '14 at 0:16
  • $\begingroup$ One sometimes considers the subset of $\Bbb C $ where $a$ and $b$ are restricted to be integers or rationals; the former set is called the Gaussian integers. $\endgroup$ – MJD Aug 4 '14 at 0:22
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They are almost always assumed to be from $\mathbb{R}$

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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$.

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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.

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