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I'm wondering why complex numbers represent coordinates without being on the form of a tuple (a,b). The complex numbers come in the form: $a+bi$ where $a$ denotes the real part and $bi$ denotes the imaginary part. This doesn't represent an ordered pair in my eyes, but more like a sum. How does it represent coordinates on a grid?

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    $\begingroup$ Well, there is a very clear 1-1 correspondence between ordered pairs $(a,b)$ and numbers of the form $a+ib$. $\endgroup$
    – Old John
    Commented Jan 23, 2014 at 23:30
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    $\begingroup$ The difference between (a,b) and a + bi is just notation. $\endgroup$ Commented Jan 23, 2014 at 23:30
  • $\begingroup$ It is the same thing, it is usually more convenient (and clear) to write in the form $a+ib$, for example, $e^{i\theta}$ would be $e^{(0,1)\theta}$. $\endgroup$
    – copper.hat
    Commented Jan 23, 2014 at 23:36

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One could actually define the complex numbers as $\mathbb R^2$ with the usual sum of vectors, and an additional multiplication which is: $(a,b)(c,d)=(ac-bd,bc+ad)$. With this we already have the complex numbers, however the multiplication is more naturally visualized if we write $$(a+bi)(c+di)=ac+(ad+bc)i+bdi^2=(ac-bd)+(ad+bc)i$$

So basically, the rule for multiplication is also verified if we write $(x,y)$ as $x+yi$, and multiply using the normal rules (distributive, associative, etc.), therefore we are justified in using this notation, and we do so because it's more comfortable. Of course the definition given is a modern construct, made in order to have a solid, rigorous theory; the historical definition was the $x+yi$ notation, with the particularity that $i^2=-1$. Also, other definitions exist.

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