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Why does rectangular form serve as an accurate description of a complex number? Why not $a * bi$(multiplication) or another operation? Why does addition describe the relationship between the real part and complex part? For example, polar form describes the relation on a imaginary and real plane. What concept is rectangular form based on?

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I don't really understand how to answer this question. One answer might just be "this is how complex numbers are defined" but I imagine that won't do. What kind of answer are you looking for? – Qiaochu Yuan May 13 '12 at 23:54
@QiaochuYuan I am looking for some kind of basis of the relationship. Most math relationships are based on something - some kind of concept or fact. Like I mentioned above polar form is based on its position on the real and imaginary planes. How was this specific formed arrived at? – user26649 May 13 '12 at 23:57
@Farhad: Why don't you think writing a complex number as $a+bi$ describes it on the plane? The real part of the complex number, namely $a$, is the "$x$-coordinate", and the imaginary part $b$ is the "$y$-coordinate". – Zev Chonoles May 13 '12 at 23:57
@ZevChonoles But when coordinates are written, they are separated by a comma not addition. For example on the Cartesian plane, if I was referring to the point $X$, I would describe its position through (x-coordinate, y-coordinate). If $a$ was the x-coordinate and $b$ the y, why isn't the form $(a,bi)$? – user26649 May 14 '12 at 0:00
@ZevChonoles Thanks! Do you mind reposting the comment as an answer? – user26649 May 14 '12 at 0:13
up vote 1 down vote accepted

Reposting some comments as an answer:

Writing a complex number as $a+bi$ does describe it on the plane - the real part of the complex number, namely $a$, is the "$x$-coordinate", and the imaginary part $b$ is the "$y$-coordinate".

Your comments indicate that you're used to writing vectors, or points on a plane, with coordinates like $(a,b)$. But one can just as easily decide to express a vector as a sum - you've probably seen expressions like $$v=a\hat{\imath}+b\hat{\jmath}+c\hat{k}$$ for example. This is just an alternative way of writing $$v=(a,b,c)$$ because $$\hat{\imath}=(1,0,0)$$ $$\hat{\jmath}=(0,1,0)$$ $$\hat{k}=(0,0,1)$$ and thus $$a\hat{\imath}+b\hat{\jmath}+c\hat{k}=a(1,0,0)+b(0,1,0)+c(0,0,1)$$ $$=(a,0,0)+(0,b,0)+(0,0,c)=(a,b,c)$$ It's the same idea with complex numbers. Think of $1$ as shorthand for $(1,0)$ and $i$ as shorthand for $(0,1)$. Then $$a+bi=a\cdot 1+b\cdot i = a(1,0)+b(0,1)=(a,b)$$

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Thanks again Zev! – user26649 May 14 '12 at 0:18

You have $\mathbb{R}$ which is a field, i.e. you can add and multiply, and every nonzero elements has a multiplicative inverse.

Now you want to somehow have square roots for all real numbers. So you include (this can be done properly, but it's not the point here) another "number", that we call $i$ and has the property that $i^2=-1$.

Now you still want to have a field; in particular you want to add and multiply, and so you need to make sense of expressions of the form $a+bi$. But it turns out that $$ \mathbb{C}=\{a+bi:\ a,b\in\mathbb{R}\} $$ is already a field, i.e. the smallest field that contains both $\mathbb{R}$ and $i$.

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"What concept is rectangular form based on?" On the concept of vectors I guess. Imagine that you have to walk through some city where there are pavement blocks of two different sizes and you can't express size of one of them by size of another - you can't compare them. Next lets assume that you walk some distance on the pavement blocks of the first kind, say x pavment blocks and than some distance on the pavement of the second kind - say y pavement blocks. Now if you were to answer question what is the distance of your walk you can only say that you had walked x pavment blocks of the first kind and y of the second. This is the sketch of the idea behing rectangular description. The real axis is one kind of pavment blocks and imaginary axis the second kind.

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