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When we write complex numbers we write in $a+ib$ form why we don't write it as we write in Cartesian coordinate system like $(x,y)$ and how the idea of a complex plane emerge

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There is a book called "A History of Vector Analysis" by Crowe which addresses this and many more topics.

http://worrydream.com/refs/Crowe-HistoryOfVectorAnalysis.pdf

It is also published as a book by Dover.

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The Form $a+ib$ is easier to work with. For example, the product $z_1\cdot z_2$ and the ratio $z_1/z_2$ for $z_1,z_2 \in \mathbb{C}$ is better to calculate using $a+ib$ or $r\mathrm{e}^{i\phi}$.

The idea of the complex plane probalby emerged because of the following formula:

$$z=a+ib=\sqrt{a^2+b^2}\left(\frac{a}{\sqrt{a^2+b^2}}+i\frac{b}{\sqrt{a^2+b^2}}\right).$$

In which $\sqrt{a^2+b^2}\neq 0$ can be interpreted as the distance from the origin / hypotenuse of triangle and $a/\sqrt{a^2+b^2}$ and $b/\sqrt{a^2+b^2}$ can be interpreted as the cosine and sine of an angle (see geometric definition of trig functions). So the similarity in the description of the complex numbers and the geometrical interpretation (e.g. $z_1,z_2 \in \mathbb{C}$ then the sum $z_1+z_2$ can be interpreted as a vector sum) made it a useful interpretation of the complex numbers.

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