Why imaginary numbers axis is plotted perpendicular to the real numbers axis? Negative numbers axis is plotted to the opposite side of the positive real number axis that make sense but i do not understand why imaginary numbers are plotted perpendicular to the real numbers axis. 
 A: The most elementary complex number $ i$  equals $ e^{i \pi/2}  $ by Euler's theorem. So it is natural to take $ \theta = \pi/2 $ line for imaginary number axis on a line perpendicular to real axis where real component=0 or origin.
A: Vey nice question. Think of the solution of the equation $x^2+1=0$, there is not real solution in the real line (real-world), but the polynomial has to have a root! so the solution must be in a different world, so mathematicians extended the real world to infinitely many real worlds $x$ shifted by some non real world $yi$. So we got $x+yi$ where our actual world (real line) just a single slice of the extended world (real + imaginary).
Now, why do we plot it perpendicular? If we do not do that means for every real number can be represented by an imaginary numbers (or vise versa) by using Pythagorean relation which means that the imaginary and real numbers are just the same which is against the task of extending the real line to complex plane.
A: One (not the only) good reason: if you define the norm of $z\in\mathbb C$ as $$||z|| = \sqrt{z\bar z}$$
then $\mathbb C$, as a normed vector space over $\mathbb R$, is isomorphic to $\mathbb R^2$.
A: Sorry dudes the best answer that i have got is that just as -1 itself can be thought of as 1 at an angle of 180 degrees so The square root of -1 just can be thought of as 1 at an angle of 90 degrees. Is not that one is simplest ? 
A: In modern mathematics, the abstract Cartesian Coordinate Plane is a an underlying set
$\quad \Bbb R \times \Bbb R$
with additional properties that can 'flesh it out' and give it more structure.
In set theory there is no such thing as plotting points, but every math student has graphed functions and relations on graph paper using a pencil; every point on the paper corresponds to an element of the set $\Bbb R \times \Bbb R$.
Without a doubt, the student knows that $x\text{-axis}$ numbers increase as you move  right and the $y\text{-axis}$ numbers increase as you move up; that is just the convention. But in this wiki article you'll read

(However, in some computer graphics contexts, the ordinate axis may be
oriented downwards.)

When if comes to the complex numbers, the student learns that,
$\quad \Bbb C = \{x + yi \mid \text{where } x \in \Bbb R \text{ and } y \in \Bbb R \}$
and
$\quad a + bi = c + di \text{ iff } a = c \land b = d$
But then the complex numbers can be plotted on graph paper - plot $x + yi$ to correspond to the element $(x,y) \in \Bbb R \times \Bbb R$.
We could also use a different bijective correspondence,
$\quad x +iy \mapsto (x,-y) \quad$
but we agree, by convention, to have $i$ point orthogonally up from the $x\text{-axis}$
(note also that both $i$ and $-i$ satisfy $x^2 = -1).$
As the student learns how to graph the binary operations of addition and multiplication of complex numbers, in no time at all they will see that the identification of $\Bbb C$ with $2\text{-dimensional space}$ has its merits.
A: The Complex plane is technically $\mathbb{R}^{2}$ equipped with a new operation and elements. Such is why $\mathbb{C}$ is plotted on the familiar $y$-axis.
One can construct $\mathbb{C}$ as follows:
Let vectors $\vec{e}_{1}=(1, 0)$ and $\vec{e}_{2}=(0, 1)$. Write every vector $\vec{z}\in \mathbb{R}^{2}$ as $\vec{z} = (x, y) = x\vec{e}_{1} + y\vec{e}_{2}$, such that $\vec{e}_{1}$ and $\vec{e}_{2}$ form a new basis.
Finally, define the multiplication operation on our constructed plane with $\vec{e}_{1}^{2} = \vec{e}_{1}$, $\vec{e}_{2}^{2} = -\vec{e}_{1}$, $\vec{e}_{1}\vec{e}_{2} = \vec{e}_{2}\vec{e}_{1} = \vec{e}_{2}$. We denote $\vec{e}_{1}$ by $1$, and $\vec{e}_{2}$ by $i$.
The standard complex number addition and multiplication operations follow from these definitions.
