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.

Solution 1:

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.

Solution 2:

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 $\mathbb R^2$.

Solution 3:

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.

Solution 4:

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 ?