# Is it necessary for the Imaginary-axis to be perpendicular to the Real-axis?

The Real number line is in one dimension. If you want to map a complex number, you would have to add a second dimension to that number line- the Imaginary-axis.

The Imaginary-axis is always perpendicular to the Real-axis. Here is my question:

Would you still be able to use the complex plane if the imaginary-axis wasn't perpendicular to the Real-axis? In other words, is it still possible to prove theorems involving complex numbers (in a geometrical way) if the Imaginary-axis wasn't perpendicular to the Real-axis?

For example: Could you prove Euler's formula if the Imaginary-axis was tilted to a 45 degree angle?

• There is no notion of perpendicular in $\Bbb C$ unless you invent one. You chose the inner product such that $\langle 1,i\rangle = 0$, but maybe I'll choose one such that $\langle 1, i\rangle = 3$. Since we never actually use this inner product anywhere in complex analysis until we start talking about conformal maps, this choice doesn't matter at all. – user98602 Aug 26 '15 at 16:20

• Euler's formula is an equality of power series and doesn't use anything more than the algebraic structure (and norm, to define convergence) of $\Bbb C$. – user98602 Aug 26 '15 at 16:36