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

Question

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?

Answer 1

In the complex plane, the multiplication by i can be represented as a rotation by 90º counterclockwise, and, therefore, if you want multiplication to have that meaning, it is required that i·i=-1 and therefore the imaginary axis intersects the real axix with an angle of 180º/2=90º.

If the axes are not perpendicular, you can not satisfy the equality a_α·b_β = (a·b)_{(α + β)}, because i·i = 1_φ·1_φ!=1_90º·1_90º=1_180º=-1

However, you can still represent addition and substraction if your axes are not orthogonal.

Answer 2

Sure, you could construct the complex numbers with some other "basis" besides $1$ and $i=\sqrt{-1}$. What about $1$ and $k={-1}^{1/4}= \frac{1+i}{\sqrt{2}}.$ Addition will still work and you will have some other rule of multiplication:

For example, with $z_1= a_1+ k b_1$ and $z_2=a_2 + k b_2$,

$$\begin{aligned} z_1 z_2 & =(a_1 + k b_1)(a_2 + k b_2) \\& = a_1a_2+k (a_1 b_2+ a_2 b_1)+ k^2 b_1 b_2 \\& = a_1 a_2 + k (a_1b_2 + a_2b_1)+(\sqrt{2}k-1)b_1b_2\\&= a_1a_2-b_1b_2+k(a_1b_2+a_2b_1+\sqrt{2}b_1b_2)\end{aligned} $$

So this other "complex number system" is closed under addition and multiplication.

Similarly, we can show that the "polar form" is okay too.