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?

  • 2
    $\begingroup$ 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. $\endgroup$ – user98602 Aug 26 '15 at 16:20

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.

  • $\begingroup$ 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 $\Bbb C$. $\endgroup$ – user98602 Aug 26 '15 at 16:36
  • $\begingroup$ Sorry, that's right. I get confused when anything like trigonometry is involved. $\endgroup$ – gonthalo Aug 26 '15 at 16:41

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