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I'm not a mathematician, this question is out of curiosity.

When I watch a lot of math explained videos, a lot of the usage for complex number is around operation on them which was visualized in a grid-like with $y$ as the imaginary part and $x$ as the real part. The operation on them was intuitive and I'm wondering when drawing and operating on a "Cartesian plane", does the imaginary number needs to be $i^2 = -1$?

Can the complex number just treated like a Cartesian plane with the $i$ just an arbitrary letter that operates just like complex number but without it having the property of an imaginary number?

Is there any example where the math breaks down when operating on a complex number where $i^2 ≠ -1$ and rather just the letter $i$?

ConfusedButterfly
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    Well, if $i^2\ne-1$ you're not using complex numbers, are you? You get this if $i^2=1$ (in which case $j$ is the preferred symbol) or this if $i^2=0$ (in which case $\varepsilon$ is the preferred symbol). – J.G. Mar 08 '23 at 17:13
  • You may be interested in the field of fractions of the polynomial ring $\mathbb{R}[i]$, where $i$ is an indeterminant. – L. F. Mar 08 '23 at 17:14

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Interesting question. Multiplying $i$ to itself is indeed $-1$. But when performing operations in the complex plane it is important to note that it behaves like $\mathbb{R}^2$ but with a different notation. This means that $i$ behaves more like a vector, so when multiplying $i$ by $i$ you are rotating the complex plane. But by what angle and in what direction is the spoiler? I recommend you try finding it first since it is a simple exercise.

$90^\circ$ counterclockwise since $i$ maps to $-1$

Kamal Saleh
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