u/ConstrainedMind

We are normally introduced to complex numbers with the equation x^(2) = -1 and told that it does not have solution in real numbers. This does not have solution in real due to the fact of multiplication rule, saying multiplication of a number with itself will be a positive number.

p - positive num, n - negative num

we have rules

p x p = p and n x n = p

But what if we change the rule to

p x p = p and n x n = n

and suddenly in new rule the solution to x^(2) = -1 exist as -1.

I know this new rule may not make sense and may be its wrong, but I want to understand does complex numbers arise because we have induced certain rules on multiplication, or they will still exist in the new rule, if they exist how would they look like ?