Take a number, square it, the result is non-negative. Because positive * positive = positive and negative * negative is positive. Or
But someone wanted to take square roots of negative numbers, so they did, and called it 'i'.
The solution, to put imaginary and complex numbers on a solid foundation is something called a ring quotient. What you do is you start with the ring (meaning number system) of polynomials over the real numbers
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$1, 2 3.5, \pi$ etc. -
$i, i^2, 0.3 + 9.5 i + 23 i^3$ and so on.