No ordinary, "real" number can give a negative result when squared. Hence the equation $x^2+4=0$ has no solution.

This doesn't seem like such a problem, but the development of the general solution to the cubic equation uses quantities of this type, which then subsequently all cancel out leaving just real solutions. Somehow these "imaginary" quantities seem to be useful.

Can they be added, subtracted, multiplied and divided like normal numbers? Certainly they can be added and subtracted, but multiplying two imaginary numbers gives a real number, so things only really work when imaginary numbers are combined with the "normal" numbers, the so-called "real numbers.".

This combination of real number with imaginary number gives what are called the complex numbers.

The imaginary number $i$ has the property $i^2=-1$