Why is the square root of a negative number imaginary?

Answer 1

Any real number, when squared will give a positive real number. This is true weather you have a negative real number and square it, or positive. The concept of imaginary numbers was invented to handle equations which needed to use the square root of a negative number, in order to solve them.

At first, they were not considered useful beyond this, so they were named imaginary numbers. Through the work of Euler and others, it became evident that imaginary and complex numbers could be useful, especially when analyzing waves, such as electrical waveforms.