Why is there root of -1?

Question

Always have we heard that there isn't a root of a negative number. That's why we call the root of -1 an imaginary number. But why, why do we need it and how even did they discover this case? I would want a good answer, not someone that says "we need it in engineering"(someone said me that). Thanks.

Answer 1

Imaginary number, like many aspects of mathematics, are tools made up by mathematicians to solve abstract problems (like what are solutions to $x^2=-1$), and which scientists then found more concrete uses for. In this sense, imaginary numbers were not discovered (in the same sense that one discovers laws of the universe), but rather invented to serve a particular purpose. Because this purpose has many interesting extensions, consequences, and applications, we keep them around as a fundamental aspect of math. Raphael Bambelli, a 16th century mathematician, was the first to formalize the use of imaginary numbers, so I suppose you could say it was he who invented them.

Imaginary numbers are useful first because not all polynomial equations of the form $a_0+a_1 x+...a_n x^n=0$ (where $a_n$ are real) have roots in the reals. But they always have complex roots. Restricting ourselves to real numbers makes some problems very difficult to solve. For instance, the motion of a mass on a spring obeys the law

$$y''(t)=-ky(t),$$ where $y$ is an unknown function. Without imaginary numbers, it is difficult to solve equations like this, but by introducing them we can use them as a tool to find real solutions.

I connection with the pendulum problem, complex numbers next become useful because of beautiful relation discovered by Euler that $$e^{i x}=\cos(x)+i\sin(x).$$ This shows us how elementary functions behave when given complex inputs. We can use this result to simplify problems that are very difficult when restricted to real algebra. When someone says that complex numbers are useful in engineering, they are probably referring to engineers using this formula in one way or another. Particular areas that I know of which make heavy use of imaginary numbers are fluid dynamics, electricity and magnetism, optics, animation, quantum mechanics, acoustics, and I'm sure there are many others.

Finally, imaginary numbers are interesting to mathematicians because it opens up an entirely new dimension of numbers which in some ways are more fundamental than the reals themselves. Results of analysis (which are the subject of an entire career of study, and I will not discuss here) are often more beautiful, interesting, robust, and fundamental in the complex plane than on the real line alone.