# Formulation of the Complex Plane

by IxRxPhysicist
Tags: complex, formulation, plane
 Mentor P: 18,346 I think it's fair to say that the origins were fairly theoretical. In fact, complex numbers first showed up with the work of Cardano. Cardano tried to solve polynomials of the third degree. He (and others) eventually found a nice method to solve those equations. Now, if you solve a quadratic equation $ax^2 + bx + c=0$, then you usually define the discriminant $D= b^2 - 4ac$. We know now that if D is negative, then there are complex solutions. But back in Cardano's day, they did not know complex numbers. So they simply said that if D is negative, then the quadratic has no solutions. Now, Cardano's method also involved some discriminant. And this discriminant can be used to find the solutions to the cubic equation. And in the solution, you will have to take the square root of the discriminant (as with the quadratic equation). So you can think that the same happens with the quadratic: if the discriminant is negative, then the square root doesn't exist, so there are no solutions. But they discovered that this is not at all the case. The found out that even if the discriminant is negative, that there are still real solutions. And it was even more weird, those real solutions could be found by exactly the same formula where you take the square root of a negative number. So they introduced expressions such as $\sqrt{-1}$ as tools to help them solve equations. They were not called numbers though, they were treated as mere tricks that happened to work. It was close to mathematical heresy to call the square roots of negative numbers actual numbers. Remember, this is a time where even negative numbers were treated suspicious. Of course, if there is anything that mathematicians have learned by now, then that is that there are no coincidences in math. In fact, coincidences are usually symptoms of a deeper underlying structure that is yet to be discovered. After a long time, the mathematicians in history also discovered this and started to look at complex numbers as actual numbers. But it took until (just before) 1800 that mathematicians started to do actual work with complex numbers. This was the birth of complex analysis.