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## The Complex Plane

Gentlemen, this is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means.  But we have proved it, and therefore we know it must be the truth. -Benjamin Pierce, American mathematician

As we saw in the last post, using the Real Number system can lead to equation whose solution is not a real number.

For example:

x²=-1

We know that if we square a positive number, we get a positive number.

And, if we square a negative number, we also get a positive number.

So, what kind of number will give us a negative answer when we square it?

A COMPLEX NUMBER!

The book An Imaginary Tale, by Paul Nahin is a great source of information about complex numbers and their applications, although it can be somewhat technical for non-mathematicians.

One of the first mathematicians noted to have worked successfully with complex numbers was Girolamo Cardano, a 16th century Italian mathematician.  Cardano was one of several Italian mathematicians involved in the efforts to find formulas that would solve cubic and quartic equations in the same way that the quadratic formula is used to solve quadratic equations.

In the process of using the formula to solve cubic equations, Cardano found a step in the solution process in which negative numbers appeared under square root symbols.  Unsure as to how to proceed, Cardano figured that these numbers (if they were to interact with the Real Number system) must work in a similar way.

In An Imaginary Tale, Paul Nahin says

The fact that Cardan[o] had no fear of imaginaries themselves is quite clear from the famous problem he gives in Ars Magna, that of dividing ten into two parts whose product is forty.  He calls this problem “manifestly impossible” because it leads immediately to the quadratic equation x²-10x+40=0, where x and 10-x are the two parts, an equation with the complex roots…of 5+√(-15) and 5-√(-15).  Their sum is obviously 10, because the [square roots] cancel, but what of their product?  Cadan[o] boldly wrote “nevertheless we will operate” and formally calculated

(5+√(-15))(5-√(-15)) = (5)(5)-(5)√(-15)+(5)√(-15) -√(-15)√(-15)=25-(-15)=40

The geometric interpretation of complex numbers was developed in the centuries following Cardano’s work and led to the conception of two number lines crossed with each other to create the complex number plane.  This is one of the defining characteristics of a complex number –

it is two dimensional.