Thursday, January 5, 2017
On the Surreal Character of Complex Numbers
Not every quadratic equation has real solutions. When we assume a quadratic equation has solutions and follow the rules of algebra in some cases we end up with something surreal with a weird displacement from the mean of the two solutions. This conclusion is logically consistent with the initial assumptions. We have to extend the real numbers to allow for the imaginary numbers in order to guarantee a solution in all cases. The weirdness can be factored out by using the imaginary number i to represent it. Multiplying the two monomial factors for the roots takes us back to the original quadratic equation when we assume i2=-1. We can plot the solutions of the quadratic equation on a plane with a real axis and an imaginary axis. Like Gibbs we can say that real and imaginary numbers are apples and oranges. The situation with Hamilton's quaternions is even weirder.