Just a quick one today, as I get caught back up from Thanksgiving. We all know and love the very basic quadratic function. Any second-order polynomial will give you a nice little parabola, which of course is ubiquitous in physics. We all know what that looks like. But what if we’re willing to square complex numbers instead of just real numbers? Traditionally we denote complex numbers with z instead of x, so our Sunday Function is:
Ok, so what happens when we square a complex number? Well, we can write any complex number as (a + bi), where “a” and “b” are real numbers. “a” is the real part and “b” is the imaginary part. Keeping in mind that “i” squared is -1, we can go ahead and square our generic expression for any complex number:
The first term (a^2 – b^2) is the real part of the number z^2 and 2ab is the imaginary part of z^2. As such we’re done if we just want to calculate numerical values. But we would like a bit better of a theoretical understanding as well. First, we see that the real part is zero if and only if a and b are equal. The imaginary part is zero if and only if one or both of a and b are also zero. So positive real numbers are sent to positive real numbers, imaginary numbers are sent to negative real numbers, and negative real numbers are sent to positive real numbers. Complex numbers will do something in between. In fact if we plot the arg(z^2) [Note: If you think of a complex number as a point on the complex plane, arg(z) represents the angle between the real axis and that point.], we’ll get this:
If you think of the complex plane as a rubber sheet, this suggests that the function f(z) = z^2 both stretches the sheet radially and bends it in a counterclockwise direction. To verify this, we’ll need to use the polar representation of complex numbers. That’ll be a job for next week.