Complex numbers are often written in their rectangular form \(a + bi\) where \(a\) and \(b\) are real numbers and \(i\) is an imaginary square root of \(-1\). If we visualize the real number line and the imaginary number line as the orthogonal axes of a Cartesian plane, we could say that \((a,b)\) are the complex number's Cartesian coordinates.
Another useful way to represent complex numbers is via their polar form \(re^{\theta{i}}\), which emphasizes the vector-like nature of complex numbers. In this form, \(r\) is a real number denoting the magnitude of the vector and \(\theta\) is the angle between the positive real axis and the vector. This form is related to another coordinate system: the polar coordinates; the polar coordinates in this case are \((r,\theta)\). We can convert a complex number's polar form to its rectangular form with the help of Euler's formula: \(e^{i\theta}=\cos(\theta) + i\sin(\theta)\).
This post is about a third form for complex numbers. This time, it's based on a type of homogeneous coordinates.
Definitions
By "homogeneous coordinates" I specifically mean coordinates that still represent the same point after each coordinate is scaled by the same amount. Meaning that for a set of, say, three coordinates \((a:b:c)\), we consider them to be homogeneous when \((xa:xb:xc)=(a:b:c)\) for any non-zero positive real number \(x\).
Examples of homogeneous coordinate systems are projective coordinates and barycentric coordinates. But here we will talk about something a bit different from those two.
Given \(a \in{\Bbb{R}}\) and \(b,c \in{\Bbb{R}_{>0}}\), the homogeneous coordinates \((a:b:c)\) can be represented by a complex number written in this form:
\[a*b^{\omega}*c^{\omega^2}\]
where \(\omega=-\frac{1}{2} + i\frac{\sqrt{3}}{2}\) and \(\omega^2=-\frac{1}{2} - i\frac{\sqrt{3}}{2}\) are the primitive third roots of unity.
The third roots of unity have this nice property:
\[1+\omega+\omega^2=0\]
Which means for any non-zero positive number \(x\),
\[x*x^{\omega}*x^{\omega^2}=x^{1+\omega+\omega^2}=1\]
\[\therefore{(xa)*(xb)^{\omega}*(xc)^{\omega^2}=a*b^{\omega}*c^{\omega^2}}\]
In other words, scaling all three components by the same amount does not change the value of the number. This is exactly what we expect from homogeneous coordinates \((a:b:c)\).
(The reason why I'm limiting \(b\) and \(c\) to non-zero positive reals here is because there's no widely accepted method for evaluating complex powers of a negative number or of zero. Readers could try playing with negative denominators if they wish using their own methods. I also have a section below for cases where \(b\) and/or \(c\) are complex numbers, but I want to start this off with the simpler case of real numbers.)
Trirational numbers
For convenience, we'll also use the notation \(a\unicode{x25B6}b\unicode{x25B6}c\), which we'll call a "
trirational number" as an analogy to the rational numbers (see the "Comparison to fractions" section below for more on this analogy).
\[a\unicode{x25B6}b\unicode{x25B6}c\overset{\text{def}}{=}a*b^{\omega}*c^{\omega^2}\]
We'll call \(a\) the "numerator" of the trirational number while \(b\) and \(c\) are the two positive "denominators" of the trirational number.
We can also write trirational numbers vertically:
\[\begin{matrix}
a \\[-5pt]
\unicode{x25BC} \\[-5pt]
b \\[-5pt]
\unicode{x25BC} \\[-5pt]
c \\[-5pt]
\end{matrix}\quad\text{or}\quad\begin{matrix}a \\
\unicode{x25B6}b \\
\unicode{x25B6}c \\
\end{matrix}\]
