Complex arithmetic using Clifford algebra
In clifford: Arbitrary Dimensional Clifford Algebras

knitr::opts_chunk$set(echo = TRUE)
library("clifford")

knitr::include_graphics(system.file("help/figures/clifford.png", package = "clifford"))

To cite the clifford package in publications please use @hankin2025_clifford_rmd. This short document shows how complex arithmetic may be implemented using Clifford algebra (of course, if one really wants to use complex numbers, base R is much more efficient and uses nicer idiom than the methods presented here). Recall that complex numbers are a two-dimensional algebra over the reals, with $(a,b)\cdot(c,d)=(ac-bd,ad+bc)$; we usually write $(a,b)$ as $a+bi$. There are two natural ways to identify complex numbers with Clifford objects; but because they use different signatures it is more convenient to treat them separately.

First method

We use $\operatorname{Cl}(0,1)$, so $e_1^2=-1$. Package idiom is straightforward; to coerce complex numbers to Clifford objects and vice versa, we need a couple of functions:

signature(0,1)
options(maxdim=1) # paranoid-level safety measure
complex_to_clifford <- function(z){Re(z) + e(1)*Im(z)}
clifford_to_complex <- function(C){const(C) + 1i*getcoeffs(C,1)}
clifford_to_complex <- function(C){const(C) + 1i*coeffs(Im(C))}

Then numerical verification is immediate. First we choose some complex numbers:

z1 <- 35 + 67i
z2 <- -2 + 12i

Then, for example:

z1
complex_to_clifford(z1)

Checking that the coercion is a homomorphism is easy:

complex_to_clifford(z1) * complex_to_clifford(z2) == complex_to_clifford(z1*z2)

Above, note that the * on the left is the geometric product, while the * on the right is the usual complex multiplication. And because the map is invertible we can check the other way too:

(C1 <- 23 + 7*e(1))
clifford_to_complex(C1)
C2 <- 2  - 8*e(1)
clifford_to_complex(C1)*clifford_to_complex(C2) == clifford_to_complex(C1*C2)

Second method

We use $\operatorname{Cl}(2)$, so $e_1^2=e_2^2=1$, and identify the imaginary unit $i$ with $e_{12}$ (thus $e_{12}^2=e_{12}e_{12}=e_{1212}=-e_{1122}=-e_1^2e_2^2=-1$). A general complex number $z=x+iy$ maps to Clifford object $x + ye_{12}$.

options(maxdim=2)  # paranoid-level safety measure
signature(2)
complex_to_clifford <- function(z){Re(z) + e(1:2)*Im(z)}
clifford_to_complex <- function(C){const(C) + 1i*coeffs(Im(C))}

Then numerical verification:

z1 <- 35 + 67i
z2 <- -2 + 12i
complex_to_clifford(z1) * complex_to_clifford(z2) == complex_to_clifford(z1*z2)

C1 <- 23 + 7*e(1:2)
C2 <- 2  - 8*e(1:2)
clifford_to_complex(C1)*clifford_to_complex(C2) == clifford_to_complex(C1*C2)

Note

The identification $x+iy\longrightarrow x+ye_{12}$ is a homomorphism whenever $e_1^2e_2^2=1$; above we used $\operatorname{Cl}(2,0)$ so $e_1^2=e_2^2=1$. However, the relation is also satisfied if $e_1^2=e_2^2=-1$, so we can equally well use $\operatorname{Cl}(0,2)$:

signature(0,2)
c(
complex_to_clifford(z1)*complex_to_clifford(z2) == complex_to_clifford(z1*z2),
clifford_to_complex(C1)*clifford_to_complex(C2) == clifford_to_complex(C1*C2)
)