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Section 2.1 Quaternions

The quaternions were discovered by Sir William Rowan Hamilton in 1843, after struggling unsuccessfully to construct an algebra in three dimensions. On 16 October 1843, as Hamilton was walking along a canal in Dublin, he realized how to construct an algebra in four dimensions instead. In perhaps the most famous act of mathematical graffiti of all time, Hamilton carved the multiplication table of the quaternions,

\begin{equation} i^2 = j^2 = k^2 = ijk = -1\tag{2.1.1} \end{equation}

onto the base of the Brougham Bridge as he passed it.

(The material in this section was originally published in [1].)

Subsection 2.1.1 Introduction

What happens if we add another, independent, square root of \(-1\text{?}\) Call it \(j\text{.}\) Then the big question is, what is \(ij\text{?}\)

Hamilton eventually proposed that \(k=ij\) should be yet another square root of \(-1\text{,}\) and that the multiplication table should be cyclic, that is

\begin{align} ij \amp= k = -ji\tag{2.1.2}\\ jk \amp= i = -kj\tag{2.1.3}\\ ki \amp= j = -ik\tag{2.1.4} \end{align}

We refer to \(i\text{,}\) \(j\text{,}\) and \(k\) as imaginary quaternionic units. Notice that these units anticommute!

Figure 2.1.1. The quaternionic multiplication table.

This multiplication table is shown schematically in Figure 2.1.1. Multiplying two of these quaternionic units together in the direction of the arrow yields the third; going against the arrow contributes an additional minus sign.

The quaternions are denoted by \(\HH\text{;}\) the “H” is for Hamilton.  1  They are spanned by the identity element \(1\) and three imaginary units, that is, a quaternion \(q\) can be thought of as four real numbers (\(q_1\text{,}\)\(q_2\text{,}\)\(q_3\text{,}\)\(q_4\)), usually written

\begin{equation} q = q_1 + q_2 i + q_3 j + q_4 k\tag{2.1.5} \end{equation}

which can be thought of as a point or vector in \(\RR^4\text{.}\) Since this can be written in the form

\begin{equation} q = (q_1 + q_2i) + (q_3 + q_4i)j\tag{2.1.6} \end{equation}

we see that a quaternion can be thought of as a pair of complex numbers \((q_{\CC1},q_{\CC2})=(q_1+q_2i,q_3+q_4i)\) or equivalently that we can write

\begin{equation} \HH = \CC \oplus \CC j\tag{2.1.7} \end{equation}

in direct analogy to the construction of \(\CC\) from \(\RR\text{.}\)

Subsection 2.1.2 Algebra

The quaternionic multiplication table is almost, but not quite, the vector cross product. The only difference is that imaginary quaternions square to a negative number, whereas the cross product of a vector with itself is zero.

This is not a coincidence. Making the obvious identification of vectors \(\vv\text{,}\) \(\ww\) with imaginary quaternions \(v\text{,}\) \(w\text{,}\) namely

\begin{equation} \vv = v_x \ii + v_y \jj + v_z \kk \longleftrightarrow v = v_x i + v_y j + v_x k\tag{2.1.8} \end{equation}

(and similarly for \(\ww\)), then the imaginary part of the quaternionic product \(vw\) is the cross product \(\vv\times\ww\text{,}\) that is

\begin{equation} \vv\times\ww \longleftrightarrow \Im(vw)\tag{2.1.9} \end{equation}

while the real part is just (minus) the dot product \(\vv\cdot\ww\text{,}\) that is

\begin{equation} -\vv\cdot\ww = \Re(vw)\tag{2.1.10} \end{equation}

so that the quaternionic product can be thought of as a combination of the dot and cross products! In fact, the use of \(\,\ii\text{,}\) \(\jj\text{,}\) \(\kk\) for Cartesian basis vectors originates with the quaternions, which predate the use of vectors [4].

We define quaternionic conjugation to be the (real) linear map which reverses the sign of each imaginary unit. Thus, the (quaternionic) conjugate \(\bar{q}\) of a quaternion \(q\) is

\begin{equation} \bar{q} = q_1 - q_2 i - q_3 j - q_4 k\tag{2.1.11} \end{equation}

if \(q\) is given by (2.1.5). This leads directly to the \textit{norm} of a quaternion \(|q|\text{,}\) defined by

\begin{equation} |q|^2 = q\bar{q} = q_1^2 + q_2^2 + q_3^2 + q_4^2\tag{2.1.12} \end{equation}

Again, the only quaternion with norm \(0\) is \(0\text{,}\) and every nonzero quaternion has a unique inverse, namely

\begin{equation} q^{-1} = {\bar{q} \over ~|q|^2}\tag{2.1.13} \end{equation}

Quaternionic conjugation satisfies the identity

\begin{equation} \bar{pq} = \bar{q}\>\bar{p}\tag{2.1.14} \end{equation}

from which it follows that the norm satisfies:

\begin{equation} |pq| = |p| |q|\tag{2.1.15} \end{equation}

This identity means that the quaternions form a division algebra, that is, not only are there inverses, but there are no zero divisors --- if a product is zero, one of the factors must be zero.

The symbol \({\mathbb Q}\) is used to denote the \textit{rational} numbers, and is therefore not available for the quaternions.