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Representations of the Quaternions
The quaternion algebra in terms of more familiar entities

Prerequisites

To understand this, you will need knowledge of the complex numbers, as is often taught in a high school second-year algebra course.

Introduction

This page will concentrate on the variety of ways quaternions may be presented, as opposed to what they are and how they came to be.

The literature of quaternions contains primarily three equivalent algebraic representations of the quaternions in terms of more familiar entities. These are:

four real numbers
This is the form in which Hamilton originally conceived the quaternions. Often quaternion calculations reduce to arithmetic in terms of four real quantities. Very often, this is written in a special notation involving quaternion units. However, for many purposes these direct calculations are very unwieldy.
one real number and one three-vector
This form is of use in physics, where for instance the first (real) component of the quaternion may be specialized as time, and the other components specialized as three spatial coordinates. It facilitates the connection between quaternion and vector-algebraic formulas. However, some symmetries are not obvious here.
two complex numbers
This is the most compact and symmetric representation of quaternions. It is also useful in showing the place held by quaternions in the progression from lower to higher order hypercomplex numbers.

It is also possible to represent the quaternion algebra as algebras of square matrices. See Complex numbers and Quaternions as Matrices.

The literature also contains many purely geometrical definitions. I will give one here.

Quaternions as four real numbers

Complex numbers are often represented in a component-wise form, like

a + b i

where a and b are real numbers, and i is that wonderful quantity that satisfies

i2 = −1.

Hamilton first conceived of the quaternions in a similar component-wise form.

q = a + b i + c j + d k

where a, b, c, and d are real numbers, and where i, j, and k, (called imaginary units) satisfy the multiplication table

* 1 i j k
1 1 i j k
i i −1 k −j
j j −k −1 i
k k j −i −1

The order of multiplication is important here: left factors are read from the left column, right factors are read from the top row.

The multiplication table can be summarized compactly as

i2 = j2 = k2 = i j k = −1 .

The symbols i, j, and k

We can think of i, j, and k as being special quaternions. In the notation representing quaternions as ordered pairs of complex numbers,

1 = ( 1, 0 ),
i = ( i, 0 ),
j = ( 0, 1 ),
k = ( 0, i ) .

The notation is a little abusive: for instance, it is questionable to use the symbol “i” to represent both a complex number and a quaternion, in the same expression. But then again, we use the symbol “1” to represent both a real number and a complex number, so the abuse isn’t unprecedented.

Quaternion as a real number and a three-vector

When making a connection with physics formulas, it is often convenient to represent a quaternion as the combination of a real value t and a three-vector v = ( x, y, z ):

q = ( t, v )

In this form, quaternion multiplication of q with

r = ( u, w )

is expressed in the Heaviside dot-cross notation as

q r = ( t u − v · w, t w + u v + v × w ) ,

and the conjugate of q is simply

q* = ( t, −v ) .

Hamilton also coined the terms scalar, vector, and tensor in this context, although the terms have been re-packaged to mean very different things in nowadays. Hamilton's vector was only what is now known as a 3-vector; one vector was transformed to another by a quaternion, and in this sense the quaternion was the quotient of two vectors. His tensor was a stretching factor between the vectors--and he also had versor to denote a turning factor.

Quaternion as a pair of complex numbers

The neatest way to demonstrate the connection between the quaternions and the complex and other hypercomplex numbers is to write them as a pair of complex numbers.

q = ( a, b ) ,

with addition defined in the component-wise way, and with multiplication defined by

(a, b) (c, d) = ( ac − bd*, bc* + da)

and with an operation called conjugation, defined by

(a, b)* = (a*, −b) .

This definition yields a norm, via

(a, b)* (a, b) = ( a*a + b*b, 0) = |(a, b)|2,

and a multiplicative inverse

q−1 = q* / |q|2 .

These pages favour the complex number pair, for its compactness and symmetry. See especially the Cayley-Dickson construction.

Another notation for the complex pair representation uses two complex components and the quantity j:

q = a + bj ,

where j has the properties j2 = i j i j = −1.

geometrical definition

Tait used a geometrical definition. In An Elementary Treatise on Quaternions, Chapter II, § 47, he writes:

Thus it appears that the ratio of two vectors, or the multiplier required to change one vector into another, in general depends upon four distinct numbers, whence the name Quaternion.

A quaternion q is thus defined as expressing a relation

β = q α

between two vectors α, β. By what precedes, the vectors α, β, which serve for the definition of a given quaternion, must be in a given plane, at a given inclination to each other, and with their lengths in a given ratio ; but it is to be noticed that they may be any two such vectors. [Inclination is understood to include sense, or currency, of rotation from a to β.]

Such geometrical interpretations are currently used in many applications. See Geometry of Quaternions.

See also

William Rowan Hamilton, On Quaternions

P. G. Tait An Elementary Treatise on Quaternions, Cambridge U. Press, 1890