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Component-wise Representation of the Quaternions
The quaternion algebra as entities with four real components

Prerequisites

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

Introduction

A traditional representation of a complex numbers is component-wise, like

a + b i

where a and b are real numbers, and i is that wonderful entity with the property

i² = −1.

What follows is a similar representation for the quaternions.

The quaternions

Hamilton first conceived of the quaternions in this 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

This table is summarized by the rules

i² = j² = k² = i j k = −1 .

These were the rules Hamilton said he carved on Broome bridge in Dublin when the idea of quaternions came to him.

The symbols i, j, and k

The symbols i, j, and k are just 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, the symbol “1” commonly represents both a real number and a complex number, so the abuse isn’t unprecedented.

Representation with just i and j

Since k = i j, any quaternion may be written as

a + b i + c j + d i j = a + b i + ( c + d i ) j

which is the same as the representation as an ordered pair of complex numbers.