Number Theory: An Introduction to Mathematics

6 Quaternions and Octonions 51

A quaternionAis said to bepureifA ̄=−A. Thus any quaternion can be uniquely
represented as the sum of a real number and a pure quaternion.
It follows from the multiplication table for the unitsI,J,KthatA=α 0 +α 1 I+
α 2 J+α 3 Khas norm

n(A)=α 02 +α^21 +α^22 +α 32.

Consequently the relationn(A)n(B)=n(AB)may be written in the form

(α^20 +α^21 +α^22 +α 32 )(β 02 +β 12 +β^22 +β 32 )=γ 02 +γ 12 +γ 22 +γ 32 ,

where

γ 0 =α 0 β 0 −α 1 β 1 −α 2 β 2 −α 3 β 3 ,

γ 1 =α 0 β 1 +α 1 β 0 +α 2 β 3 −α 3 β 2 ,

γ 2 =α 0 β 2 −α 1 β 3 +α 2 β 0 +α 3 β 1 ,

γ 3 =α 0 β 3 +α 1 β 2 −α 2 β 1 +α 3 β 0.

This ‘4-squares identity’ was already known to Euler (1770).
An important application of quaternions is to the parametrization of rotations in
3-dimensional space. In describing this application it will be convenient to denote
quaternions now by lower case letters. In particular, we will writei,j,kin place
ofI,J,K.
Letube a quaternion with normn(u)=1, and consider the mappingT:H→H
defined by

Tx=uxu−^1.

Evidently

T(x+y)=Tx+Ty,

T(xy)=(Tx)(Ty),

T(λx)=λTx ifλ∈R.

Moreover, sinceu−^1 = ̄u,

Tx ̄=Tx.

It follows that

n(Tx)=n(x),

since

n(Tx)=TxTx=TxTx ̄=T(xx ̄)=n(x)T 1 =n(x).

Furthermore,T maps pure quaternions into pure quaternions, sincex ̄ =−x
implies

Tx=Tx ̄=−Tx.