Number Theory: An Introduction to Mathematics

50 I The Expanding Universe of Numbers

where thenorm n(A)andtrace t(A)are both real:

n(A)=aa ̄+bb ̄, t(A)=a+ ̄a.

Moreover,n(A)>0ifA=0. It follows that any quaternionA=0 has a multiplica-
tive inverse: ifA−^1 =n(A)−^1 A ̄,then

A−^1 A=AA−^1 = 1.

Norms and traces have the following properties:for all A,B∈H,

t(A ̄)=t(A),

n(A ̄)=n(A),

t(A+B)=t(A)+t(B),

n(AB)=n(A)n(B).

Only the last property is not immediately obvious, and it can be proved in one line:

n(AB)=ABAB=B ̄AAB ̄ =n(A)BB ̄ =n(A)n(B).

Furthermore, for anyA∈Hwe have

A^2 −t(A)A+n(A)= 0 ,

since the left side can be written in the formA^2 −(A+A ̄)A+AA ̄. (The relation is actu-
ally just a special case of the ‘Cayley–Hamilton theorem’ of linear algebra.) It follows
that the quadratic polynomialx^2 +1 has not two, but infinitely many quaternionic roots.
If we put

I=

(

01

− 10

)

, J=

(

0 i

i 0

)

, K=

(

i 0

0 −i

)

,

then

I^2 =J^2 =K^2 =− 1 ,

IJ=K=−JI, JK=I=−KJ, KI=J=−IK.

Moreover, any quaternionAcan be uniquely represented in the form

A=α 0 +α 1 I+α 2 J+α 3 K,

whereα 0 ,...,α 3 ∈R. In fact this is equivalent to the previous representation with

a=α 0 +iα 3 , b=α 1 +iα 2.

The corresponding representation of the conjugate quaternion is

A ̄=α 0 −α 1 I−α 2 J−α 3 K.

HenceA ̄=Aif and only ifα 1 =α 2 =α 3 =0andA ̄=−Aif and only ifα 0 =0.