Number Theory: An Introduction to Mathematics

54 I The Expanding Universe of Numbers

(ij)ε=kε=( 0 ,k), i(jε)=i( 0 ,j)=( 0 ,−k).

It is for this reason that we defined octonions as ordered pairs, rather than as matrices.
It should be mentioned, however, that we could have used precisely the same con-
struction to define complex numbers as ordered pairs of real numbers, and quaternions
as ordered pairs of complex numbers, but the verification of the associative law for
multiplication would then have been more laborious.
Although multiplication is non-associative,Odoes inherit some other properties
fromH.Ifwedefinetheconjugateof the octonionα=(a 1 ,a 2 )to be the octonion
α ̄=(a 1 ,−a 2 ), then it is easily verified that

α+β= ̄α+β, ̄ αβ=β ̄α, ̄ α ̄ ̄=α.

Furthermore,

αα ̄= ̄αα=n(α),

where thenorm n(α) =a 1 a 1 +a 2 a 2 is real. Moreovern(α) >0ifα =0, and
n(α) ̄ =n(α).
It will now be shown that ifα,β∈Oandα=0, then the equation

ξα=β

has a unique solutionξ∈O. Writingα=(a 1 ,a 2 ),β=(b 1 ,b 2 )andξ=(x 1 ,x 2 ),we
have to solve the simultaneous quaternionic equations

x 1 a 1 −a 2 x 2 =b 1 ,

a 2 x 1 +x 2 a 1 =b 2.

If we multiply the second equation on the right bya 1 and replacex 1 a 1 by its value
from the first equation, we get

n(α)x 2 =b 2 a 1 −a 2 b 1.

Similarly, if we multiply the first equation on the right bya 1 and replacex 2 a 1 by its
value from the second equation, we get

n(α)x 1 =b 1 a 1 +a 2 b 2.

It follows that the equationξα=βhas the unique solution

ξ=n(α)−^1 βα. ̄

Since the equationαη=βis equivalent toη ̄α ̄ =β ̄, it has the unique solution
η=n(α)−^1 αβ ̄. ThusOis adivision algebra. It should be noted that, sinceOis non-
associative, it is not enough to verify thatevery nonzero element has a multiplicative
inverse.
It follows from the preceding discussion that,for allα,β∈O,

(βα)α ̄ =n(α)β=α(αβ). ̄