Definisi Setiap oktonion terdiri dari kombinasi linear riil unit oktonion :
{ 1 , i , j , k , l , i l , j l , k l } {\displaystyle \{1,i,j,k,l,il,jl,kl\}\,} dan suatu oktonion dituliskan dengan persamaan berikut:
x = x 0 + x 1 i + x 2 j + x 3 k + x 4 l + x 5 i l + x 6 j l + x 7 k l {\displaystyle x=x_{0}+x_{1}i+x_{2}j+x_{3}k+x_{4}l+x_{5}il+x_{6}jl+x_{7}kl\,} dengan koefisien riil { x i } {\displaystyle \{x_{i}\}}
Sama halnya dengan kuaternion, pertambahan dan pengurangan oktonion dilakukan dengan menambahi dan mengurangi setiap unit serupa. Sementara perkalian dilakukan dengan mengalikan setiap unit oktonion pertama dengan setiap unit oktonion kedua, dan lalu menjumlahi hasil perkalian. Berikut tabel perkalian unit-unit oktonion:
× 1 i j k l il jl kl 1 1 i j k l il jl kl i i −1 k −j il −l −kl jl j j −k −1 i jl kl −l −il k k j −i −1 kl −jl il −l l l −il −jl −kl −1 i j k il il l −kl jl −i −1 −k j jl jl kl l −il −j k −1 −i kl kl −jl il l −k −j i −1
Penjumlahan x + y = ( x 0 + x 1 i + x 2 j + x 3 k + x 4 l + x 5 i l + x 6 j l + x 7 k l ) + ( y 0 + y 1 i + y 2 j + y 3 k + y 4 l + y 5 i l + y 6 j l + y 7 k l ) = ( x 0 + y 0 ) + ( x 1 + y 1 ) i + ( x 2 + y 2 ) j + ( x 3 + y 3 ) k + ( x 4 + y 4 ) l + ( x 5 + y 5 ) i l + ( x 6 + y 6 ) j l + ( x 7 + y 7 ) k l {\displaystyle {\begin{aligned}&x+y\\&=(x_{0}+x_{1}i+x_{2}j+x_{3}k+x_{4}l+x_{5}il+x_{6}jl+x_{7}kl)+(y_{0}+y_{1}i+y_{2}j+y_{3}k+y_{4}l+y_{5}il+y_{6}jl+y_{7}kl)\\&=(x_{0}+y_{0})+(x_{1}+y_{1})i+(x_{2}+y_{2})j+(x_{3}+y_{3})k+(x_{4}+y_{4})l+(x_{5}+y_{5})il+(x_{6}+y_{6})jl+(x_{7}+y_{7})kl\end{aligned}}}
Pengurangan x − y = ( x 0 + x 1 i + x 2 j + x 3 k + x 4 l + x 5 i l + x 6 j l + x 7 k l ) − ( y 0 + y 1 i + y 2 j + y 3 k + y 4 l + y 5 i l + y 6 j l + y 7 k l ) = ( x 0 − y 0 ) + ( x 1 − y 1 ) i + ( x 2 − y 2 ) j + ( x 3 − y 3 ) k + ( x 4 − y 4 ) l + ( x 5 − y 5 ) i l + ( x 6 − y 6 ) j l + ( x 7 − y 7 ) k l {\displaystyle {\begin{aligned}&x-y\\&=(x_{0}+x_{1}i+x_{2}j+x_{3}k+x_{4}l+x_{5}il+x_{6}jl+x_{7}kl)-(y_{0}+y_{1}i+y_{2}j+y_{3}k+y_{4}l+y_{5}il+y_{6}jl+y_{7}kl)\\&=(x_{0}-y_{0})+(x_{1}-y_{1})i+(x_{2}-y_{2})j+(x_{3}-y_{3})k+(x_{4}-y_{4})l+(x_{5}-y_{5})il+(x_{6}-y_{6})jl+(x_{7}-y_{7})kl\end{aligned}}}
Perkalian x × y = ( x 0 y 0 − x 1 y 1 − x 2 y 2 − x 3 y 3 − x 4 y 4 − x 5 y 5 − x 6 y 6 − x 7 y 7 ) + ( x 0 y 1 + x 1 y 0 + x 2 y 3 − x 3 y 2 + x 4 y 5 − x 5 y 4 − x 6 y 7 − x 7 y 6 ) i + ( x 0 y 2 − x 1 y 3 + y 0 x 2 + x 3 y 1 + x 4 y 6 + x 5 y 7 − x 6 y 4 − x 7 y 5 ) j + ( x 0 y 3 + x 1 y 2 − x 2 y 1 + x 3 y 0 + x 4 y 7 − x 5 y 6 + x 6 y 5 − x 7 y 4 ) k + ( x 0 y 4 − x 1 y 5 − x 2 y 6 − x 3 y 7 + x 4 y 0 + x 5 y 1 + x 6 y 2 + x 7 y 3 ) l + ( x 0 y 5 + x 1 y 4 − x 2 y 7 + x 3 y 6 − x 4 y 1 + x 5 y 0 − x 6 y 3 + x 7 y 2 ) i l + ( x 0 y 6 + x 1 y 7 + x 2 y 4 − x 3 y 5 − x 4 y 2 + x 5 y 3 + x 6 y 0 − x 7 y 1 ) j l + ( x 0 y 7 − x 1 y 6 + x 2 y 5 + x 3 y 4 − x 4 y 3 − x 5 y 2 + x 6 y 1 + x 7 y 0 ) k l {\displaystyle {\begin{aligned}&x\times y=\\&(x_{0}y_{0}-x_{1}y_{1}-x_{2}y_{2}-x_{3}y_{3}-x_{4}y_{4}-x_{5}y_{5}-x_{6}y_{6}-x_{7}y_{7})+\\&(x_{0}y_{1}+x_{1}y_{0}+x_{2}y_{3}-x_{3}y_{2}+x_{4}y_{5}-x_{5}y_{4}-x_{6}y_{7}-x_{7}y_{6})i+\\&(x_{0}y_{2}-x_{1}y_{3}+y_{0}x_{2}+x_{3}y_{1}+x_{4}y_{6}+x_{5}y_{7}-x_{6}y_{4}-x_{7}y_{5})j+\\&(x_{0}y_{3}+x_{1}y_{2}-x_{2}y_{1}+x_{3}y_{0}+x_{4}y_{7}-x_{5}y_{6}+x_{6}y_{5}-x_{7}y_{4})k+\\&(x_{0}y_{4}-x_{1}y_{5}-x_{2}y_{6}-x_{3}y_{7}+x_{4}y_{0}+x_{5}y_{1}+x_{6}y_{2}+x_{7}y_{3})l+\\&(x_{0}y_{5}+x_{1}y_{4}-x_{2}y_{7}+x_{3}y_{6}-x_{4}y_{1}+x_{5}y_{0}-x_{6}y_{3}+x_{7}y_{2})il+\\&(x_{0}y_{6}+x_{1}y_{7}+x_{2}y_{4}-x_{3}y_{5}-x_{4}y_{2}+x_{5}y_{3}+x_{6}y_{0}-x_{7}y_{1})jl+\\&(x_{0}y_{7}-x_{1}y_{6}+x_{2}y_{5}+x_{3}y_{4}-x_{4}y_{3}-x_{5}y_{2}+x_{6}y_{1}+x_{7}y_{0})kl\end{aligned}}}
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