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Octonion — Definition, Formula & Examples

An octonion is an 8-dimensional number that extends the quaternions, written as a linear combination of one real unit and seven imaginary units e1,e2,…,e7e_1, e_2, \ldots, e_7. Octonions form the largest normed division algebra over the reals, but unlike quaternions, their multiplication is not associative.

The octonions O\mathbb{O} form an 8-dimensional, non-commutative, non-associative, normed division algebra over R\mathbb{R}. Every octonion has the form a0+a1e1+a2e2+⋯+a7e7a_0 + a_1 e_1 + a_2 e_2 + \cdots + a_7 e_7 where ai∈Ra_i \in \mathbb{R} and the imaginary units satisfy ei2=−1e_i^2 = -1, with products among distinct units determined by the Fano plane mnemonic. The octonions satisfy the alternative law — x(xy)=x2yx(xy) = x^2 y and (yx)x=yx2(yx)x = y x^2 — but not full associativity.

Key Formula

x=a0+a1e1+a2e2+a3e3+a4e4+a5e5+a6e6+a7e7x = a_0 + a_1 e_1 + a_2 e_2 + a_3 e_3 + a_4 e_4 + a_5 e_5 + a_6 e_6 + a_7 e_7
Where:
  • a0,…,a7a_0, \ldots, a_7 = Real coefficients of the octonion
  • e1,…,e7e_1, \ldots, e_7 = Imaginary basis units satisfying e_i^2 = -1

How It Works

Each octonion is an 8-tuple of real numbers attached to basis elements {1,e1,e2,…,e7}\{1, e_1, e_2, \ldots, e_7\}. You add octonions component-wise, just like vectors. Multiplication follows specific rules encoded by the Fano plane: for example, e1e2=e3e_1 e_2 = e_3 but e2e1=−e3e_2 e_1 = -e_3, showing non-commutativity. The key surprise is non-associativity: in general (ab)c≠a(bc)(ab)c \neq a(bc), so you must be careful about parenthesization. Every nonzero octonion still has a multiplicative inverse, which is what makes O\mathbb{O} a division algebra.

Worked Example

Problem: Verify that octonion multiplication is non-associative by computing (e1e2)e4(e_1 e_2) e_4 and e1(e2e4)e_1 (e_2 e_4) using the standard Fano-plane products: e1e2=e3e_1 e_2 = e_3, e2e4=e6e_2 e_4 = e_6, e3e4=e5e_3 e_4 = e_5, and e1e6=−e5e_1 e_6 = -e_5.
Step 1: Compute the left grouping: first find e1e2e_1 e_2, then multiply by e4e_4.
(e1e2)e4=e3⋅e4=e5(e_1 e_2) e_4 = e_3 \cdot e_4 = e_5
Step 2: Compute the right grouping: first find e2e4e_2 e_4, then left-multiply by e1e_1.
e1(e2e4)=e1⋅e6=−e5e_1 (e_2 e_4) = e_1 \cdot e_6 = -e_5
Step 3: Compare the two results.
e5≠−e5e_5 \neq -e_5
Answer: (e1e2)e4=e5(e_1 e_2)e_4 = e_5 but e1(e2e4)=−e5e_1(e_2 e_4) = -e_5, confirming that octonion multiplication is non-associative.

Why It Matters

Octonions appear in theoretical physics, particularly in string theory and models of exceptional Lie groups like G2G_2. By the Hurwitz theorem, R\mathbb{R}, C\mathbb{C}, H\mathbb{H}, and O\mathbb{O} are the only normed division algebras over the reals, making octonions the final step in that chain.

Common Mistakes

Mistake: Assuming octonion multiplication is associative because quaternion multiplication is.
Correction: Quaternions are associative but not commutative. Octonions lose associativity as well — you must always respect parenthesization when multiplying three or more octonions.

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