Example of a non-associative algebra

This page presents and discusses an example of a non-associative over the real numbers.

The multiplication is defined by taking the of the usual multiplication: $$a*b=\overline{ab}$$. This is a commutative, non-associative division algebra of dimension 2 over the reals, and has no unit element.

Proof that (C,*) is a division algebra
For a proof that $$\mathbb{R}$$ is a field, see real number. Then, the complex numbers themselves clearly form a vector space.

It remains to prove that the binary operation given above satisfies the requirements of a division algebra for all scalars a and b in $$\mathbb{R}$$ and all vectors x, y, and z (also in $$\mathbb{C}$$).
 * (x + y)z = x z + y z;
 * x(y + z) = x y + x z;
 * (a x)y = a(x y); and
 * x(b y) = b(x y);

For distributivity:


 * $$x*(y+z)=\overline{x(y+z)}=\overline{xy+xz}=\overline{xy}+\overline{xz}=x*y+x*z,$$

(similarly for right distributivity); and for the third and fourth requirements
 * $$ (ax)*y=\overline{(ax)y}=\overline{a(xy)}=\overline{a}\cdot\overline{xy}=\overline{a}(x*y).$$

Non associativity of (C,*)

 * $$a * (b * c) = a * \overline{b c} = \overline{a \overline{b c}} = \overline{a} b c $$
 * $$(a * b) * c = \overline{a b} * c = \overline{\overline{a b} c} = a b \overline{c} $$

So, if a, b, and c are all non-zero, and if a and c do not differ by a real multiple, $$a * (b * c) \neq (a * b) * c$$.