# Example of a non-associative algebra

Template:Mergeto {{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} This page presents and discusses an example of a non-associative division algebra over the real numbers.

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

## Proof that ${\displaystyle (\mathbb {C} ,*)}$ is a division algebra

For a proof that ${\displaystyle \mathbb {R} }$ 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

• (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 all scalars a and b in ${\displaystyle \mathbb {R} }$ and all vectors x, y, and z (also in ${\displaystyle \mathbb {C} }$).

For distributivity:

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

(similarly for right distributivity); and for the third and fourth requirements

${\displaystyle (ax)*y={\overline {(ax)y}}={\overline {a(xy)}}={\overline {a}}\cdot {\overline {xy}}={\overline {a}}(x*y).}$

## Non associativity of ${\displaystyle (\mathbb {C} ,*)}$

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