# Example of a non-associative algebra

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: . 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

editFor a proof that 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 and all vectors **x**, **y**, and **z** (also in ).

For distributivity:

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

## Non associativity of (C,*)

editSo, if *a*, *b*, and *c* are all non-zero, and if *a* and *c* do not differ by a real multiple, .