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.
For 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:
![{\displaystyle x*(y+z)={\overline {x(y+z)}}={\overline {xy+xz}}={\overline {xy}}+{\overline {xz}}=x*y+x*z,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0edeba20326263c7f697b1147778e533cb64cd4a)
(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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4964a6cf271c95a27254cf5cfc0f340d0acafc7)
![{\displaystyle a*(b*c)=a*{\overline {bc}}={\overline {a{\overline {bc}}}}={\overline {a}}bc}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90e419a03a85b04252117b5a6008685b28504da1)
![{\displaystyle (a*b)*c={\overline {ab}}*c={\overline {{\overline {ab}}c}}=ab{\overline {c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdfa06a20e426791a5bdd29480e7604bc426d1ec)
So, if a, b, and c are all non-zero, and if a and c do not differ by a real multiple,
.