Example of a non-associative algebra

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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 is a division algebra[edit | edit source]

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:

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

Non associativity of [edit | edit source]

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