Real Normed Algebra
A real normed algebra, also called a composition algebra, is a multiplication 
on 
that respects
the length of vectors, i.e., 
for
.
The only real normed algebras with a multiplicative identity are the real numbers 
, complex
numbers 
, quaternions 
, and octonions 
(Koecher and Remmert 1988).
Hurwitz (1898) proved that a real normed algebra must have dimension 
, 2, 4, or 8.
There are four real normed algebras of dimension 2: the complex
numbers and three others (Koecher and Remmert 1988).
Real normed algebras have no zero divisors since the equation 
implies that 
.
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