Hypercomplex Numbers - Two-dimensional Real Algebras

Two-dimensional Real Algebras

Theorem: Up to isomorphism, there are exactly three 2-dimensional unital algebras over the reals: the ordinary complex numbers, the split-complex numbers, and the dual numbers.

proof: Since the algebra is closed under squaring, and it has but two dimensions, the non-real basis element u squares to an arbitrary linear combination of 1 and u:

with arbitrary real numbers a₀ and a₁. Using the common method of completing the square by subtracting a₁u and adding the quadratic complement a₁²/4 to both sides yields

so that

The three cases depend on this real value:

• If 4a₀ = −a 2
  1 , the above formula yields ũ2 = 0. Hence, ũ can directly be identified with the nilpotent element of the Dual numbers' basis .
• If 4a₀ > −a 2
  1 , the above formula yields ũ2 > 0. This leads to the split-complex numbers which have normalized basis with . To obtain j from ũ, the latter must be divided by the positive real number which has the same square as ũ.
• If 4a₀ < −a 2
  1 , the above formula yields ũ2 < 0. This leads to the complex numbers which have normalized basis with . To yield i from ũ, the latter has to be divided by a positive real number which squares to the negative of ũ2.

The complex numbers are the only two-dimensional hypercomplex algebra that is a field. Algebras such as the split-complex numbers that include non-real roots of 1 also contain idempotents and zero divisors, so such algebras cannot be division algebras. However, these properties can turn out to be very meaningful, for instance in describing the Lorentz transformations of special relativity.