A split-complex number, also called a hyperbolic number, has the form
 |
where 
and 
. Addition and multiplication make the split-complex
numbers a two-dimensional commutative algebra
over the real numbers. This algebra
is not a field, since 
, so 
and 
are zero divisors.
The map 
identifies the split-complex algebra
with the direct sum 
. The image is an ordered
pair, not a choice between the two numbers 
and 
, and the inverse map
sends the ordered pair 
to 
. Under this identification,
addition and multiplication are performed componentwise. In particular, 
maps to 
, which is distinct from
both 
and 
even though 
. The tessarine and
bicomplex number systems each contain a copy
of the split-complex numbers.
