A bicomplex number is an element of the four-dimensional commutative real algebra generated by two distinct imaginary
units 
and 
satisfying 
and 
.
It has the form
 |
where 
and 
belong to the embedded copy of the complex numbers
generated by 
.
Here 
is a formal generator adjoined to that copy of 
, not a variable restricted to
. Thus 
is not another name for 
, the other complex solution of 
. It is an additional square
root of 
in the larger bicomplex algebra.
This distinction is possible because the bicomplex algebra contains zero divisors and therefore is not a field. In particular, 
, although neither factor is zero. Thus
the factorization does not imply 
. Under the isomorphism
with the direct sum 
, an ordinary complex
number 
corresponds to 
,
while 
corresponds to 
and 
corresponds to 
.
It follows directly that 
but 
. In fact, the four bicomplex solutions of 
are 
and 
.
The elements 
,
, 
, and 
form a basis over the real
numbers. On writing 
, every bicomplex number has the real-coordinate form 
, with
 |
Bicomplex multiplication is commutative and associative. The elements 
and 
give another pair of zero divisors, since 
. After relabeling its generating units, the system
is isomorphic, as a real algebra,
to the tessarine system.
