The set of all complex numbers is denoted by 
.
The square root operation is closed over [8].
As was the case with real numbers,
having a closed square root operation is only partly
responsible for the importance of complex numbers.
Each complex number can be specified as a pair of real numbers:
.
Since there is a simple homomorphism 
,
The algorithms for computing with complex numbers are more intricate than those for real numbers. Even with the more intricate algorithms, this number system has many of the popular properties of the real number system. Addition and multiplication have inverses (partial for multiplication), are associative and commutative, and jointly satisfy the distributive law. Most common operators are closed over the complexes. However, there is no natural ordering relation for complex numbers.
The construction of the complex numbers from the real numbers
can be viewed as an application of a general ``doubling procedure'',
a procedure which creates number systems whose elements
are represented as 
-tuples of real numbers.
This same procedure can be used to construct the quaternion
and Cayley number systems [33].
Common Practice
Complex numbers are very useful in modelling some phenomena. The same difficulties are encountered in computing directly with complex numbers as are encountered computing with real numbers. This is clear since the real numbers are homomorphic to the complex numbers; and conversely, the complex numbers are built from the real numbers in a remarkably simple way. All of the number systems built by application(s) of the doubling procedure can be emulated directly, using the real numbers as the base number system.
| Jeff Tupper | March 1996 |