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2.4 Complex Numbers

The set of all complex numbers is denoted by   tex2html_wrap_inline32165 . The square root operation is closed over tex2html_wrap_inline32165 [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:

figure6220

The first element of each pair is the ``real'' part of a complex number, while the second element of each pair is the ``imaginary'' part of a complex number. The pairs can be written as above, with the imaginary part written as a real multiplied by i, tex2html_wrap_inline32307 .

Since there is a simple homomorphism tex2html_wrap_inline32309 ,

math6229

the real numbers are often viewed as a subset of the complex numbers [64].

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 tex2html_wrap_inline32311 -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.


next up previous notation contents
Next: 2.5 Floating Point Up: 2 Numbers Previous: 2.3 Real Numbers
Jeff TupperMarch 1996