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Division in a binary representation for complex numbers

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Abstract:

Computer operations involving complex numbers, essential in such applications as Fourier transforms or image processing, are normally performed in a 'divide-and-conquer' approach dealing separately with real and imaginary parts. A number of proposals have treated complex numbers as a single unit but all have foundered on the problem of the division process without which it is impossible to carry out all but the most basic arithmetic. This paper resurrects an early proposal to express complex numbers in a single 'binary' representation, reviews basic complex arithmetic and is able to provide a fail-safe procedure for obtaining the quotient of two complex numbers expressed in the representation. Thus, while an outstanding problem is solved, recourse is made only to readily accessible methods. A variety of extensions to the work requiring similar basic techniques are also identified. An interesting side-line is the occurrence of fractal structures, and the power of the 'binary' representation in analysing the structure is briefly discussed.

Document Type: Research Article

DOI: http://dx.doi.org/10.1080/0020739031000108592

Affiliations: 1: School of Mathematics & Physics University of Tasmania Private Bag 1-360 Launceston Tasmania 7250 Australia, Email: D.Blest@utas.edu.au 2: Department of Electrical and Electronics Engineering Sultan Qaboos University Al-Khod 123 Muscat Sultanate of Oman, Email: tjamil@squ.edu.om

Publication date: July 1, 2003

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