Search

Publisher Summary This chapter presents concurrent binary relations that occur in various parts of analysis and shows how they can be applied successfully with the use of enlargements. It discusses simultaneous approximation theorems of Dirichlet and Kronecker and applies them to obtain an interesting boundedness condition for finite sets of complex numbers.

Computer operations with complex numbers are usually performed by dealing with the real and imaginary parts separately and combining the two as a final operation. It might be an advantage in some problems to treat a complex number as a unit and to carry out all operations in this form. The number system to be described permits the representation of a complex number as a single binary number to a degree of accuracy limited only by the capacity of the computer. It is binary in that only the two symbols 1 and 0 are used; however, the base is not 2, but the complex number 1 + i. The quantity 1 i would be equally suitable, and, in fact, for real numbers it is immaterial which of these two we consider the base. The first few powers of 1 + i are