1. The rapid development of electronic computers with real operands only highlights the modest results available for computers manipulating complex numbers, although complex variables play a truly unique role in modern science and engineering, including many directions crucial for scientific-technical progress. Today, like 30 years ago, virtually the entire complex arithmetic is merely add-on software to the basic operations of real arithmetic. Existing hardware implementations of complex aritl'anetic (processors for discrete or fast Fourier transforms, digital filters, signal processors), like their software analogues, manipulate pairs of real operands representing the real and the imaginary part of a complex variable. The reason for this state of affairs, in our view, is the formal transfer from mathematics [1] to computer science of the interpretation of complex numbers as a pair (a, b) of real numbers with binary operations of additive (addition, subtraction) and multiplicative (multiplication and division) groups [2].This "coordinate" thinking, while quite proper for mathematical proofs, has done a poor service for computational mathematics. It has limited the domain of search for efficient engineering solutions and slowed down the improvement of appropriate hardware tools. Curiously, many authors in computer science regard it as an achievement if they can get rid of an imaginary or complex variable in some algorithm. The development of the well-known Hartley transform also leads to sad thoughts in this context [3]. The point is that this transform, while only partially solving the "superproblem" of the Fourier transform -the algebraization of integral convolution, is presented by the developers as having the advantage of not involving complex numbers.Our point of view on this topic, if not diametrically opposite, is at least essentially different. It is impossible to ignore the multitude of facts when the embedding of a real problem into an equivalent complex problem makes the original problem much more meaningful and leads to a much better solution. There is really no other approach if the problem is inherently complex-valued.Overcoming stereotypes in the development of algorithms and applied programs is a topical issue, which however requires separate consideration. In this article we focus on some issues directly related with development of hardware for complex arithmetic.Mathematics provides three forms of representation of complex numbers [2]: algebraic a + jb, trigonometric r(cosU + jsinU), and exponential rexp(jU). The trigonometric and exponential forms are sometimes identified. There is also matrix representation of complex numbers. In all these representations, the complex number is a two-dimensional vector with rectangular (a, b) or polar (r, U) coordinates. We refer to these forms as vector or coordinate representations of complex numbers.However, alongside vector representations, we can also devise an alternative representation of complex values as numbers in some positional system. This opens...
