What are the main differences of symbolic computation from numerical and why did the term computer algebra?

When talking about computational methods, we believe that all computations are performed in the field of real or complex numbers. In fact, every computer program deals only with a finite set of rational numbers, because such numbers are represented in computer. For writing integer is usually 16 or 32 binary symbols (bits) to real 32 or 64 bits. This set is not closed relative to the arithmetic operations that can be expressed in different overflows, for example, multiplying rather large numbers or division by a small number. Even more significant feature of computational mathematics is that arithmetic operations on these numbers are performed by a computer different from the arithmetic operations in the field of rational numbers, moreover, for computer operations are not met the basic field axioms (associativity, distributivity). These features of computation are expressed in terms of error or accuracy calculations. The error estimate is one of the major problems of computational mathematics. Each problem you want to solve using available computer resources for the foreseeable future, with a given accuracy.

In computer algebra are not applied, bedesteni complex numbers, but widely used algebraic numbers. Algebraic number is defined as its minimal polynomial, and sometimes required to specify specify the interval on the straight line or region in a complex plane, which contains the only root of this polynomial. Polynomials play a character computing a critical role. On the use of polynomial arifmeticheskoe theoretical methods of analytical mechanics; they are used in many areas of mathematics, physics and other Sciences. In addition, computer algebra deals with objects such as differential fields (function fields), allowing for exponential, logarithmic, trigonometric functions, matrix rings (elements of the matrix belong to the rings sufficiently General form), and others. Even in arithmetic operations over such objects is swell information for recording intermediate results of computations requires a significant amount of computer memory.