Symbolic computing
The process of manipulating numbers and variables according to the rules of mathematical logic. Variables are used to represent a real number or a set of real numbers to exact precision. Numbers can be added, subtracted, multiplied, and divided. Numerical calculations or numerical computations involve performing these operations. For example, if the symbol * is used to represent multiplication and / is used to represent division, then the equation 
represents a numerical computation whose result is 2. In mathematics, it is also possible to use symbols to represent numbers and to do computations with the symbols. For example,
In this calculation or computation, the mathematical rules for subtraction and division are obeyed. The symbol x is treated as an unknown or variable that is to be solved for, and a and b as parameters representing unknown but fixed numbers. In the 1990s, mathematical software packages became available that could manipulate symbols according to the rules of mathematics. They are also called symbolic computational packages and are programming environments, since any real number has a decimal expansion. These packages have the ability to obtain exact answers as symbols. For example, if 
were entered into a symbolic computation package, the software package would return the answer 
The software package also allows the user to then ask for an approximation to 2 to any number of digits. The decimal units for 2 are determined using an algorithm that performs numerical calculation (called a numerical algorithm). Symbolic computation packages allow both symbolic and numerical computation. See Algorithm, Computer programming, Mathematical software
The capability of small computers and hand-held calculators to do extremely complicated calculations in a small amount of time led to the development of mathematical software packages that could do symbolic manipulation. These symbolic manipulation packages (also called computer algebra systems or symbolic computation systems) can do algebra, trigonometry, number theory, calculus, ordinary and partial differential equations, matrix algebra, and other areas of mathematics by manipulating the symbols according to the rules of mathematics much faster than armies of mathematicians. The answers are given exactly in symbolic form and can then be approximated numerically using routines in the package.
The advent of visualization on the computer has made it possible for these packages to plot data or functions in many forms. The functions are entered symbolically, and the software evaluates them numerically so that numbers can be plotted. The graphs generated by the plot routines can be labeled, rotated, or animated, among many other possibilities.
Hand-held calculators can perform symbolic manipulation, graph data or functions, and be programmed to carry out a sequence of operations. Mathematical software packages have been developed that can perform symbolic manipulation, graph data or functions, and be programmed using routines in the package. These packages run on most commonly used operating systems. They can also be called by computer programs to perform manipulations in the program. There are packages that do statistics, computational finance, computational physics, and computational chemistry. See Operating system, Programming languages
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