Mathematics as a science.
Mathematics is a science which subject-matter is spatial forms and quantitative relationships of the real world. This classic definition of mathematics is being constantly expanded and enriched due to the development of science and engineering that leads to the appearance of new spatial forms and quantitative relationships.
As a part of science mathematics consists of several branches. For the sake of convenience they may be united into two large groups. The first group deals with quantitative relations, the other treats of spatial forms and relations. The former includes mathematical analysis, the theory of functions, algebra, numerical methods, etc.
There are branches of mathematics for which both analytical and geometrical approaches are equally important. All the mathematical sciences are closely connected, their development is interdependent.
Each branch of mathematics is built up on the basis of the same logical patterns; first basic objects, relations and operations are introduced, then axioms are formulated. Axioms are statements about basic properties connecting the introduced objects, relations and operations. New objects grow from the basic ones. Their properties are studied with the help of axioms and theorems which are their logical consequences. The properties of any mathematical theory depend to a large extent on its basic axioms.
An outstanding characteristic of mathematics is the use of its own specific symbolic language. With the help of symbols and signs one can write in a compact form expressions containing much information.
The totality of all abstract mathematical sciences is called pure mathematics. The totality of all particular interpretations of mathematical theories is called applied mathematics. Practical applications are based on solid theoretical foundations, on the other hand acute problems of science and engineering require elaboration of new abstract theories. That is why mathematicians solve not only applied problems but also theoretical ones, which lead to the development of mathematics itself. Pure and applied mathematics together constitute a unified science which is called mathematics.
Mathematics is proved to be extremely useful in solving scientific problems. The range of sciences in which mathematics is used is wide.
Uses and types of computers
The computer is a universal information-processing machine. A little over a decade ago they were large, not very reliable and comparatively slow in operation. Since then they were developed significantly. Now computers are million times faster than humans in performing computing operations.
Computers can perform different tasks. They can “remember” information, store it in their “memory” until it is needed, sort through the stored facts and choose the proper ones, work with lightning speed and so on. However, we cannot call the computer an “electronic brain”, because in order to perform any task it must be supplied with a complete and detailed set of operating instructions.
The most important property of a computer is its many-sidedness. Almost everyday a new use is found for these devices to help man. One and the same computer may be required on different occasions to help in the design of airplane wing sections, to detect errors in the design of a computer, to calculate the production of some large factory, to control the mixture of ingredients in some chemical process, or even to play a game of chess. In fact, a computer may be said to perform any task provided that the method of performance can be described in complete detail. It is important, however, to realize at the outset that the term “electronic brain” popularly applied to an automatic electronic digital computer is a misnomer. The computer is only automatic in the sense that it can deal with explicit instructions which tell it exactly what to do it cannot in itself, take steps, deal with difficulties, which have not been foreseen by the person who presents the problem. The computer must therefore be supplied with a complete and detailed set of operating instructions to solve a given problem together with the numerical values of the quantities which are to be operated upon. A set of such operating instructions causing the computer to perform a particular calculation on any values off numerical data presented to it, and to print the results, of the calculation, is called a programme.