Практикум для подготовки к экзамену по английскому языку (для студентов математического факультета I и II курсов). Жилина Л.В. - 13 стр.

mathematics, despite apparent paradoxes and numerous objections (see            1. Read and translate the text
Hilbert). His program was virtually abandoned in the 1930s when Kurt                                                                        Topic 6
Godel demonstrated that for any general axiomatic system there are
always theorems that cannot be proved or disproved (see Godel).                                           Algebra
       Hilbert's followers, known as formalists, view mathematics in              Algebra from arabic means the reunion of broken parts. The
terms of abstract structures. The oldest philosophy of mathematics is     branch of mathematics, which deals in the most general way with prop-
usually ascribed to Plato. Platonism asserts the existence of eternal     erties and relations of numbers. The first known use of the word is in
truths, independent of the human mind. In this philosophy the truths of   the title of a book by Muhammad ibn-Musa al-Khwarizmi, one of the
mathematics arise from an abstract, ideal reality.                        most important Arab mathematicians of the 9-th century. The story of
                                                                          algebra begins from 1800 B.C. when any problem that we should now
      2. Retell the text                                                  solve by algebra was solved by guessing or by some cumbersome
                                                                          arithmetic process. Then Alexandrian school had appeared and geomet-
                                                                          ric method was in use. In the 3rd century mathematicians began to use
                                                                          symbols to write an equation. And in 17th century algebra already al-
                                                                          lows us the equation ax2+bx+c=0.
                                                                                  Algebra is a generalization of arithmetic. Each state of arithmetic
                                                                          deals with particular numbers.
                                                                                  The square of the sum of any two numbers a and b can be com-
                                                                          puted by the rule (a+b)2 = a +2ab+b. This is a general rule which re-
                                                                          mains true no matter what particular numbers may replace the symbols
                                                                          a and b. A rule of this kind is often called a formula. Algebra is the sys-
                                                                          tem of rules concerning the operations with numbers. The operations of
                                                                          addition, subtraction, multiplication, division, raising to a power are
                                                                          called algebraic expressions. Algebraic expressions consisting of more
                                                                          than one term are called multinomials.
                                                                                  Algebra is the base of modern science, so formulas and expres-
                                                                          sions are widely used not only in mathematics but also in physics,
                                                                          chemistry, biology and others. Many branches of science need to han-
                                                                          dle formulas and expressions automatically. That's why computer alge-
                                                                          bra is the rapidly developed branch of algebra
                                                                                  For numerical calculations the digital computer is now thor-
                                                                          oughly established as a device that can greatly ease the human burden
                                                                          of work. It is less generally appreciated that there are computer pro-
                                                                          grams equally well adapted to the manipulation of algebraic expres-
                                                                          sions. Algebraic programs have three advantages over purely numerical
                                                                          ones. First it is frequently more economical of computer time to sim-
                                                                          plify an expression algebraically before evaluating it numerically. Sec-

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