Algebra

Algebra originated in the Middle East. Earlier than 1000 BC, the Babylonians developed an arithmetical system for solving problems that could be written algebraically. This was in advance of other systems, notably that of the Ancient Egyptians, who were able to solve the same problems, but did so by using geometry. The word algebra comes from Arabic and translates into English as reunion. It describes a system of mathematics which performs calculations by firstly rewriting, that is, transposing them, and then reducing them to their simplest form.

Algebra is the branch of mathematics which studies the structure of things, the relationship between things and quantity. It looks different from arithmetic when it is written. Arithmetic uses numbers and the four operators (plus, minus, multiply and divide). Algebra uses symbols, usually letters, and the operators. Actually, it is not very different from arithmetic; what can be done in algebra can be done in arithmetic. There, are good mathematical reasons, however, why algebra is used instead of arithmetic.

Firstly, by not using numbers, mathematicians are able to set out arithmetical laws. In this way they are able to understand the system of numbers more clearly. Secondly, by using algebra, mathematicians are able to perform calculations where unknown quantities are involved. This unknown is usually represented by x. Solutions can then be applied not just to the immediate problem, but to all problems of the same nature by the use of a formula. A common algebraic problem to solve in school exams would be, for example: find x where 3x + 8 = 14. A third reason for the use of algebra rather than arithmetic is that it allows calculations which involve change in the relationship between what goes into the problem and what comes out of it, that is, between input and output. It is an algebraic formula which allows a business to calculate its potential profit (or loss) over any period of time.

It is possible to classify algebra by dividing it into four areas. Firstly, there is elementary algebra in which symbols (such as x and y, or a and b) are used to denote numbers. In this area, the rules that control the mathematical expressions and equations using these symbols are studied. Then, there is abstract or modem algebra in which mathematical systems consisting of a set of elements and several rules (axioms) for the interaction of the elements and the operations are defined and researched. Thirdly, there is linear algebra (linear equations) in which linear transformations and vector spaces, including matrices, are studied. Finally, there is universal algebra in which the ideas common to all algebraic structures are studied.

Like all branches of mathematics, algebra has developed because we need it to solve our problems. By avoiding the use of numbers we are able to generalise both the problem and the solution.

Vocabulary

to originate	происходить, брать начало
to be in advance	опережать
notably	исключительно, особенно, весьма
reunion	воссоединение
to transpose	переносить в другую часть уравнения с обратным знаком
to reduce	сводить к ч.-л. (зд.)
to look different from	отличаться от
instead of	вместо
to set out laws	излагать законы (правила)
in this way	таким образом
rather than	а не; скорее … чем
to allow	разрешать, позволять
to denote	обозначать
to consist of	состоять из
to avoid	избегать

 

1. Answer the following questions:

1) Where and when did algebra originate?

2) In what way does it differ from geometry?

3) What does algebra describe?

4) How can we classify algebra?

5) When did you begin learning algebra and how did you do in algebra at school?

6) What does linear algebra deal with?

7) What is modern algebra concerned with?

8) Is algebra or geometry more difficult for you?

 

2. Find the English equivalents for the Russia ones in the right column.

состоять из	to apply to the immediate problem
решать проблему	common to all structures
несколько правил	in advance of other systems
общие для всех структур	over any period of time
применять к непосредственной проблеме	to consist of
отношения между вещами	relationships between things
раньше других систем	to calculate profits and losses
за какой-нибудь период времени	to perform calculations
высчитать прибыль и потери	to solve a problem
сделать вычисления	several rules

 

3. Try to remember the verbs:

to solve, to include, to consist of, to come out of, to perform, to avoid, to denote, to define, to describe, to apply, to represent, to allow, to interact, to use, to transpose, to involve.

 

4. Pair work. Discuss the text. Tell each other what interesting and unknown to you things you have found in the text.