Equivalent fractions

Multiplying the numerator and denominator of a fraction by the same (non-zero) number, the results of the new fraction is said to be equivalent to the original fraction. The word equivalent means that the two fractions have the same value. That is, they retain the same integrity – the same balance or proportion. This is true because for any number n, multiplying by is really multiplying by one, and any number multiplied by one has the same value as the original number. For instance, consider the fraction : when the numerator and denominator are both multiplied by 2, the result is , which has the same value (0.5) as . To picture this visually, imagine cutting the example cake into four pieces; two of the pieces together ( ) make up half the cake ( ).

For example: , , and are all equivalent fractions.

Dividing the numerator and denominator of a fraction by the same non-zero number will also yield an equivalent fraction. This is called reducing or simplifying the fraction. A fraction in which the numerator and denominator have no factors in common (other than 1) is said to be irreducible or in its lowest or simplest terms. For instance, is not in its lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, is in lowest terms – the only number that is a factor of both 3 and 8 is 1.

Any fraction can be fully reduced to its lowest terms by dividing both the numerator and denominator by their greatest common divisor.For example, the greatest common divisor of 63 and 462 is 21, therefore, the fraction can be fully reduced by dividing the numerator and denominator by 21:

.

Reciprocals and the ‘invisible denominator’

The reciprocal of a fraction is another fraction with the numerator and denominator reversed. The reciprocal of , for instance, is .

Since any number divided by 1 results in the same number, it is possible to write any whole number as a fraction by using 1 as the denominator: 17 = (1 is sometimes referred to as the ‘invisible denominator’). Therefore, except for zero, every fraction or whole number has a reciprocal. The reciprocal of 17 is .

In pairs, look at the highlighted words and phrases. Try to guess what they mean from the context. Then check with your dictionary or teacher. Work out the list of the terms involved, make a kind of glossary.

2. In the text find the definition of:

a. equivalent fractions

b. reducing a fraction

c. irreducible fraction

d. reciprocal

Explain what fractions out of these seven are equivalent.

Say if the following fractions are reducible or in their lowest terms. Explain why.

Unit 7

ALGEBRA

Reading and Vocabulary

1. Match the definitions/explanations in A (1–5) with the words in B (a-e):

Read the article below and complete it with a word from the task 1 (column B).

Algebra is the 1 _____ of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures.

The word algebra comes from the Arabic language (al-jabr, literally, restoration) and much of its methods from Arabic/Islamic mathematics. Its roots can be traced to ancient Indian mathematics, which had a direct influence on Muhammad ibn M sā al-khwārizmī (c. 780–850). He learned Indian mathematics and introduced it to the Muslim world through his famous arithmetic text, Book on Addition and Subtraction after the method of the Indians. He later wrote The Compendious Book on Calculation by Completion and Balancing, which established algebra as a mathematical discipline that is independent of geometry and arithmetic.

Algebra in its simplest2 _____ is a generalized form of arithmetic. Thus algebra usage includes all the definitions and processes of arithmetic.

In arithmetic all the numbers we use are expressed by means of the digits 0, 1, 2, … 9, each of which has a single definite 3 _____.

In algebra besides the ordinary arithmetical numbers we use symbols which usually have not a single definitevalue. Letters of the Latin alphabet are generally used to represent numbers. A number represented by algebraic symbols is called an algebraic expression. An algebraic expression is an expression in which several numbers represented by letters (or by letters and figures) are connected by means of signs. These signs indicate the operations to which the number must be subjected and the 4 _____ of these operations.

Algebra deals with the operations of rational and irrational numbers, algebraic expressions, equations, logarithms, functions, graphs and complex numbers.

The turning point in the history of algebra was the 16-th century. In the 16-th century the French mathematician Viet and later Descartes introduced the systematic use of the first letters of the alphabet for given quantities and the last letters for the unknown. Just as the discovery of zero created the arithmetic of today so did the literal 5 _____ ushered in a new era in the history of algebra.

The names of many other famous mathematicians are connected with the development of algebra because of their great contributionto this branch of mathematics.

The first Russian book containing a certain information on algebra was ‘Arithmetic’ by Magnitsky. The elementary algebra has been taught as a subject at school in Russia since the XVIII century. Lobachevsky, Bunyakovsky, Chebishev touched the problem of teaching algebra too.

Algebra is one of the most rapidly changing areas of mathematics. It is sensitive to the trends which originate in all other branches of mathematics. The most important new demands in algebra come from topology, analysis and algebraic geometry.