Work in pairs. Take turns to give examples of everyday use of numbers in our lives. Look at the pictures and say the time.

READING

3. Read the text and find the expressions which mean:

a. the totality of the rational and irrational numbers;

b. a positional numeral system employing 10 as the base and requiring 10 different numerals, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and a dot (decimal point);

c. the positive and negative integers and fractions, and the number 0.

Numbers

Number is a word or symbol used to designate quantities.

RATIONAL NUMBERS. The simplest numbers are the natural numbers, 1, 2, 3,.... The natural numbers are also called the whole numbers, or positive rational integers. The natural numbers are closed with respect to addition and multiplication—that is, the sum and product of two natural numbers are always natural numbers. Because the quotient (the result of dividing) of two natural numbers, however, is not always a natural number, it is convenient to introduce the positive fractions to represent the quotient of any two natural numbers. The natural number n is identified with the fraction n /1. Furthermore, because the difference of two positive fractions is not always a positive fraction, it is expeditious to introduce the negative fractions and the number zero (0). The positive and negative integers and fractions, and the number 0, comprise the rational number system.

IRRATIONAL NUMBERS. The development of geometry indicated the need for more numbers; the length of the diagonal of a square with sides one unit long cannot be expressed as a rational number. Similarly, the ratio of the circumference to the diameter of a circle is not a rational number. These and other needs led to the introduction of the irrational numbers. For example, Ã = 1.4142135623... and p = 3.1415926535... are irrational numbers. The totality of the rational and irrational numbers makes up the real number system.

NUMBER SYSTEM is any of various sets of symbols and the rules for using them to express quantities as the basis for counting, comparing amounts, performing calculations, determining order, making measurements, etc.

Our modern decimal, or base-10, number system is a positional numeral system employing 10 as the base and requiring 10 different numerals, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and a dot (decimal point).

The origins of the decimal number system can be traced to ancient Egyptian, Babylonian (Sumerian), and Chinese roots. The bulk of the credit for the base-10 system, however, complete with a symbol for zero, goes to the Hindu mathematicians of the fifth and sixth centuries. After their discovery of the system it was transmitted to mathematicians in the Islamic world who developed it to include decimal fractions during the period from the 9th to 11th centuries AD. The introduction of the system into the West took place with the translation of the treatise of Muḥammad ibn Mūsāal-Khwārizmī (780–850) in Spain in the twelfth century.

 

LANGUAGE DEVELOPMENT

 

4. Find Ukrainian equivalents to the following English words and word combinations:

to designate quantities, natural number, positive rational integer, addition, multiplication, quotient, positive fraction, difference, expeditious, negative fraction, rational number system, irrational number, introduction, totality, real number system, performing calculations, determining order, making measurements, digit, decimal point, ancient, century, transmit, translation.

 

5. Match the English phrases with their Ukrainian equivalents:

1. natural number a) ірраціональне число

2. positive fraction b) система чисел

3. rational integer c) крапка у десятковій дробі

4. decimal point d) додатна дріб

5. number system e) натуральне число

6. rational number f) раціональне ціле число

7. irrational number g) раціональне число

 

6. Complete the following sentences using the information from the text:

1. The bulk of the credit for the base-10 system …….

2. The development of geometry indicated the …….

3. Number system is any of various sets of …….

4. The natural numbers are closed with …….

5. Our modern decimal, or base-10, number system is …….

6. Because the result of dividing of two natural numbers …….

7. The introduction of the system into the West …….

 

7. Read the clues and complete the crossword:

^1

^2

^3

^4

^5

^6

^7

^8

 

Across:   The method of combining two numbers by adding one of them to itself as many times as the other states.   The act of adding numbers together.   The fact of being measurable; an amount or number.   A letter, sign or figure which expresses a sound, operation, number, chemical substance, etc.   A total amount; sum.   The name of the sign 0 and of the number it stands for.

Down:   The act of measuring; a length, height, found by measuring.   A division or part of a whole number.   A number which is the result when one number is divided by another.   A group of related parts which work together forming a whole; an ordered set of ideas, methods or ways of working.   A whole number.

 

8. Decide whether the statements below are true or false:

1. The origins of the real number system can be traced to ancient Egyptian, Babylonian (Sumerian), and Chinese roots.

2. The natural numbers are also called the whole numbers, or positive irrational integers.

3. The natural number n is identified with the positive fraction n /1.

4. The totality of the rational and irrational numbers puts together the real number system.

5. Number is a letter or symbol used to designate quantities.

6. Similarly, the ratio of the circumference to the radius of a circle is not a rational number.

7. The positive and negative integers and fractions, and the number 0, form the rational number system.

T	T	T	T	T	T	T
F	F	F	F	F	F	F

 

LISTENING

Listen and practise.

Money

£ 400 50 p? 9.40? 47.99 ¥ 5,000 $ 100

Fractions

1/4 3/4 2/3 7/8 12 ½

Decimal and percentage

6.2 17.25 50% 75.7% 100%

Dates

1995 2020 1789 15/7/94 30/10/02

Phone numbers

01865-556890 800 451-7545 919 677-1303