Lesson 3 number systems

1. Number names were among the first words used when people began to talk, but it has taken thousands of years for people to learn how to use numbers or the written figures which we call "numerals".

2. In early days people often counted on a scale of three or four instead of ten and sometimes other number scales were used. Later they found it more convenient to count by tens, using the fingers of both hands. We still use this "scale" in counting; that is, we count to ten; then to ten tens; then to ten times ten tens; and so on.

3. When people first began to use numbers they knew only one way to work with them; that was to count. Later they found out how to add, subtract, and multiply. They invented special devices to make computations easier, especially in dealing with large numbers.

4. Division was not often needed in ancient times in compar­ison with multiplication, and it is far more rarely employed than addition and subtraction at the present time. In ancient times it was very difficult to use fractions" because they did not know any easy way of writing them. Decimal fractions were introduced in the 16th century and are now much more commonly used than common fractions.

5. Besides the decimal number system which we commonly use there are other systems of numeration such as the binary, octal, etc. In the number systems we use digits. The Latin word digiti means "fingers".

6. The binary number system can best be described by analogy to the decimal number system. The decimal number system utilizes the arabic numerals 0 to 9 in a shorthand notation for certain arithme­tic operations. Each digit in a decimal number is actually multiplied by a power of 10; the various products are then summed together to yield the decimal quantity.

7. In the binary number system, the arabic numerals 0 and 1 are the only digits used. By combining these binary digits, or bits, in the proper manner, decimal numbers can be represented. For the pre­sent analysis, only integer decimal values will be considered. In order to represent a decimal number, the bits 0 and 1 are written in a se­quence. The various bits in a binary sequence are multiplied by power of 2, and the position of a bit determines the power of 2 by which the bit is to be multiplied. The sequence is often written from right to left, with the most significant bit to the extreme left.

Exercise1. Find the synonyms from the text to the following words:

1. start (1) 6. calculations (3)

2. sign (1) 7. seldom (4)

3. calculate (2) 8. to use (6)

4. comfortable (2) 9. to give (math.) (6)

5. machine tool, apparatus (3) 10. important (7).

Exercise 2. Give the English equivalents to the following phrases:

1. минуло тисячi років, коли люди дізналися, як користуватися цифрами

2. у давні часи

3. лічити десятками

4. у порівнянні з

5. зазвичай використовуються

6. зробити обчислювання більш

7. десяткова кількість

8. десяткова величина

Exercise 3. Say if the statements are true or false:

1. In early days people counted by tens using their hands.

2. People invented special devices to make counting easier. These machines learnt them to add, to subtract and multiply.

3. There are only two number systems: the decimal and the binary ones.

4. The decimal quantity is yielded by summing together various products.

5. The Arabic numerals from 0 to 9 are the only digits used in the binary number system.

Exercise 4. Complete the following sentences choosing the right variant:

1. In ancient times people use… rarely.

a). multiplication

b). addition

c). division.

2. They avoided using fractions, because…

a). couldn’t write them properly

b). couldn’t count them properly

c). couldn’t read them properly.

3. The binary system utilizes...

a). the Arabic numerals 0 to 9

b). the Arabic numerals 0 and 1

c). the Arabic numerals 0 and 10.

4. The digits in a decimal number are multiplied …

a). by a power of 2

b). by a power of 10

c). by a power of 0.

5. The most significant bit in a binary sequence is…

a). at the beginning

b). at the end

Exercise 5 Arrange the following facts in chronological order:

1. People introduced decimal fractions.

2. People found out how to add, subtract, multiply.

3. They found it more convenient to count by tens, using fingers of both hands.

4. People used numbers when they began to talk.

5. They learnt how to use written figures.

6. People counted on a scale of three or four.

7. They invented special devices to make calculations easier.

Exercise 6. Answer the following questions:

1. What scales were used by people in ancient times?

2. What number systems are commonly used?

3. Why are decimal fractions much more commonly used than common fractions? When were decimal fractions introduced?

4. What numerals does the decimal/ binary system utilize?

5. How can decimal numbers be represented?

Exercise 7. Describe the history of number usage.

Exercise 8. Explain why decimal fractions were introduced only in the 16^th century.

Exercise 9. Define the types of number systems. Analyze their differences and similarities.

Exercise 10. Discuss the text according to the points:

1. The history of number usage.

2. The main types of number systems.

3. Digits, digit sequences in the decimal and the binary systems.

Notes:

1. figure = number = digit = sign

2. numerals – числівники

3. scale – шкала

4. finger – палець

5. comparison – порівняння

6. rarely employed = seldom used

7. common / decimal fractions – прості та десяткові дробі

8. notation = concept

9. power ( by a power) – ступінь

10. product – добуток

11. to yield – створювати

12. sequence - послідовність.