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Various number system used in digital computers

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Recap:

Number systems

Importance of binary number system

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Prerequisite

What are various number systems used in computers?

Binary number system. Octal number system.Hexadecimal number system.

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DECIMAL NUMBER SYSTEM

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Decimal number system:

The number system that we use in our daily life is called the decimal number system.

The decimal number system is a base-10 system.

It uses 10 digits, from 0 to 9.

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Decimal number system (contd...) :

Example – (1234)10 subscript 10 denotes decimal number.

The base or radix of a number system is defined as the number system uses ten digits.

The weight of each digit of a decimal number depends on its relative position within the number.

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Weight of each digit of a decimal number:

6498 = 6000 + 400 + 90 + 8

= 6*103 + 4*102 + 9*101 + 8*100

The weight of the 1st digit of the number from right hand

side = 1st digit * 100

The weight of the 2nd digit of the number from right hand

side = 2nd digit * 101

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Weight of each digit of a decimal number (contd) :

The weight of the 3rd digit of the number from right hand

side = 3rd digit * 102

The weight of the 4th digit of the number from right hand

side = 4th digit * 103

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Weight of each digit of a decimal number (contd) :

General expression

The weight of the nth digit of the number from right hand side

= nth digit * 10n-1

= nth digit * (Base)n-1

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BINARY NUMBER SYSTEM

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Binary number system:

Binary number system is used in digital computers.

The base (or radix) of binary system is 2.

It uses only two digits 0 or 1 and these digits are abbreviated as bits.

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Binary number system (contd) :

Example – (11010)2 = (26)10

The weight of each digit of a binary number depends on its relative position within the number.

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Weight of each bit of a binary number:

1101 (binary number) = 1*23 + 1*22 + 0*21 + 1*20

= 8 + 4 + 0 + 1

= 13 (decimal number)

The weight of the 1st bit of the binary number from right hand side = 1st bit * 20

The weight of the 2nd bit of the number from right hand

side = 2nd bit * 21

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Weight of each bit of a binary number (contd) :

The weight of the 3rd bit of the number from right hand

side = 3rd bit * 22

The weight of the 4th bit of the number from right hand

side = 4th bit * 23

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Weight of each bit of a binary number (contd) :

General expression

The weight of the nth bit of the number from right hand side

= nth bit * 2n-1

= nth bit * (Base)n-1

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Binary equivalent of decimal numbers:

Decimal No. Binary equal Decimal No. Binary equal

0 0 11 1011

1 01 12 1100

2 10 13 1101

3 11 14 1110

4 100 15 1111

5 101 16 10000

6 110 31 11111

7 111 32 100000

8 1000 63 111111

9 1001 64 1000000

10 1010 128 10000000

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OCTAL NUMBER SYSTEM

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Octal number system:

Octal means 8.

The base of octal number system is 8.

It uses eight digits 0,1,2,3,4,5,6, and 7.

The decimal number 8 is represented by 10, 9 by 11, 10 by 12 and so in octal number system.

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Octal number system (contd) :

As 8 = 23, each octal digit is represented by a group of three binary bits.

This system was issued to provide a shorthand way to deal with long strings of 0s and 1s created in binary.

Example – (2056)8 = (1070)10

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Binary representation of octal digits:

Decimal Octal number

0

1

2

3

4

5

6

7

8

10

15

0

1

2

3

4

5

6

7

10

12

17

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Conversion of octal to binary:

To convert a binary number to octal.

Group binary digits in group of three.

Convert each group to its equivalent octal digit.

Example – (101110)8= 56

101 11065

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Hexadecimal number system

Hexadecimal means 16(6+10)

It contains 16 symbols

Its base is 16.

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Hexadecimal number system (contd)

The first ten digits are represented by digits 0 to 9 and remaining six by the letters A through F.

Each hexadecimal digit is represented by four binary bits.

Example – (A3)16 = (1010 0011)2

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EQUIVALENT VALUES OF DIFFERENT SYSTEMS

DECIMAL BINARY OCTAL HEXADECIMAL

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

10000

0

1

2

3

4

5

6

7

10

11

12

13

14

15

16

17

20

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

10

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Summary:

In this class, you have learnt about

Various number systems used in Various number systems used in digital computerdigital computer

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Frequently Asked Questions

1. What is meant by base of a number system ?

2. What is the value of the base for decimal, binary, octal and hexadecimal number systems?

3. What are octal and hexadecimal number systems ?

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Quiz

1. The decimal number system is a

a. base 2

b. base 16

c. base 10

Answer: c

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Quiz

2. The base of octal number system is

a. 8

b. 16

c. 2

Answer: a

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Quiz

3. The first ten digits in hexa decimal system are represented by digits 0 to 9 and remaining six by the letters

a. 10 to 16

b. A to F

c. A1 to A6

Answer: b