Introduction to Number System

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Number System

When we type some letters or words, the computer translates them in binary numbers as computers can understand only binary numbers.

Decimal number system has base 10 as it uses 10 digits from 0 to 9. In decimal number system, the successive positions to the left of the decimal point represent units, tens, hundreds, thousands and so on.

A value of each digit in a number can be determined using

The digit Symbol value (is the digit value 0 to 9) The position of the digit in the number Increasing Power of the base (i.e. 10) occupying

successive positions moving to the left

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Example

Decimal number (592):

Number Symbol Value

Position from the right end

Positional Value

Decimal Equivalent

5 9 22

9

5

0

1

2

100

101

102

2*100 = 2

9*101 = 90

5*102 = 500 592eITnotes.com

Binary number system

Uses two digits, 0 and 1. Also called base 2 number system

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(110011)2 = (51)10

Number Symbol Value

Position from the right end

Positional Value

Decimal Equivalent

1 1 0 0 1 1

1

1

0

0

1

1

0

1

2

3

4

5

20

21

22

23

24

25

1*0 = 1

1*2 = 2

0*4 = 0

0*8 = 0

1*16= 16

1*32= 32 51

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Cont…

A Decimal number can converted into binary number by the following methods:

Double-Dabble Method Direct Method

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Double-Dabble Method

Divide the number by 2 Write the dividend under the number

. This become the new number Write the remainder at the right in

column Repeat these three steps until a ‘0’ is

produced as a new number Output (bottom to top).

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Convert decimal 17 into binary number

Step Remainder

1 Divide 17 by 2 2 17 8

1

2 Divide 8 by 2 2 8 4

0

3 Divide 4 by 2 2 4 2

0

4 Divide 2 by 2 2 2 1

0

5 Divide 1 by 2 2 1 0

1

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Direct Method

Write the positional values of the binary number

…. 26 25 24 23 22 21 20

…. 64 32 16 8 4 2 1 Now compare the decimal number with position

value listed above. The decimal number lies between 32 and 64. Now place 1 at position 32.

64 32 16 8 4 2 1

1 Subtract the positional value to the decimal

number i.e ( 45-32=13)

45

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Cont..

64 32 16 8 4 2 1

1 45-32 =13

1 1 13-8=5

1 1 1 5-4=1 1 1 1 1 1-1=0Place 0 at the rest of position value 0 1 0 1 1 0 1 (45)10=(101101)2

45

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Decimal number to fractional Binary number Multiply the decimal fraction by 2 Write the integer part in a column The fraction part become a new

fraction Repeat step 1 to 3 until the fractional

part become zero. Once the required number of digits

(say 4) have been obtained , we can stop.

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Example

Decimal number is (0.625)

Ans: (0.625)10= (0.101)2

Fractional decimal number

Operation

Product Fractional part of product

Integer part of product

0.625 Multiply by 2

1.250 .250 1

0.250 -do- 0.500 .500 0

0.500 -do- 1.000 0 1

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Questions

Convert decimal 89 into equivalent binary number by using Double-Dabble Method

(89)10= (1011001)2

Convert decimal 89 into equivalent binary number by using Direct Method

(89)10= (1011001)2

Convert decimal 0.8125 into fractional binary number

(0.8125)10 = (0.1101)2

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Convert Binary to Decimal Direct Method Double Dabble Method

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Direct Method

Binary Number

Positional value

operation

1 1 1 0 0 1 0 1 1

0

1

0

0

1

1

1

1*20

0*21

1*22

0*23

0*24

1*25

1*26

1*27

1

0

4

0

0

32

64

128 = 229eITnotes.com

Double Dabble Method

Multiply left most digit by 2 add to the next digit and so on.

1 1 0 1

2+ 1 0 1 3 0 1

6+ 0 1

6 1

12+ 1 13

(1101)2= (13)10

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Example

Convert Binary number 10111011 to decimal

(10111011)2 = (187)10

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Convert fractional Binary number to Fractional Decimal number Write out the binary number as (-)ve

power of two. The various digits positions after binary points are 1,2,3,4…..and so on.

Convert each power of two into its decimal equivalent

Add these to give the decimal number

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Example

. 1 0 1 1

1*2-1 0*2-2 1*2-3 1*2-4

0.5 + 0 + 0.125 + 0.0625

= 0.6875

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Questions

Convert the fractional binary number to decimal number

(0.1101) ans= 0.8125 (0.1011) ans= 0.6875

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Octal number notation

Octal is base 8 counting system having digit values 0 through 7

The octal system groups three binary bits together into one digit symbol.

Octal Binary

0 000

1 001

2 010

3 011

4 100

5 101

6 110

7 111eITnotes.com

Convert binary number into octal Divide the given binary number into

group of three bits (from right to left) Replace each group by its octal

equivalent Examples: 11001 101010001110

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Convert decimal to octal

Divide the number by 8 Write the dividend under the

number. This become the new number

Write the remainder at the right in a column

Repeat steps 1 to 3 until a ‘0’ is produced as a new number

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Question

Convert decimal 17 to octal number Ans= (17)10 = (21)8

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Convert octal to decimal number

Write out the octal digits as power of 8

Convert each power of 8 into its decimal equivalent term

Add these terms to produce the required decimal number

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Example

(721)8= (465)10

Ques: Convert the octal 131 to its equivalent decimal number

ans: 89

7 2 1

=7*82

=448

465

2*81

16

1*80

1

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Hexadecimal

Hexadecimal number system is a base 16 counting system

It uses 16 Symbols: 0 to 9 and the capital letter A,B…F.

Each Hexadecimal is equivalent to a group of 4 binary bits.

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Hexadecimal

Binary Hexadecimal

Binary

0 0000 8 1000

1 0001 9 1001

2 0010 A 1010

3 0011 B 1011

4 0100 C 1100

5 0101 D 1101

6 0110 E 1110

7 0111 F 1111

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Convert binary to Hexadecimal

Divide the given binary number into groups of 4 bits each(from right to left).

Replace each group by its hexadecimal Equivalent.

Questions:1.Convert (101111100001)2 into its

hexadecimal. Ans: (BEI)16.

2. Convert (10101111.0010111)2 into its hexadecimal.

Ans: (AF.2E)16eITnotes.com

Convert Decimal to Hexadecimal Divide the number by 16. Write the dividend under the number.

This become the new number. Write the remainder at the right in a

column. Repeat steps 1 to 3 until a ‘0’ is

produced as a new number.Question: Convert the Decimal 87 to

hexadecimal number. (87)10= (57)16

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Convert hexadecimal to Decimal Write out the Hexadecimal digits as

power of 16. Convert each power of 16 into its

decimal equivalent term. Add these terms to produce the

required decimal number.Question: (A2D)16=(2605)10

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Data Representation

We known that computer work with binary numbers and therefore the numbers, letters, and other symbols have to be converted into their binary equivalents.

However, this is not enough in the sense that still we do not know how to store this binary information so that it become suitable for computer processing.

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Cont..

The Representation of a positive integer number is quite straight forward but we are interested to represent positive as well as negative numbers.

For a Positive number , the sign bit set to 0 and for negative number the sign bit is set to 1.

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Integer Representation

An integer can be represented by fixed point representation

The left most bit is considered as sign bit.

The magnitude of the number can be represented in following three ways:

1. Signed magnitude representation.

2. Signed 1’s complement representation.

3. Signed 2’s complement representation.

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Signed Magnitude

In this representation , if n bit of storage is available then 1 bit is reserved for sign and n-1 bits for the magnitude.

The Disadvantage of this representation is that during addition and Subtraction, the sign bit has to be considered along with the magnitude.

Sign

bitmagnitude

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Signed 1’s Compliment

The 1’s Compliment of a binary integer can be obtained by simply replacing the digit 0 by 1 and digit 1 by 0

Example: 00001100 is 11100111

0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0

(+0)1

0

(-0)10

0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1

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Signed 2’s Compliment The 2’s Compliment of a binary number is

obtained by adding 1 to 1’s Compliment. Example: (+12)10= 1100

11110011 1’s Compliment 1 11110100 2,s Compliment

Therefore, Positive integer 2’s compliment is the negative integer

0 0 0 0 1 1 0 0

1 1 1 1 0 0 1 1 1’s

1 1 1 1 0 1 0 0(-

12)10

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Question

Express the following in signed magnitude form, 1’s Compliment, 2’s Compliment:

(35)10 = 100011

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Floating point representation We can represent a floating point binary number in

the following form: ±M * 2±e

Where M : is the mantissa or significant e : is the exponent Example: 101.11 10111 * 2-2

101.11 * 20

10.111 *21

1.0111 *22

.10111 * 23

.010111 * 24

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Cont.. .10111 * 23

M e The Mantissa part of the number is suitably

shifted (left or right) to obtain a non zero digit at a most significant position. The activity is known as normalization.

In a 16 bit representation, let us assume that 10 bits are reserved for mantissa and 6 for exponent.

Sign Sign

Mantissa exponent0 1 0 1 1 1 0 0 0 0 0 0 0 0 1 1

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Question

Represent floating point binary number in 16 bit representation (1110.001)

The normalization number is = .1110001 * 24

16 bit representation: Sign Sign

0 111000100 0 00100 M e

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