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DATA REPRESENTATION

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  • Computer works on 1’s and0’s , in other words data stored inside a computer gets stored in digital form.

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  • 1 and 0 together constitute a bit, which is the smallest piece of information computer can store.

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  • If there is a single wire, there exist the following states: ON/OFF, YES/NO, TRUE/FALSE, 1/0

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1 wire = 2 combinations i.e 0,1

2 1 = 2

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2 wires = 2 2 = 4

4 combinations i.e 00,01,10,11

3 wires = 2 3 = 8 combinations

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number system :

  • Number systems are the technique to represent numbers in the computer system architecture.

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  • When this data is fed to the computer it is converted to numeric format according to the ASCII

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  • American Standard Code for Information Interchange

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  • In digital number system are classified into digits

* Decimal System

* Binary System

* Octal System

* Hexadecimal System

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  • Each digit in the number system depends upon :

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i) Face value of digits

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ii) Base of number system

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iii) Position of number system

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decimal number system

Decimal system has only 10 digits from 0 to 9

Base for Decimal is 10

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Eg : The number 425 is represented as

400 + 20 + 5

4 x 102 + 2 x 101 + 5 x 100

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decimal number system

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Eg : The number 4791.83 is represented as :

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= 4x103 + 7x102 + 9x101 + 1x100 + 8x10-1 + 3x10 -2

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binary number system

Binary system has only 2 digits i.e 0 and 1.

The base for binary number is 2

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So we will learn how to convert Decimal to Binary

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Convert (29)10 to (X)2

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(29)10 = (11101)2 οΏ½

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Convert the following :

1. (43)10 to (X)2

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Convert the following :

  1. (101)10 to (X)2
  2. (200)10 to (X)2
  3. (156)10 to (X)2
  4. (197)10 to (X)2
  5. (255)10 to (X)2

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​

​

​

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Convert the following :

  1. (101)10 to (X)2 - 1100101
  2. (200)10 to (X)2 - 11001000
  3. (156)10 to (X)2 - 10011100
  4. (197)10 to (X)2 - 11000101
  5. (255)10 to (X)2 - 11111111

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​

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7. (3.15)10 to (X)2

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8. (14.625)10 to (X)2

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8. (10.16)10 to (X)2

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Convert binary to decimal

1 . (1100)2 to (X)10

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1 x 23 + 1 x 22 + 0 x 21 + 0 x 20

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8 + 4 + 0 + 0 = 12

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Ans : (1100)2 to (12)10

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Convert binary to decimal

  1. (10101)2 to (X)10
  2. (1100101)2 to (X)10
  3. (10101111)2 to (X)10
  4. (11100101)2 to (X)10

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​

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Convert binary to decimal

  1. (10101)2 to (X)10 - 21
  2. (1100101)2 to (X)10 101
  3. (10101111)2 to (X)10 175
  4. (11100101)2 to (X)10 229

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​

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Convert binary to decimal

4. (101.110)2 to (X)10

1 x 22 + 0 x 21 + 1 x 20 + 1 x 2-1 + 1 x 2-2 + 0 x 2-3

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= 4 + 0 + 1 + 0.5 + 0.25 + 0

= 5.75

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Ans : (101.110)2 to (5.75)10

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2.(1010.0101) 2 ---(10.3125) 10

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OCTAL NUMBER SYSTEM

  • Octal Number System has base 8

and numbers from (0 – 7)

Convert from decimal to octal

1. (125)10 to (X)8

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2. (670)10 to (X)8οΏ½οΏ½οΏ½

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2. (670)10 to (1236)8οΏ½

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3. (0.218)10 to (x)8οΏ½

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3. (0.218)10 to (x)8οΏ½

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Decimal

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​

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Binary

Octal

Hexa decimal

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4. (670)8 to (X)2

X

Y

Z

N

0

0

0

0

0

0

1

1

0

1

0

2

0

1

1

3

1

0

0

4

1

0

1

5

1

1

0

6

1

1

1

7

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4. (670)8 to (X)2

  • 110111000

X

Y

Z

N

0

0

0

0

0

0

1

1

0

1

0

2

0

1

1

3

1

0

0

4

1

0

1

5

1

1

0

6

1

1

1

7

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5. (347)8 to (X)2

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6. (56.27)8 to (X)2

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7. (101110)2 to (X)8

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8. (1110)2 to (X)8

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9. (10111.001)2 to (X)8

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10. (10110110.0101)2 to (X)8

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5. (347)8 to (X)2 011 100 111

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6. (56.27)8 to (X)2 101110.010111

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7. (101110)2 to (X)8 56

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8. (1110)2 to (X)8 16

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9. (10111.001)2 to (X)8 27.1

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10. (10110110.0101)2 to (X)8 266.24

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HEXADECIMAL NUMBER SYSTEM

  • A Hexadecimal system has 16 values from 0-9 and A-F.

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  • 0 1 2 3 4 5 6 7 8 9 A B C D E F

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A

B

C

D

N

0

0

0

0

0

0

0

0

1

1

0

0

1

0

2

0

0

1

1

3

0

1

0

0

4

0

1

0

1

5

0

1

1

0

6

0

1

1

1

7

1

0

0

0

8

1

0

0

1

9

1

0

1

0

A

1

0

1

1

B

1

1

0

0

C

1

1

0

1

D

1

1

1

0

E

1

1

1

1

F

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convert the following

  1. (4AF)16 to (X)2

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  • (101101.10)2 to (X)16

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  • (BAD)16 to (X)8

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  • (345)8 to (X)16

5. (179)16 to (X)10

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convert the following

  1. (4AF)16 to (X)2 010010101111
  2. (101101.10)2 to (X)16 2D.8
  3. (BAD)16 to (X)8 5655
  4. (345)8 to (X)16 E5
  5. (179)16 to (X)10 377

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BINARY ADDITION

  • ALU doesn’t directly work with decimal numbers, they work with binary numbers
  • There are 5 basic cases of binary addition

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CASE 1 : 0 + 0 = 0

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CASE 2 : 0 + 1 = 1

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CASE 3 : 1+ 0 = 1

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CASE 4 : 1 + 1 = 10 equals 0 with a carry 1

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CASE 5 : 1 + 1 + 1 = 11 equals 1 with a carry 1

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ADD THE FOLLOWING USING BINARY ADDITION

  1. 101 + 110

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1 0 1

1 1 0

10 1 1

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ADD THE FOLLOWING USING BINARY ADDITION

  1. 1011 and 110
  2. 15 + 27
  3. 01010111 and 00110101
  4. 11110 and 11
  5. 11.01 and 101.11

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  1. 1011 and 110 - 10001
  2. 15 + 27 - 101010
  3. 01010111 and 00110101 - 10001100
  4. 11110 and 11 - 100001
  5. 11.01 and 101.11 - 1001.00