NCERT Computer Science Class 11 - Chapter 2 Solutions - part1

 Chapter 2 - Encoding Schemes and Number System

Exercise Solutions

1. Write base values of binary, octal and hexadecimal number system.

The base values of different number systems are as follows:

Binary Number System:	Base 2
Octal Number System:	Base 8
Hexadecimal Number System:	Base 16

These systems use different sets of symbols to represent values, with binary using 0 and 1, octal using digits 0-7, and hexadecimal using digits 0-9 and letters A-F.

2. Give full form of ASCII and ISCII.

ASCII	American Standard Code for Information Interchange
ISCII	Indian Script Code for Information Interchange

3. Try the following conversions:

(i) (514)₈ = (?)₁₀

To convert the octal number (514)₈ to decimal, we can use the positional value of each digit in the octal system, which is base 8. Each digit is multiplied by 8 raised to the power of its position (counting from right to left, starting at 0).

Conversion Steps:

- Identify the digits:  5, 1, 4

- Assign positional values:

• 5 is in the 8² place
• 1 is in the 8¹ place
• 4 is in the 8⁰ place

-  Calculate the decimal value:

(514)₈  =  5×8²  +  1×8¹  +  4×8⁰

            =  5×64  +  1×8  +  4×1

            =  320  +  8  +  4

            =  332

(ii) (220)₈   =   (?)₂

Lets convert from Octal to Decimal first:

(220)₈​  =   2×8²   +   2×8¹   +   0×8⁰

            =   2×64   +  2×8     +   0×1

            =   128     +  16       +      0

            =   144

Now, convert to Binary:

To convert the decimal number 144 to binary, you can use the method of successive division by 2.

Here’s how it works:

- Divide the number by 2 and record the quotient and the remainder.

- Continue dividing the quotient by 2 until the quotient becomes 0.

- The binary equivalent is the remainders read in reverse order (from bottom to top).

Division	Quotient	Remainder
144 ÷ 2	72	0
72 ÷ 2	36	0
36 ÷ 2	18	0
18 ÷ 2	9	0
9 ÷ 2	4	1
4 ÷ 2	2	0
2 ÷ 2	1	0
1 ÷ 2	0	1

Now, read the remainders from bottom to top:  10010000

(iii) (76F)₁₆   =   (?)₁₀

Follow below steps:

Identify the values of each digit:

• 7 corresponds to 7
• 6 corresponds to 6
• F corresponds to 15

Apply the positional values:

Each digit in a hexadecimal number is multiplied by 16ⁿ, where n is the position of the digit from right to left, starting at zero:

• For 7

                7 × 16²  =  7 × 256  =  1792

• For 6

                6 × 16¹  = 6 × 16 = 96

• Fo F

                15 × 16⁰  =  15 × 1  =  15

Sum the results

                  1792  +  96  +  15  =  1903

Thus, the decimal equivalent of (76F)₁₆​ is 1903

(iv)  (4D9)₁₆    =   (?)₁₀

Identify the values of each digit:

• 4 corresponds to 44
• D corresponds to 13
• 9 corresponds to 9

Apply the positional values:

Each digit in a hexadecimal number is multiplied by 16ⁿ, where n is the position of the digit from right to left, starting at zero:

• For 4

            4  ×  16²  =  4  ×  256  =  1024

• For D

            13  ×  16¹  =  13 × 16  =  208

• For 9

            9  ×  16⁰  =  9×1  =  9

Sum the results

            1024  +  208  +  9  =  1241

            

Thus, the decimal equivalent of (4D9)₁₆​ is 1241.

(v)  (11001010)₂   =   (?)₁₀

Step 1: Identify the values of each digit

Each digit in a binary number represents a power of 2, starting from the rightmost digit (which is 2⁰).

Postion from right	7	6	5	4	3	2	1	0
Binary Digit	1	1	0	0	1	0	1	0

Step 2: Calculate the decimal value

        1  ×  2⁷  =  1  ×  128  =  128

        1  ×  2⁶  =  1  ×  64  =  64

        0  ×  2⁵  =  0  ×  32  =  0

        0  ×  2⁴  =  0  ×  16  =  0

        1  ×  2³  =  1  ×  8  =  8

        0  ×  2²  =  0  ×  4  =  0

        1  ×  2¹  =  1  ×  2  =  2

        0  ×  2⁰  =  0  ×  1  =  0

Step 3: Sum the results

        128  +  64  +  0  +  0  +  8  +  0  +  2  +  0  =  202

Thus, the decimal equivalent of the binary number 11001010 is 202.

(vi) (1010111)₂  =  (?)₁₀

Step 1: Identify the values of each digit

Each digit in a binary number represents a power of 2, starting from the rightmost digit (which is 2⁰).

Position from right	6	5	4	3	2	1	0
Binary Digit	1	0	1	0	1	1	1

Step 2: Calculate the decimal value

        1  ×  26  =  1  ×  64  =  64

        0  ×  25  =  0  ×  32  =  0

        1  ×  24  =  1  ×  16  =  16

        0  ×  23  =  0  ×  8  =  0

        1  ×  22  =  1  ×  4  =  4

        1  ×  21  =  1  ×  2  =  2

        1  ×  20  =  1  ×  1  =  1

Step 3: Sum the results

        64  +  0  +  16  +  0  +  4  +  2  +  1  =  87

Thus, the decimal equivalent of the binary number 1010111 is 87. 

4. Try the following conversions:

(i) (54)₁₀  =   (?)₂

Step-by-Step Conversion

54 ÷ 2 = 27, remainder 0

27 ÷ 2 = 13, remainder 1

13 ÷ 2 = 6, remainder 1

6 ÷ 2 = 3, remainder 0

3 ÷ 2 = 1, remainder 1

1 ÷ 2 = 0, remainder 1

Thus 54 in binary is: 110110

(ii)  (120)₁₀  =  (?)₂

Step-by-Step Conversion

120 ÷ 2 = 60, remainder 0

60 ÷ 2 = 30, remainder 0

30 ÷ 2 = 15, remainder 0

15 ÷ 2 = 7, remainder 1

7 ÷ 2 = 3, remainder 1

3 ÷ 2 = 1, remainder 1

1 ÷ 2 = 0, remainder 1

Thus 120 in binary is 1111000

(iii) (76)₁₀  =  (?)₈

Step-by-Step Conversion

76 ÷ 8 = 9, remainder 4

9 ÷ 8 = 1, remainder 1

1 ÷ 8 = 0, remainder 1

This 76 in Octal is 114

(iv) (889)₁₀  =  (?)₈

Step-by-Step Conversion

889 ÷ 8 = 111, remainder 1

111 ÷ 8 = 13, remainder 7

13 ÷ 8 = 1, remainder 5

1 ÷ 8 = 0, remainder 1

Thus 889 in Octal is 1571

(v)  (789)₁₀  =  (?)₁₆

Step-by-Step Conversion

789 ÷ 16 = 49, remainder 5

49 ÷ 16 = 3, remainder 1

3 ÷ 16 = 0, remainder 3

Thus 789 in hexadecimal is 315

(vi)  (108)₁₀  =  (?)₁₆

Step-by-Step Conversion

108 ÷ 16 = 6, remainder 12 (which is represented as C in hexadecimal)

6 ÷ 16 = 0, remainder 6

Thus 108 in Hexadecimal is 6C

5. Try the following conversions from Octal to Decimal:

(i) 145

Steps:

        ( 1 × 82 ) + (4 × 81 ) + ( 5 × 80 )

        = ( 1 × 64 ) + ( 4 × 8 ) + ( 5 × 1 )

        = 64 + 32 + 5

        = 101

Thus 145 in Octal is 101 in Decimal

(ii) 6760

Steps:

        ( 6 × 8³ ) + ( 7 × 8² ) + ( 6 × 8¹ ) + ( 0 × 8⁰ )

        = ( 6 × 512 ) + ( 7 × 64 ) + ( 6 × 8 ) + ( 0 × 1 )

        = 3072 + 448 + 48 + 0

        = 3568

Thus 6760 in Octal is 3568 in Decimal.

(iii) 455

Steps:

( 4 × 8² ) + ( 5 × 8¹ ) + ( 5 × 8⁰ )

= ( 4 × 64 ) + ( 5 × 8 ) + ( 5 × 1 )

= 256 + 40 + 5

= 301

Thus 455 in Octal is 301 in Decimal

(iv) 10.75

Steps:

Calculate the Decimal value:

( 1 × 8¹ ) + ( 0 × 8⁰ )

= ( 1 × 8 ) + ( 0 × 1 )

= 8 + 0

        = 8

 

Convert the fractional part (0.75)

( 7 × 8⁻¹ ) + ( 5 × 8⁻² )

= ( 7 × 1/8 ​) + ( 5 × 1/64​ )

= 7/8  +   5/64

= 61/64

Now add both parts:

8  +  61/64

= 8.953125