Chapter 2 of 11

Updated 6 August 2026

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Encoding Schemes and Number System — NCERT Solutions

CBSE · Class 11 · Computer Science

NCERT Solutions for Encoding Schemes and Number System, CBSE Class 11 Computer Science: 17 textbook questions solved step by step.

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EXERCISE — Encoding Schemes and Number System

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

The base (radix) of a number system equals the total number of unique digits (symbols) it uses.

Number System	Symbols Used	Base Value
Binary	0, 1	2
Octal	0, 1, 2, 3, 4, 5, 6, 7	8
Hexadecimal	0–9, A, B, C, D, E, F	16

Binary base = 2
Octal base = 8
Hexadecimal base = 16

2Give full form of ASCII and ISCII.Show solution

ASCII — American Standard Code for Information Interchange

ISCII — Indian Script Code for Information Interchange

ASCII is a 7-bit (or 8-bit extended) encoding scheme used to represent English characters and control characters. ISCII is an 8-bit encoding scheme developed to represent characters of Indian scripts.

3Try the following conversions.
(i)

(514)_8 = (?)_{10}

(ii)

(220)_8 = (?)_2

(iii)

(76F)_{16} = (?)_{10}

(iv)

(4D9)_{16} = (?)_{10}

(v)

(11001010)_2 = (?)_{10}

(vi)

(1010111)_2 = (?)_{10}

Show solution

(i) $(514)_8 = (?)_{10}$

Formula: Multiply each digit by its positional power of 8.
$5 \times 8^2 + 1 \times 8^1 + 4 \times 8^0$
$= 5 \times 64 + 1 \times 8 + 4 \times 1$
$= 320 + 8 + 4 = \boxed{332}$
$\therefore (514)_8 = (332)_{10}$

(ii) $(220)_8 = (?)_2$

Method: Replace each octal digit with its 3-bit binary equivalent.
$2 \rightarrow 010, \quad 2 \rightarrow 010, \quad 0 \rightarrow 000$
$\therefore (220)_8 = (010\,010\,000)_2 = \boxed{(010010000)_2}$

(iii) $(76F)_{16} = (?)_{10}$

Note: $F = 15$
$7 \times 16^2 + 6 \times 16^1 + 15 \times 16^0$
$= 7 \times 256 + 6 \times 16 + 15 \times 1$
$= 1792 + 96 + 15 = \boxed{1903}$
$\therefore (76F)_{16} = (1903)_{10}$

(iv) $(4D9)_{16} = (?)_{10}$

Note: $D = 13$
$4 \times 16^2 + 13 \times 16^1 + 9 \times 16^0$
$= 4 \times 256 + 13 \times 16 + 9 \times 1$
$= 1024 + 208 + 9 = \boxed{1241}$
$\therefore (4D9)_{16} = (1241)_{10}$

(v) $(11001010)_2 = (?)_{10}$

Assign positional values (right to left, starting from $2^0$ ):
$1\times2^7 + 1\times2^6 + 0\times2^5 + 0\times2^4 + 1\times2^3 + 0\times2^2 + 1\times2^1 + 0\times2^0$
$= 128 + 64 + 0 + 0 + 8 + 0 + 2 + 0 = \boxed{202}$
$\therefore (11001010)_2 = (202)_{10}$

(vi) $(1010111)_2 = (?)_{10}$

$1\times2^6 + 0\times2^5 + 1\times2^4 + 0\times2^3 + 1\times2^2 + 1\times2^1 + 1\times2^0$
$= 64 + 0 + 16 + 0 + 4 + 2 + 1 = \boxed{87}$
$\therefore (1010111)_2 = (87)_{10}$

4Do the following conversions from decimal number to other number systems.
(i)

(54)_{10} = (?)_2

(ii)

(120)_{10} = (?)_2

(iii)

(76)_{10} = (?)_8

(iv)

(889)_{10} = (?)_8

(v)

(789)_{10} = (?)_{16}

(vi)

(108)_{10} = (?)_{16}

Show solution

(i) $(54)_{10} = (?)_2$

Method: Repeated division by 2, read remainders bottom to top.
$54 \div 2 = 27 \text{ R } 0$
$27 \div 2 = 13 \text{ R } 1$
$13 \div 2 = 6 \text{ R } 1$
$6 \div 2 = 3 \text{ R } 0$
$3 \div 2 = 1 \text{ R } 1$
$1 \div 2 = 0 \text{ R } 1$
Reading remainders from bottom to top: $\boxed{(110110)_2}$

(ii) $(120)_{10} = (?)_2$

$120 \div 2 = 60 \text{ R } 0$
$60 \div 2 = 30 \text{ R } 0$
$30 \div 2 = 15 \text{ R } 0$
$15 \div 2 = 7 \text{ R } 1$
$7 \div 2 = 3 \text{ R } 1$
$3 \div 2 = 1 \text{ R } 1$
$1 \div 2 = 0 \text{ R } 1$
Reading bottom to top: $\boxed{(1111000)_2}$

(iii) $(76)_{10} = (?)_8$

Method: Repeated division by 8.
$76 \div 8 = 9 \text{ R } 4$
$9 \div 8 = 1 \text{ R } 1$
$1 \div 8 = 0 \text{ R } 1$
Reading bottom to top: $\boxed{(114)_8}$

(iv) $(889)_{10} = (?)_8$

$889 \div 8 = 111 \text{ R } 1$
$111 \div 8 = 13 \text{ R } 7$
$13 \div 8 = 1 \text{ R } 5$
$1 \div 8 = 0 \text{ R } 1$
Reading bottom to top: $\boxed{(1571)_8}$

(v) $(789)_{10} = (?)_{16}$

Method: Repeated division by 16.
$789 \div 16 = 49 \text{ R } 5$
$49 \div 16 = 3 \text{ R } 1$
$3 \div 16 = 0 \text{ R } 3$
Reading bottom to top: $\boxed{(315)_{16}}$

(vi) $(108)_{10} = (?)_{16}$

$108 \div 16 = 6 \text{ R } 12 \quad (12 = C)$
$6 \div 16 = 0 \text{ R } 6$
Reading bottom to top: $\boxed{(6C)_{16}}$

5Express the following octal numbers into their equivalent decimal numbers.
(i) 145 (ii) 6760 (iii) 455 (iv) 10.75Show solution

(i) $(145)_8 = (?)_{10}$

$1 \times 8^2 + 4 \times 8^1 + 5 \times 8^0$
$= 64 + 32 + 5 = \boxed{101}$

(ii) $(6760)_8 = (?)_{10}$

$6 \times 8^3 + 7 \times 8^2 + 6 \times 8^1 + 0 \times 8^0$
$= 6 \times 512 + 7 \times 64 + 6 \times 8 + 0$
$= 3072 + 448 + 48 + 0 = \boxed{3568}$

(iii) $(455)_8 = (?)_{10}$

$4 \times 8^2 + 5 \times 8^1 + 5 \times 8^0$
$= 256 + 40 + 5 = \boxed{301}$

(iv) $(10.75)_8 = (?)_{10}$

Integer part: $1 \times 8^1 + 0 \times 8^0 = 8$

Fractional part: $7 \times 8^{-1} + 5 \times 8^{-2} = \dfrac{7}{8} + \dfrac{5}{64} = 0.875 + 0.078125 = 0.953125$

$\therefore (10.75)_8 = \boxed{(8.953125)_{10}}$

6Express the following decimal numbers into hexadecimal numbers.
(i) 548 (ii) 4052 (iii) 58 (iv) 100.25Show solution

(i) $(548)_{10} = (?)_{16}$

$548 \div 16 = 34 \text{ R } 4$
$34 \div 16 = 2 \text{ R } 2$
$2 \div 16 = 0 \text{ R } 2$
Reading bottom to top: $\boxed{(224)_{16}}$

(ii) $(4052)_{10} = (?)_{16}$

$4052 \div 16 = 253 \text{ R } 4$
$253 \div 16 = 15 \text{ R } 13 \quad (13 = D)$
$15 \div 16 = 0 \text{ R } 15 \quad (15 = F)$
Reading bottom to top: $\boxed{(FD4)_{16}}$

(iii) $(58)_{10} = (?)_{16}$

$58 \div 16 = 3 \text{ R } 10 \quad (10 = A)$
$3 \div 16 = 0 \text{ R } 3$
Reading bottom to top: $\boxed{(3A)_{16}}$

(iv) $(100.25)_{10} = (?)_{16}$

Integer part (100):
$100 \div 16 = 6 \text{ R } 4$
$6 \div 16 = 0 \text{ R } 6$
Integer part in hex = $64$

Fractional part (0.25):
$0.25 \times 16 = 4.00 \rightarrow \text{digit} = 4$
Fractional part in hex = $.4$

$\therefore (100.25)_{10} = \boxed{(64.4)_{16}}$

7Express the following hexadecimal numbers into equivalent decimal numbers.
(i) 4A2 (ii) 9E1A (iii) 6BD (iv) 6C.34Show solution

(i) $(4A2)_{16} = (?)_{10}$

$A = 10$
$4 \times 16^2 + 10 \times 16^1 + 2 \times 16^0$
$= 1024 + 160 + 2 = \boxed{1186}$

(ii) $(9E1A)_{16} = (?)_{10}$

$E = 14,\ A = 10$
$9 \times 16^3 + 14 \times 16^2 + 1 \times 16^1 + 10 \times 16^0$
$= 9 \times 4096 + 14 \times 256 + 1 \times 16 + 10$
$= 36864 + 3584 + 16 + 10 = \boxed{40474}$

(iii) $(6BD)_{16} = (?)_{10}$

$B = 11,\ D = 13$
$6 \times 16^2 + 11 \times 16^1 + 13 \times 16^0$
$= 1536 + 176 + 13 = \boxed{1725}$

(iv) $(6C.34)_{16} = (?)_{10}$

$C = 12,\ 3 = 3,\ 4 = 4$

Integer part:
$6 \times 16^1 + 12 \times 16^0 = 96 + 12 = 108$

Fractional part:
$3 \times 16^{-1} + 4 \times 16^{-2} = \frac{3}{16} + \frac{4}{256} = 0.1875 + 0.015625 = 0.203125$

$\therefore (6C.34)_{16} = \boxed{(108.203125)_{10}}$

8Convert the following binary numbers into octal and hexadecimal numbers.
(i) 1110001000 (ii) 110110101 (iii) 1010100 (iv) 1010.1001Show solution

Method for Octal: Group bits in sets of 3 from right (for integer part) and from left (for fractional part). Replace each group with its octal digit.

Method for Hexadecimal: Group bits in sets of 4 from right (for integer part) and from left (for fractional part). Replace each group with its hex digit.

(i) $(1110001000)_2$

To Octal (groups of 3):
$001\; 110\; 001\; 000$
$= 1\quad 6\quad 1\quad 0$
$\therefore \boxed{(1610)_8}$

To Hexadecimal (groups of 4):
$0011\; 1000\; 1000$
$= 3\quad 8\quad 8$
$\therefore \boxed{(388)_{16}}$

(ii) $(110110101)_2$

To Octal (groups of 3):
$110\; 110\; 101$
$= 6\quad 6\quad 5$
$\therefore \boxed{(665)_8}$

To Hexadecimal (groups of 4):
$0001\; 1011\; 0101$
$= 1\quad B\quad 5$
$\therefore \boxed{(1B5)_{16}}$

(iii) $(1010100)_2$

To Octal (groups of 3):
$001\; 010\; 100$
$= 1\quad 2\quad 4$
$\therefore \boxed{(124)_8}$

To Hexadecimal (groups of 4):
$0101\; 0100$
$= 5\quad 4$
$\therefore \boxed{(54)_{16}}$

(iv) $(1010.1001)_2$

To Octal:
Integer: $001\; 010 \rightarrow 1\; 2$
Fractional: $100\; 1\mathbf{00} \rightarrow 100\; 100 \rightarrow 4\; 4$ (pad right with zeros)
$\therefore \boxed{(12.44)_8}$

To Hexadecimal:
Integer: $1010 \rightarrow A$
Fractional: $1001 \rightarrow 9$
$\therefore \boxed{(A.9)_{16}}$

9Write binary equivalent of the following octal numbers.
(i) 2306 (ii) 5610 (iii) 742 (iv) 65.203Show solution

Method: Replace each octal digit with its 3-bit binary equivalent.

Octal–Binary table:

Octal	Binary
0	000
1	001
2	010
3	011
4	100
5	101
6	110
7	111

(i) $(2306)_8$
$2 \to 010,\quad 3 \to 011,\quad 0 \to 000,\quad 6 \to 110$
$\therefore \boxed{(010011000110)_2}$

(ii) $(5610)_8$
$5 \to 101,\quad 6 \to 110,\quad 1 \to 001,\quad 0 \to 000$
$\therefore \boxed{(101110001000)_2}$

(iii) $(742)_8$
$7 \to 111,\quad 4 \to 100,\quad 2 \to 010$
$\therefore \boxed{(111100010)_2}$

(iv) $(65.203)_8$
$6 \to 110,\quad 5 \to 101 \;.\; 2 \to 010,\quad 0 \to 000,\quad 3 \to 011$
$\therefore \boxed{(110101.010000011)_2}$

10Write binary representation of the following hexadecimal numbers.
(i) 4026 (ii) BCA1 (iii) 98E (iv) 132.45

Method: Replace each hexadecimal digit with its 4-bit binary equivalent.

Key hex–binary values: $A=1010,\ B=1011,\ C=1100,\ D=1101,\ E=1110,\ F=1111$

(i) $(4026)_{16}$
$4 \to 0100,\quad 0 \to 0000,\quad 2 \to 0010,\quad 6 \to 0110$
$\therefore \boxed{(0100000000100110)_2}$

(ii) $(BCA1)_{16}$
$B \to 1011,\quad C \to 1100,\quad A \to 1010,\quad 1 \to 0001$
$\therefore \boxed{(1011110010100001)_2}$

(iii) $(98E)_{16}$
$9 \to 1001,\quad 8 \to 1000,\quad E \to 1110$
$\therefore \boxed{(100110001110)_2}$

(iv) $(132.45)_{16}$
$1 \to 0001,\quad 3 \to 0011,\quad 2 \to 0010 \;.\; 4 \to 0100,\quad 5 \to 0101$
$\therefore \boxed{(000100110010.01000101)_2}$

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11How does computer understand the following text? (hint: 7 bit ASCII code).
(i) HOTS (ii) Main (iii) CaSe

The computer converts each character to its 7-bit ASCII code. The decimal ASCII values and their 7-bit binary equivalents are given below.

Standard ASCII decimal values (selected):

A=65, B=66, C=67, D=68, E=69, F=70, G=71, H=72, I=73, J=74, K=75, L=76, M=77, N=78, O=79, P=80, Q=81, R=82, S=83, T=84, U=85, V=86, W=87, X=88, Y=89, Z=90
a=97, e=101, i=105, m=109, n=110, s=115

(i) HOTS

Character	ASCII (Decimal)	7-bit Binary
H	72	1001000
O	79	1001111
T	84	1010100
S	83	1010011

The computer understands HOTS as: 1001000 1001111 1010100 1010011

(ii) Main

Character	ASCII (Decimal)	7-bit Binary
M	77	1001101
a	97	1100001
i	105	1101001
n	110	1101110

The computer understands Main as: 1001101 1100001 1101001 1101110

(iii) CaSe

Character	ASCII (Decimal)	7-bit Binary
C	67	1000011
a	97	1100001
S	83	1010011
e	101	1100101

The computer understands CaSe as: 1000011 1100001 1010011 1100101

Note: Uppercase and lowercase letters have different ASCII codes, so the computer treats 'C' and 'c' as different characters.

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12The hexadecimal number system uses 16 literals (0-9, A-F). Write down its base value.

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13Let X be a number system having B symbols only. Write down the base value of this number system.

Given: A number system X uses exactly $B$ unique symbols.

Concept: The base (radix) of any number system is equal to the total count of distinct symbols (digits/literals) it uses.

$\text{Base of number system X} = B$

For example:

If $B = 2$ (symbols: 0, 1) → Binary system with base 2
If $B = 8$ (symbols: 0–7) → Octal system with base 8
If $B = 16$ (symbols: 0–9, A–F) → Hexadecimal system with base 16

Therefore, the base value of number system X is $\boxed{B}$ .

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14Write the equivalent hexadecimal and binary values for each character of the phrase given below.
"हम सब एक"

The phrase "हम सब एक" is in Hindi (Devanagari script). It is encoded using Unicode (specifically UTF-16 or Unicode code points).

The Unicode (hexadecimal) code points for Devanagari characters are in the range U+0900 to U+097F.

Character	Unicode Code Point (Hex)	Binary (16-bit)
ह	U+0939	0000 1001 0011 1001
म	U+092E	0000 1001 0010 1110
(space)	U+0020	0000 0000 0010 0000
स	U+0938	0000 1001 0011 1000
ब	U+092C	0000 1001 0010 1100
(space)	U+0020	0000 0000 0010 0000
ए	U+090F	0000 1001 0000 1111
क	U+0915	0000 1001 0001 0101

Each Devanagari character is represented by its Unicode code point in hexadecimal, and the corresponding 16-bit binary value is obtained by converting that hex value to binary.

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15What is the advantage of preparing a digital content in Indian language using UNICODE font?

Advantages of preparing digital content in Indian languages using Unicode:

Universal Compatibility: Unicode is a universal encoding standard. Content created using Unicode can be read and displayed correctly on any device, operating system, or platform (Windows, Mac, Linux, Android, iOS) without any special font installation.

Web and Internet Support: Unicode-encoded Indian language content is fully supported by web browsers, search engines, and the internet. It can be indexed and searched by search engines like Google.

Interoperability: Unicode allows seamless exchange of Indian language documents between different applications (MS Word, Google Docs, email clients, etc.) without encoding errors or garbled text.

Single Standard for All Scripts: Unicode supports all Indian scripts (Devanagari, Tamil, Telugu, Bengali, Gujarati, etc.) under one standard, making it easy to create multilingual documents.

Long-term Preservation: Since Unicode is an internationally accepted standard, digital content created in Unicode will remain accessible and readable in the future.

No Dependency on Specific Fonts: Unlike older encoding schemes (like ISCII-based fonts), Unicode content does not require the recipient to have a specific font installed to read the text correctly.

Software Support: All modern software applications, databases, and programming languages support Unicode, making it easy to process, store, and retrieve Indian language content.

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16Explore and list the steps required to type in an Indian language using UNICODE.

Steps to type in an Indian language (e.g., Hindi) using Unicode on a Windows computer:

Step 1: Add the Indian Language Input Method

Go to Control Panel → Clock, Language, and Region → Language.
Click Add a language and select the desired Indian language (e.g., Hindi).
Click Add to install the language pack.

Step 2: Enable the Keyboard Layout

After adding the language, click on Options next to the language.
Add an Input Method (keyboard layout), such as Devanagari – INSCRIPT or Hindi Phonetic keyboard.

Step 3: Switch the Input Language

Use the Language Bar on the taskbar (or press Windows key + Space / Alt + Shift) to switch between English and the Indian language input.

Step 4: Open a Unicode-Compatible Application

Open any Unicode-compatible application such as Notepad, MS Word, or a web browser.
Ensure the font selected supports the Indian script (e.g., Mangal, Nirmala UI for Hindi/Devanagari).

Step 5: Start Typing

With the Indian language input method active, start typing. The keys will produce the corresponding Indian script characters encoded in Unicode.
For phonetic keyboards, type the Roman transliteration and the software converts it to the Indian script.

Step 6: Using Online Tools (Alternative)

Alternatively, use online Unicode typing tools such as Google Input Tools or Quillpad, which allow phonetic typing and produce Unicode output.

Note: On modern systems (Windows 10/11, macOS, Linux), Unicode support for Indian languages is built-in and the above steps may vary slightly depending on the OS version.

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17Encode the word 'COMPUTER' using ASCII and convert the encode value into binary values.

Given: Word = COMPUTER

Concept: Each character is assigned a unique 7-bit ASCII decimal value. We then convert each decimal value to its 7-bit binary equivalent.

Character	ASCII Decimal Value	7-bit Binary
C	67	1000011
O	79	1001111
M	77	1001101
P	80	1010000
U	85	1010101
T	84	1010100
E	69	1000101
R	82	1010010

Verification of a few conversions:

$C = 67$ : $64+2+1 = 2^6+2^1+2^0 \Rightarrow 1000011$
$O = 79$ : $64+8+4+2+1 = 2^6+2^3+2^2+2^1+2^0 \Rightarrow 1001111$
$M = 77$ : $64+8+4+1 = 2^6+2^3+2^2+2^0 \Rightarrow 1001101$

Result:
The word COMPUTER is encoded in ASCII and binary as:
$\text{C O M P U T E R}$
$\text{67 79 77 80 85 84 69 82}$
$\text{1000011 1001111 1001101 1010000 1010101 1010100 1000101 1010010}$

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Frequently Asked Questions

What are the important topics in Encoding Schemes and Number System for CBSE Class 11 Computer Science?

Key topics in Encoding Schemes and Number System include Encoding Schemes, ASCII, ISCII, and Unicode Examples, Number Systems and Positional Value, Binary, Octal, and Hexadecimal Number Systems. Study these first, then practise questions on each for Class 11 exams.

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