Data Representation

A computer understands only 0s and 1s, yet it displays your name, calculates your marks and shows colourful images. Data Representation explains how: this chapter covers the decimal, binary, octal and hexadecimal number systems, conversions between them, and how characters and strings are encoded using ASCII, ISCII and Unicode (UTF-8 and UTF-32).

2.1 🏁 Introduction

Computers are essentially just VERY fast switches — millions of tiny electronic switches that can only be in TWO states: ON or OFF.

graph LR
    ON["⚡ Switch ON\n= 1"]
    OFF["⚫ Switch OFF\n= 0"]
    BINARY["🔢 Combined together:\nBINARY NUMBERS\n(0s and 1s)"]

    ON --> BINARY
    OFF --> BINARY

    style ON fill:#4CAF50,color:#fff
    style OFF fill:#9E9E9E,color:#fff
    style BINARY fill:#2196F3,color:#fff

Think of it this way: A light switch can only be ON or OFF — there's no "half on." Computers work EXACTLY like this, using millions of switches (transistors) to represent EVERY piece of data — numbers, letters, images, sounds — all as combinations of 0s and 1s!

Why do we need different Number Systems?

2.2 🔢 Digital Number Systems

A Number System is a way of representing numbers using a specific set of symbols (digits) and a BASE (also called radix) — the total count of unique digits used.

graph TD
    NS["🔢 NUMBER SYSTEMS"]
    D["🔟 Decimal\nBase 10\n(0-9)"]
    B["🔵 Binary\nBase 2\n(0-1)"]
    O["🟠 Octal\nBase 8\n(0-7)"]
    H["🟣 Hexadecimal\nBase 16\n(0-9, A-F)"]

    NS --> D
    NS --> B
    NS --> O
    NS --> H

    style D fill:#4CAF50,color:#fff
    style B fill:#2196F3,color:#fff
    style O fill:#FF9800,color:#fff
    style H fill:#9C27B0,color:#fff

2.2.1 Decimal (Denary) Number System (Base 10) 🔟

The number system we humans use EVERY DAY — uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Number: 2 5 3
Position: 100s  10s  1s
Value: (2×100) + (5×10) + (3×1) = 253

Positional Value System:

253 (base 10) = 2×10² + 5×10¹ + 3×10⁰
              = 2×100 + 5×10 + 3×1
              = 200 + 50 + 3 = 253

Key Concept: In ANY number system, each digit's position represents a POWER of the base. This "positional value" concept is the foundation for understanding ALL number systems!

2.2.2 Binary Number System (Base 2) 🔵

Uses only 2 digits: 0 and 1 — the LANGUAGE computers actually speak internally!

Binary: 1 0 1 1
Position: 2³   2²   2¹   2⁰
Value: (1×8) + (0×4) + (1×2) + (1×1) = 11 (decimal)

8 + 0 + 2 + 1 = 11 → So (1011)₂ = (11)₁₀

Each binary digit is called a BIT (Binary Digit). 8 bits together = 1 BYTE!

2.2.3 Octal Number System (Base 8) 🟠

Uses 8 digits: 0, 1, 2, 3, 4, 5, 6, 7.

Octal: 3 7 2
Position: 8²    8¹    8⁰
Value: (3×64) + (7×8) + (2×1) = 192+56+2 = 250 (decimal)

Octal was popular in early computing as a SHORTHAND for binary — since 8 = 2³, exactly 3 binary digits map to ONE octal digit!

2.2.4 Hexadecimal Number System (Base 16) 🟣

Uses 16 digits: 0-9, then A, B, C, D, E, F (where A=10, B=11, C=12, D=13, E=14, F=15).

Hex: 2 A
Position: 16¹    16⁰
Value: (2×16) + (A×1) = (2×16) + (10×1) = 32+10 = 42 (decimal)

Hexadecimal is HUGELY popular in programming — since 16 = 2⁴, exactly 4 binary digits map to ONE hex digit! This makes hex a super-compact way to write binary. You've SEEN hex before: color codes like #FF5733 in CSS/web design are hexadecimal!

All Four Number Systems — Quick Reference:

2.3 🔄 Number Conversions

This section works through every conversion, step by step.

graph TD
    DEC["🔟 Decimal"]
    BIN["🔵 Binary"]
    OCT["🟠 Octal"]
    HEX["🟣 Hexadecimal"]

    DEC <--> BIN
    DEC <--> OCT
    DEC <--> HEX
    BIN <--> OCT
    BIN <--> HEX
    HEX <--> OCT

    style DEC fill:#4CAF50,color:#fff
    style BIN fill:#2196F3,color:#fff
    style OCT fill:#FF9800,color:#fff
    style HEX fill:#9C27B0,color:#fff

2.3.1 Decimal-to-Binary Conversion 🔟➡️🔵

Method: Repeated Division by 2 — divide by 2 repeatedly, note the remainders, then read BOTTOM TO TOP.

Example: Convert (25)₁₀ to Binary

25 ÷ 2 = 12  remainder 1   ↑
12 ÷ 2 = 6   remainder 0   ↑
6  ÷ 2 = 3   remainder 0   ↑
3  ÷ 2 = 1   remainder 1   ↑
1  ÷ 2 = 0   remainder 1   ↑ (Read bottom to top!)

Answer: (25)₁₀ = (11001)₂

graph TD
    A["25 ÷ 2 = 12 R 1"]
    B["12 ÷ 2 = 6 R 0"]
    C["6 ÷ 2 = 3 R 0"]
    D["3 ÷ 2 = 1 R 1"]
    E["1 ÷ 2 = 0 R 1"]
    F["Read UPWARDS:\n1-1-0-0-1 = 11001"]

    A --> B --> C --> D --> E --> F

    style F fill:#4CAF50,color:#fff

2.3.2 Binary-to-Decimal Conversion 🔵➡️🔟

Method: Multiply each digit by its positional power of 2, then ADD.

Example: Convert (11001)₂ to Decimal

1    1    0    0    1
2⁴   2³   2²   2¹   2⁰
16   8    4    2    1

(1×16) + (1×8) + (0×4) + (0×2) + (1×1)
= 16 + 8 + 0 + 0 + 1
= 25

Answer: (11001)₂ = (25)₁₀ ✅ (Matches our earlier conversion — good check!)

2.3.3 Decimal-to-Octal Conversion 🔟➡️🟠

Method: Repeated Division by 8 — same process as Decimal-to-Binary, but divide by 8!

Example: Convert (100)₁₀ to Octal

100 ÷ 8 = 12  remainder 4   ↑
12  ÷ 8 = 1   remainder 4   ↑
1   ÷ 8 = 0   remainder 1   ↑ (Read bottom to top!)

Answer: (100)₁₀ = (144)₈

2.3.4 Octal-to-Decimal Conversion 🟠➡️🔟

Method: Multiply each digit by its positional power of 8, then ADD.

Example: Convert (144)₈ to Decimal

1    4    4
8²   8¹   8⁰
64   8    1

(1×64) + (4×8) + (4×1) = 64 + 32 + 4 = 100

Answer: (144)₈ = (100)₁₀ ✅

2.3.5 Octal-to-Binary Conversion 🟠➡️🔵

Method: Convert EACH octal digit into its 3-bit binary equivalent (since 8 = 2³).

The Octal-Binary Cheat Sheet:

Example: Convert (372)₈ to Binary

3  →  011
7  →  111
2  →  010

Combine: 011 111 010

Answer: (372)₈ = (011111010)₂

2.3.6 Binary-to-Octal Conversion 🔵➡️🟠

Method: Group binary digits into sets of 3 (from the RIGHT), then convert each group to its octal digit.

Example: Convert (11111010)₂ to Octal

Step 1: Group into 3s from the RIGHT (pad with 0s on the LEFT if needed)
11 111 010  →  011 111 010

Step 2: Convert each group:
011 = 3
111 = 7
010 = 2

Answer: 372

Answer: (11111010)₂ = (372)₈

2.3.7 Decimal-to-Hex Conversion 🔟➡️🟣

Method: Repeated Division by 16 — same process, but divide by 16, and convert remainders 10-15 into A-F!

Example: Convert (250)₁₀ to Hexadecimal

250 ÷ 16 = 15  remainder 10 (=A)   ↑
15  ÷ 16 = 0   remainder 15 (=F)   ↑ (Read bottom to top!)

Answer: (250)₁₀ = (FA)₁₆

2.3.8 Hex-to-Decimal Conversion 🟣➡️🔟

Method: Multiply each digit by its positional power of 16, then ADD.

Example: Convert (FA)₁₆ to Decimal

F     A
16¹   16⁰
16    1

(F×16) + (A×1) = (15×16) + (10×1) = 240 + 10 = 250

Answer: (FA)₁₆ = (250)₁₀ ✅

2.3.9 Binary-to-Hex Conversion 🔵➡️🟣

Method: Group binary digits into sets of 4 (from the RIGHT), then convert each group to its hex digit (since 16 = 2⁴).

The Binary-Hex Cheat Sheet:

Example: Convert (11111010)₂ to Hex

Step 1: Group into 4s from the RIGHT
1111 1010

Step 2: Convert each group:
1111 = F
1010 = A

Answer: FA

Answer: (11111010)₂ = (FA)₁₆ ✅ (Matches our earlier work!)

2.3.10 Hex-to-Binary Conversion 🟣➡️🔵

Method: Convert EACH hex digit into its 4-bit binary equivalent.

Example: Convert (FA)₁₆ to Binary

F  →  1111
A  →  1010

Combine: 1111 1010

Answer: (FA)₁₆ = (11111010)₂

📐 The Complete Conversion Map — Quick Reference

graph LR
    D["Decimal"]
    B["Binary"]
    O["Octal"]
    H["Hex"]

    D -->|"÷2 repeatedly"| B
    B -->|"×2ⁿ, add"| D
    D -->|"÷8 repeatedly"| O
    O -->|"×8ⁿ, add"| D
    D -->|"÷16 repeatedly"| H
    H -->|"×16ⁿ, add"| D
    B -->|"group in 3s"| O
    O -->|"3-bit each"| B
    B -->|"group in 4s"| H
    H -->|"4-bit each"| B

    style D fill:#4CAF50,color:#fff
    style B fill:#2196F3,color:#fff

2.4 🔤 Character/String Representation

Numbers are easy — computers store them directly in binary. But how does a computer store LETTERS, symbols, and text? Through special CODING SCHEMES that assign a unique number to EVERY character!

graph TD
    CHAR["🔤 CHARACTER CODES"]
    ASCII["🇺🇸 ASCII\n(English characters,\n7-8 bits)"]
    ISCII["🇮🇳 ISCII\n(Indian language\ncharacters)"]
    UNI["🌍 Unicode\n(ALL world languages,\n16+ bits)"]

    CHAR --> ASCII
    CHAR --> ISCII
    CHAR --> UNI

    style ASCII fill:#2196F3,color:#fff
    style ISCII fill:#FF9800,color:#fff
    style UNI fill:#9C27B0,color:#fff

2.4.1 ASCII Code 🇺🇸

ASCII = American Standard Code for Information Interchange. Assigns a unique number (0-127) to English letters, digits, and symbols.

Character 'A'  →  ASCII value 65  →  Binary: 01000001
Character 'a'  →  ASCII value 97  →  Binary: 01100001
Character '0'  →  ASCII value 48  →  Binary: 00110000

Key ASCII Facts:

Fun Trick! 🧠 Notice that lowercase letters = uppercase value + 32! 'A' is 65, 'a' is 97 (65+32=97). This is why programmers can easily convert case using simple math!

2.4.2 ISCII Code 🇮🇳

ISCII = Indian Script Code for Information Interchange. Developed specifically to represent characters from INDIAN languages (Hindi, Tamil, Bengali, etc.) that ASCII cannot handle.

Analogy: If ASCII is a phrasebook that ONLY has English words, ISCII is a SPECIAL phrasebook created specifically for Indian regional languages — filling the gap ASCII couldn't cover!

2.4.3 Unicode 🌍

Unicode is a UNIVERSAL character encoding standard designed to represent EVERY character from EVERY language/script in the WORLD — plus emojis, symbols, and more!

graph LR
    PROB["😩 Problem\nASCII = English only\nISCII = Indian only\nOther countries need\nTHEIR OWN codes too!"]
    UNI["🌍 Unicode\nONE universal standard\nfor ALL languages\nWORLDWIDE"]

    PROB -->|"Solved by"| UNI

    style PROB fill:#F44336,color:#fff
    style UNI fill:#4CAF50,color:#fff

Why Unicode was needed:

Key Unicode Facts:

ASCII vs ISCII vs Unicode — The Ultimate Comparison:

2.4.4 Code Points — How Unicode Numbers Every Character 🔢

Unicode gives every character in the world its own unique number, called its code point. A code point is written as U+ followed by the number in hexadecimal.

Notice that 'A' is U+0041 = 65 — exactly its ASCII value. That is what "Unicode is backward compatible with ASCII" means.

Code points run from U+0000 to U+10FFFF, which leaves room for 1,114,112 characters.

But a code point is only a number. To store a character in memory or in a file, that number has to be turned into bytes. The rule that does this is called a Unicode encoding scheme. Your syllabus names two of them: UTF-8 and UTF-32 (UTF = Unicode Transformation Format).

graph LR
    CH["🔤 Character\n₹"]
    CP["🔢 Code point\nU+20B9"]
    U8["🟢 UTF-8\nE2 82 B9\n(3 bytes)"]
    U32["🔵 UTF-32\n00 00 20 B9\n(4 bytes)"]

    CH --> CP
    CP --> U8
    CP --> U32

    style CP fill:#9C27B0,color:#fff
    style U8 fill:#4CAF50,color:#fff
    style U32 fill:#2196F3,color:#fff

Analogy: Think of Unicode as one giant roll register for every character in the world — each character gets its own roll number (the code point). UTF-8 and UTF-32 are two different ways of writing that roll number on a form: UTF-8 uses only as many boxes as the number needs, while UTF-32 always fills in exactly four boxes.

2.4.5 UTF-8 — The Flexible Encoding 🟢

UTF-8 is a variable-length encoding. Each character is stored in 1, 2, 3 or 4 bytes (8, 16, 24 or 32 bits), depending on how big its code point is. The "8" tells you that it works in units of 8 bits (1 byte).

UTF-8 in action:

Character   Code point   UTF-8 bytes (hex)   Bytes used
A           U+0041       41                  1
é           U+00E9       C3 A9               2
₹           U+20B9       E2 82 B9            3
अ           U+0905       E0 A4 85            3
😀          U+1F600      F0 9F 98 80         4

1 byte : 0xxxxxxx
2 bytes: 110xxxxx 10xxxxxx
3 bytes: 1110xxxx 10xxxxxx 10xxxxxx
4 bytes: 11110xxx 10xxxxxx 10xxxxxx 10xxxxxx

2.4.6 UTF-32 — The Fixed-Size Encoding 🔵

UTF-32 is a fixed-length encoding. Every character is stored in exactly 4 bytes (32 bits), whatever it is. The code point is simply written as a 32-bit binary number, with leading zeros added in front.

UTF-32 in action:

Character   Code point   UTF-32 bytes (hex)   Bytes used
A           U+0041       00 00 00 41          4
₹           U+20B9       00 00 20 B9          4
😀          U+1F600      00 01 F6 00          4

For example, 'A' (65) in UTF-32 is:

00000000 00000000 00000000 01000001

Analogy: Packing a tiffin 🍱. UTF-8 puts each item in the smallest box it fits into — a laddoo gets a small box, a full meal gets a big one. UTF-32 puts every item, even a single laddoo, into the same big 4-compartment tiffin. The boxes are easy to stack and count, but a lot of space is wasted!

2.4.7 UTF-8 vs UTF-32 ⚖️

Worked example — counting storage:

Notice that नमस्ते looks like 4 letters but is stored as 6 code points — the half-letter sign ( ् ) and the matra ( े ) are separate Unicode characters.

Try it yourself — Python's encode() method turns a string into bytes, so len() of the result is the number of bytes used:

for ch in ["A", "é", "₹", "अ", "😀"]:
    u8 = ch.encode("utf-8")
    u32 = ch.encode("utf-32-be")
    print(ch, f"U+{ord(ch):04X}", len(u8), "byte(s) in UTF-8,", len(u32), "bytes in UTF-32")

Output:

A U+0041 1 byte(s) in UTF-8, 4 bytes in UTF-32
é U+00E9 2 byte(s) in UTF-8, 4 bytes in UTF-32
₹ U+20B9 3 byte(s) in UTF-8, 4 bytes in UTF-32
अ U+0905 3 byte(s) in UTF-8, 4 bytes in UTF-32
😀 U+1F600 4 byte(s) in UTF-8, 4 bytes in UTF-32

We write "utf-32-be" so that Python stores only the 4 bytes of each character. With plain "utf-32", Python adds 4 extra marker bytes at the start.

⚠️ Common Errors and Misconceptions

Decimal = Base 10 (0-9)
Binary  = Base 2  (0-1)
Octal   = Base 8  (0-7)
Hex     = Base 16 (0-9, A-F)

🎯 Sample Exam Questions

Q1: Very Short Answer [1 mark each]

a) What is the base of the Hexadecimal number system? → 16

b) What does ASCII stand for? → American Standard Code for Information Interchange

c) How many bits make up 1 byte? → 8 bits

d) What is the ASCII value of the character 'A'? → 65

e) Which number system uses digits 0-7 only? → Octal

f) How many bytes does UTF-32 use to store any one character? → 4 bytes (32 bits)

g) How many bytes does the letter 'A' take in UTF-8? → 1 byte

Q2: Conversion [2 marks]

Q: Convert (45)₁₀ to Binary.

45 ÷ 2 = 22 remainder 1
22 ÷ 2 = 11 remainder 0
11 ÷ 2 = 5  remainder 1
5  ÷ 2 = 2  remainder 1
2  ÷ 2 = 1  remainder 0
1  ÷ 2 = 0  remainder 1

Answer: (45)₁₀ = (101101)₂

Q3: Conversion [2 marks]

Q: Convert (11010)₂ to Decimal.

1    1    0    1    0
16   8    4    2    1

(1×16)+(1×8)+(0×4)+(1×2)+(0×1) = 16+8+0+2+0 = 26

Answer: (11010)₂ = (26)₁₀

Q4: Conversion [3 marks]

Q: Convert (1101 1010)₂ to Octal and Hexadecimal.

To Octal (group in 3s):

011 011 010
011 = 3, 011 = 3, 010 = 2
Answer: 332

To Hex (group in 4s):

1101 1010
1101 = D, 1010 = A
Answer: DA

Q5: Short Answer [2 marks]

Q: Differentiate between ASCII and Unicode.

ASCII is a 7-8 bit encoding covering only English letters, digits, and symbols (128-256 characters). Unicode is a universal 16+ bit encoding that represents characters from ALL world languages, including emojis — with the first 128 values matching ASCII for backward compatibility.

Q6: Short Answer [2 marks]

Q: Differentiate between UTF-8 and UTF-32.

Q7: Application [3 marks]

Q: The word INDIA₹ is to be stored. (a) How many bytes are needed in UTF-8? (b) How many in UTF-32? (c) Which encoding is better for storing mostly English text, and why?

Characters: I N D I A ₹  →  6 characters
UTF-8 : I, N, D, I, A = 1 byte each = 5 bytes; ₹ (U+20B9) = 3 bytes
        Total = 5 + 3 = 8 bytes
UTF-32: 6 characters × 4 bytes = 24 bytes

Answer: (a) 8 bytes (b) 24 bytes (c) UTF-8, because English (ASCII) characters take only 1 byte each in UTF-8 but 4 bytes each in UTF-32.

✏️ Practice Problems

1. Convert (78)₁₀ to Binary, Octal, and Hexadecimal.

2. Convert (110101)₂ to Decimal, Octal, and Hexadecimal.

3. Convert (BC)₁₆ to Decimal and Binary.

4. Convert (57)₈ to Decimal and Binary.

5. Explain why Octal-to-Hexadecimal conversion requires going through Binary as an intermediate step.

6. What is the ASCII value of 'Z'? What would the ASCII value of 'z' be, using the +32 shortcut?

7. Differentiate between ASCII, ISCII, and Unicode with one key feature of each.

8. Convert the decimal number 500 into Binary, Octal, and Hexadecimal, showing all working steps.

9. Explain why computers use the Binary number system internally, rather than Decimal.

10. A student converts (25)₁₀ to binary and gets (10011)₂ instead of the correct (11001)₂. Identify the likely mistake made during the conversion process.

11. Write the Unicode code point of 'A' and explain why it is the same as its ASCII value.

12. How many bytes are needed to store the string "Class 11" in UTF-8 and in UTF-32? (Remember that the space is also a character.)

13. Give two advantages of UTF-8 over UTF-32 and one advantage of UTF-32 over UTF-8.