Introduction
The hexadecimal (hex) number system (base-16) is widely used in computing for its compact representation of binary data. This guide covers:
✔ What is the Hexadecimal System?
✔ Hex ⇄ Binary, Decimal, Octal Conversions
✔ Real-World Applications (Memory Addressing, Color Codes)
✔ Exam Tricks & Practice Problems
1. What is the Hexadecimal System?
Base: 16
Digits:
0-9, A-F(whereA=10,B=11, ...,F=15)Uses:
Memory addressing (
0x1F).Web design (CSS colors like
#FF5733).Debugging programs.
Example:2F₁₆ = 2×16¹ + 15×16⁰ = 47₁₀ (Decimal).
2. Hexadecimal Conversions (Step-by-Step)
(A) Hexadecimal → Decimal
Method: Multiply each digit by 16ⁿ (n = position from right, starting at 0).
Example: Convert A3₁₆ to decimal.
(B) Hexadecimal → Binary
Method: Convert each hex digit to a 4-bit binary group.
Example: Convert D5₁₆ to binary.
(C) Hexadecimal → Octal
Step 1: Convert hex → binary.
Step 2: Group binary into 3-bit sets → octal.
Example: Convert E7₁₆ to octal.
3. Other Systems → Hexadecimal
(A) Binary → Hexadecimal
Method: Group binary digits into 4-bit nibbles, convert each to hex.
Example: Convert 11001101₂ to hex.
1100 | 1101 C D = **CD₁₆**
(B) Decimal → Hexadecimal
Method: Divide by 16, track remainders (≥10 becomes A-F).
Example: Convert 255₁₀ to hex.
255 ÷ 16 = 15 (Remainder 15 = F) 15 ÷ 16 = 0 (Remainder 15 = F) Read remainders **bottom-up**: **FF₁₆**
(C) Octal → Hexadecimal
Step 1: Convert octal → binary.
Step 2: Group binary into 4-bit nibbles → hex.
Example: Convert 745₈ to hex.
7 → 111 4 → 100 5 → 101 = **111 100 101₂** = **0001 | 1110 | 0101** (Pad left with 0) = **1 | E | 5₁₆** = **1E5₁₆**
4. Why Use Hexadecimal?
| Advantages | Disadvantages |
|---|---|
| Compact (1 hex digit = 4 bits) | Requires memorizing A-F |
| Easy binary conversion | Less intuitive than decimal |
Real-World Uses:
Memory Addresses:
0xFFFF(64KB limit in old systems).Color Codes:
#FFFFFF= White.Assembly Language: Register values.
5. Exam-Focused FAQs
❓ Q1: Why is hex preferred over octal in modern computing?
✅ A: Hex uses 4-bit groups (aligned with bytes), while octal uses 3 bits (obsolete for 8-bit systems).
❓ Q2: How to represent decimal 18 in hex?
✅ A: 18 ÷ 16 = 1 (Remainder 2) → 12₁₆.
❓ Q3: Convert CSS color #A52A2A to binary.
✅ A:
A → 1010 | 5 → 0101 | 2 → 0010 | A → 1010 | 2 → 0010 | A → 1010 = **101001010010101000101010₂**
6. Practice Problems
Convert
3D7₁₆to decimal.Convert
10101111₂to hex.Convert
762₈to hex.
(Answers: 983₁₀, AF₁₆, 1F2₁₆)
7. Summary
Hexadecimal = Base-16 (Digits
0-9, A-F).Conversions:
Hex ⇄ Binary (4-bit groups).
Hex ⇄ Decimal (Multiply by 16ⁿ).
Hex ⇄ Octal (Convert via binary).
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