Hexadecimal Number System Definition Base 16 Digits Conversion Method and Solved Examples
The concept of Hexadecimal Number System plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Hexadecimal Number System?
A Hexadecimal Number System is defined as a positional number system having a base of 16. It uses 16 symbols: numerals 0–9 and letters A–F, where A stands for 10, B for 11, C for 12, D for 13, E for 14, and F for 15 in decimal. You’ll find this concept applied in areas such as computer science (memory addresses, programming), digital electronics, and color coding in web design.
Key Features of the Hexadecimal Number System
- It is a base 16 system (positions are weighted by powers of 16)
- Digits used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
- Letters represent values above 9: A = 10, B = 11, ..., F = 15
- Common in computer technology due to easy conversion with binary
- Compactly represents large numbers compared to decimal or binary
Hexadecimal Numbers Table (0 to 20)
| Decimal | Hexadecimal | Binary |
|---|---|---|
| 0 | 0 | 0000 |
| 1 | 1 | 0001 |
| 2 | 2 | 0010 |
| 3 | 3 | 0011 |
| 4 | 4 | 0100 |
| 5 | 5 | 0101 |
| 6 | 6 | 0110 |
| 7 | 7 | 0111 |
| 8 | 8 | 1000 |
| 9 | 9 | 1001 |
| 10 | A | 1010 |
| 11 | B | 1011 |
| 12 | C | 1100 |
| 13 | D | 1101 |
| 14 | E | 1110 |
| 15 | F | 1111 |
| 16 | 10 | 0001 0000 |
| 17 | 11 | 0001 0001 |
| 18 | 12 | 0001 0010 |
| 19 | 13 | 0001 0011 |
| 20 | 14 | 0001 0100 |
Hexadecimal vs Other Number Systems
| System | Base | Digits Used | Example: 16 |
|---|---|---|---|
| Decimal | 10 | 0-9 | 16 |
| Binary | 2 | 0, 1 | 10000 |
| Octal | 8 | 0-7 | 20 |
| Hexadecimal | 16 | 0-9, A-F | 10 |
How to Convert Decimal to Hexadecimal
Steps to convert a decimal number to hexadecimal:
1. Divide the decimal number by 16 and write down the remainder.2. Divide the quotient again by 16; write the new remainder.
3. Repeat until the quotient is 0.
4. The hexadecimal value is the string of remainders written in reverse order. Use A–F for remainders 10–15.
Example: Convert 24210 to hexadecimal.
1. 242 ÷ 16 = 15, remainder 22. 15 ÷ 16 = 0, remainder 15 (F)
3. Collect remainders from bottom: F2
So, 24210 = F216.
How to Convert Hexadecimal to Decimal
Multiply each hex digit by the power of 16 based on its position (rightmost = 160) and add them up.
Example: Convert AB416 to decimal.
1. A = 10, B = 11, 4 = 42. 10 × 162 = 2560
3. 11 × 161 = 176
4. 4 × 160 = 4
5. Add: 2560 + 176 + 4 = 274010
Step-by-Step Conversion: Binary to Hexadecimal
Group the binary number into blocks of 4 digits (from right). Convert each block to its hex equivalent.
Example: Convert 1100011111012 to hexadecimal.
1. Group: 1 1000 1111 101 (pad leftmost group: 0001 1000 1111 1101)2. 0001 = 1, 1000 = 8, 1111 = F, 1101 = D
3. Answer: 18FD16
Real-Life and Cross-Disciplinary Usage
The hexadecimal number system is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for exams like JEE or Olympiads will see its relevance in:
- HTML/CSS color codes (e.g., #FF5733 for orange)
- Memory addresses in computers (compact representation of bytes)
- Machine-level programming and debugging
- Storing binary data efficiently
Vedantu provides easy video explanations about connections between hexadecimal, binary, and decimal for both school and competitive exams.
Speed Trick or Common Shortcut
To convert a large binary to hex quickly, split binary into 4-bit blocks (right to left) and replace each with its hexadecimal equivalent using a chart. This reduces long calculations and avoids errors.
Example Trick: 1111 11012 → F D16
Tricks like these save time in board exams and are used in many Vedantu live doubt-sessions.
Try These Yourself
- Write the hexadecimal numbers from 1 to 16.
- Convert decimal 121 to hexadecimal.
- Convert 7B516 to decimal.
- What is the hex value for 1011012?
Frequent Errors and Misunderstandings
- Forgetting that after F, the next value is 10 (not G).
- Mixing up the position value (powers of 16 are critical).
- Confusing hexadecimal with octal (base 8) or decimal (base 10).
- Using the wrong binary group size (always 4 bits for hex).
Relation to Other Concepts
The idea of Hexadecimal Number System connects closely with topics such as the Binary Number System and the Decimal Number System. Mastering this topic helps with tasks like number system conversions, programming basics, and digital electronics circuits.
Classroom Tip
A quick way to remember hexadecimal letters is to use the mnemonic "A Big Cat Doesn't Eat Fish" for A-F (A=10, ..., F=15). Vedantu’s teachers highlight such patterns in live classes for easier recall.
We explored Hexadecimal Number System—from definition, chart, examples, conversion methods, errors, and connections to other number systems. Continue practicing with Vedantu to become confident in using this concept in both exams and real-life applications.
Useful Resources and Further Reading
FAQs on Hexadecimal Number System in Mathematics
1. What is the hexadecimal number system?
The hexadecimal number system is a base-16 number system that uses sixteen symbols: 0–9 and A–F. In this system:
- The digits 0–9 represent values zero to nine.
- The letters A, B, C, D, E, F represent decimal values 10–15.
- Each place value is a power of 16 (e.g., 16⁰, 16¹, 16²).
2. How do you convert hexadecimal to decimal?
To convert hexadecimal to decimal, multiply each digit by its corresponding power of 16 and add the results. Steps:
- Write place values as powers of 16.
- Convert letters (A–F) to decimal values (10–15).
- Multiply and add.
- 2 × 16¹ = 2 × 16 = 32
- F (15) × 16⁰ = 15 × 1 = 15
- Total = 47
3. How do you convert decimal to hexadecimal?
To convert decimal to hexadecimal, repeatedly divide the number by 16 and record the remainders. Steps:
- Divide the decimal number by 16.
- Write down the remainder.
- Repeat until the quotient is 0.
- Read remainders from bottom to top.
- 75 ÷ 16 = 4 remainder 11 (B)
- 4 ÷ 16 = 0 remainder 4
4. Why is hexadecimal used in computers?
Hexadecimal is used in computers because it provides a compact and readable representation of binary numbers. Key reasons:
- One hexadecimal digit represents exactly 4 binary bits.
- It simplifies large binary values.
- Common in memory addresses, machine code, and color codes.
5. What is the difference between hexadecimal and decimal number systems?
The main difference is that hexadecimal is base-16 while decimal is base-10. Comparison:
- Decimal system: Uses digits 0–9 and powers of 10.
- Hexadecimal system: Uses digits 0–9 and A–F with powers of 16.
- Decimal example: 345 = 3×10² + 4×10¹ + 5×10⁰.
- Hex example: 2F = 2×16¹ + 15×16⁰.
6. How do you convert hexadecimal to binary?
To convert hexadecimal to binary, replace each hex digit with its 4-bit binary equivalent. Steps:
- Convert each hexadecimal digit separately.
- Use 4-bit binary representation.
- 3 = 0011
- A (10) = 1010
7. How do you convert binary to hexadecimal?
To convert binary to hexadecimal, group binary digits into sets of four from right to left and convert each group. Steps:
- Split into 4-bit groups.
- Convert each group to its hexadecimal equivalent.
- 1010 = A
- 1101 = D
8. What are the digits used in the hexadecimal number system?
The hexadecimal number system uses sixteen digits: 0–9 and A–F. Their values are:
- 0–9 represent 0 to 9.
- A = 10
- B = 11
- C = 12
- D = 13
- E = 14
- F = 15
9. Can you give an example of hexadecimal addition?
Hexadecimal addition follows base-16 rules, carrying over when the sum is 16 or more. Example: Add A + 7.
- A = 10
- 10 + 7 = 17 (decimal)
- 17 ÷ 16 = 1 remainder 1
10. What are common mistakes when working with hexadecimal numbers?
Common mistakes in the hexadecimal number system usually involve place values and letter conversions. Typical errors include:
- Forgetting that A–F represent 10–15.
- Using powers of 10 instead of powers of 16.
- Incorrect binary grouping (not using 4 bits).
- Miscalculating carry in hexadecimal addition.
