Hexadecimal Number System: Definition, Conversion & Uses

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 2

2. 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 = 4

2. 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

• Binary Number System
• Decimal Number System
• Number System Conversion
• Octal Number System