🔢 Hexadecimal Conversion

🔢 Working with Hexadecimal Numbers

Hexadecimal (base-16) is a number system commonly used in computing as a more human-friendly way to represent binary data. Understanding hexadecimal and how to convert between number systems is an essential skill in information technology.

🔍 Hexadecimal Number System

What is Hexadecimal?

• A base-16 number system
• Uses 16 digits: 0-9 and A-F (where A=10, B=11, C=12, D=13, E=14, F=15)
• Position values are powers of 16 (16^0, 16^1, 16^2, etc.)
• Often prefixed with "0x" or suffixed with "h" to indicate hexadecimal

🤔 Why Hexadecimal is Used

• Compact representation of binary data
• One hexadecimal digit represents exactly 4 binary digits (bits)
• Easier for humans to read and write than long binary sequences
• Commonly used for memory addresses, color codes, and machine code

🔄 Hexadecimal to Denary Conversion

To convert hexadecimal to denary (base-10):

1. Identify the position value of each digit (powers of 16 from right to left)
2. Convert each hexadecimal digit to its denary value
3. Multiply each digit value by its position value
4. Sum all the results

Example 1: Convert 2F₁₆ to denary

2    F
↓    ↓
16^1 16^0
16   1
↓    ↓
2×16 15×1
32   15

Sum: 32 + 15 = 47₁₀

Example 2: Convert 1A3₁₆ to denary

1    A    3
↓    ↓    ↓
16^2 16^1 16^0
256  16   1
↓    ↓    ↓
1×256 10×16 3×1
256   160   3

Sum: 256 + 160 + 3 = 419₁₀

🔄 Denary to Hexadecimal Conversion

To convert denary to hexadecimal:

1. Divide the number by 16
2. Record the remainder (0-15, using A-F for 10-15)
3. Continue dividing the quotient by 16 until the quotient becomes 0
4. Read the remainders from bottom to top to get the hexadecimal number

Example 1: Convert 250₁₀ to hexadecimal

Division    Quotient    Remainder
250 ÷ 16    15          10 (A)
15 ÷ 16     0           15 (F)

Reading remainders from bottom to top: FA₁₆

Example 2: Convert 52₁₀ to hexadecimal

Division    Quotient    Remainder
52 ÷ 16     3           4
3 ÷ 16      0           3

Reading remainders from bottom to top: 34₁₆

🔀 Binary to Hexadecimal Conversion

To convert binary to hexadecimal:

1. Group the binary digits in sets of 4, starting from the right
2. Convert each group of 4 bits to its hexadecimal equivalent
3. Combine the hexadecimal digits

Example 1: Convert 1010 1101₂ to hexadecimal

1010  1101
 ↓     ↓
 A     D

Result: AD₁₆

Example 2: Convert 11 1110 0001₂ to hexadecimal

0011  1110  0001
 ↓     ↓     ↓
 3     E     1

Result: 3E1₁₆

🔀 Hexadecimal to Binary Conversion

To convert hexadecimal to binary:

1. Convert each hexadecimal digit to its 4-bit binary equivalent
2. Combine all binary groups

Example 1: Convert C5₁₆ to binary

C    5
↓    ↓
1100 0101

Result: 1100 0101₂

Example 2: Convert 2FA₁₆ to binary

2     F     A
↓     ↓     ↓
0010  1111  1010

Result: 0010 1111 1010₂

📝 Common Hexadecimal Values to Remember

Hex	Binary	Denary		Hex	Binary	Denary
0	0000	0		8	1000	8
1	0001	1		9	1001	9
2	0010	2		A	1010	10
3	0011	3		B	1011	11
4	0100	4		C	1100	12
5	0101	5		D	1101	13
6	0110	6		E	1110	14
7	0111	7		F	1111	15

💻 Practical Applications

Hexadecimal is commonly used in:

• 🧠 Memory addresses in programming and debugging
• 🎨 Color codes in web design (e.g., #FF0000 for red)
• 🌐 MAC addresses for network interfaces
• 🔍 File signatures and hexadecimal editors
• 🖥️ Assembly language and machine code representation

Understanding hexadecimal conversion is essential for many aspects of computer science and digital systems work.