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Understanding Hexadecimal Calculations
Hexadecimal (base-16) is a numerical system widely used in computing and digital systems. It provides a more human-friendly representation of binary-coded values and is commonly used in programming, memory addressing, and color codes.
1. Hexadecimal Basics
The hexadecimal system uses 16 distinct symbols: 0-9 to represent values zero to nine, and A-F to represent values ten to fifteen.
Decimal values: 0-15
Practical Example: Color Codes
Web colors are often represented as hexadecimal triplets:
#RRGGBB where RR (red), GG (green), BB (blue)
Example: #FF0000 is pure red, #00FF00 is pure green, #0000FF is pure blue
2. Hexadecimal to Decimal Conversion
Each hex digit represents four binary digits (bits), and each position represents a power of 16.
Conversion Example: 2AF3
(2 × 16³) + (A × 16²) + (F × 16¹) + (3 × 16⁰)
= (2 × 4096) + (10 × 256) + (15 × 16) + (3 × 1)
= 8192 + 2560 + 240 + 3 = 10995
3. Decimal to Hexadecimal Conversion
Divide the decimal number by 16 repeatedly and use the remainders to build the hex value.
remainder = number % 16
hex_digit = get_hex_digit(remainder)
number = number // 16
hex_value = hex_digit + hex_value
Conversion Example: 2545
2545 ÷ 16 = 159 remainder 1 (hex digit: 1)
159 ÷ 16 = 9 remainder 15 (hex digit: F)
9 ÷ 16 = 0 remainder 9 (hex digit: 9)
Reading remainders in reverse: 9F1
4. Hexadecimal Arithmetic
Hexadecimal arithmetic follows the same rules as decimal arithmetic, but with base 16 instead of base 10.
Subtraction: When borrowing, each borrowed unit is worth 16
Multiplication/Division: Similar to decimal but with base 16
Example: Adding 1A and 2F
1A (26 in decimal) + 2F (47 in decimal) = 49 in decimal
In hex: A + F = 19 (write 9, carry 1)
1 + 1 + 2 = 4
Result: 49 (hex) which is 73 in decimal
5. Bitwise Operations
Bitwise operations work on the binary representation of numbers and are fundamental in low-level programming.
OR (|): 1 if either bit is 1
XOR (^): 1 if bits are different
NOT (~): Inverts all bits
Shift Left (<<): Multiply by 2ⁿ
Shift Right (>>): Divide by 2ⁿ
Example: Bitwise AND of 5B and 3C
5B in binary: 01011011
3C in binary: 00111100
AND result: 00011000 (18 in hex)
Common Uses of Hexadecimal
- Memory Addressing: Computer memory addresses are often represented in hex
- Color Codes: Web colors use hexadecimal notation (e.g., #FFFFFF for white)
- Debugging: Memory dumps and error codes often use hex
- Assembly Language: Machine-level programming frequently uses hex
- File Formats: Many file formats use hex for headers and signatures
Frequently Asked Questions
Q: Why is hexadecimal used in computing?
A: Hexadecimal is convenient because it can represent every byte (8 bits) as two consecutive hex digits. It's more compact than binary and easier for humans to read and remember than long binary strings.
Q: How do I know if a number is hexadecimal?
A: Hexadecimal numbers are often prefixed with "0x" (e.g., 0x1A) or suffixed with "h" (e.g., 1Ah). In contexts where it's clear (like CSS colors), they may appear without prefixes.
Q: What's the largest single hex digit?
A: The largest single hex digit is F, which represents the decimal value 15 (binary 1111).
Q: How do I convert a large decimal number to hex?
A: Divide the number by 16 repeatedly and note the remainders. The hex number is the remainders read from last to first, with 10-15 represented as A-F.
Q: What's the difference between signed and unsigned hex numbers?
A: Unsigned hex numbers are always positive. Signed hex numbers use the most significant bit to indicate sign (1 for negative). The interpretation depends on the context and number of bits.