Perform hexadecimal arithmetic operations, conversions, and bitwise calculations with real-time results, step-by-step solutions, and comprehensive validation. Supports multiple bit widths and number systems.
Maximum value: FFFFFFFF₁₆ (4,294,967,295₁₀)
Enter hex numbers (0-9, A-F)
Enter two numbers and select an operation to see real-time results.
Master hexadecimal mathematics, conversions, and computer science applications with our comprehensive guide.
The hexadecimal number system, also known as base-16 or simply "hex," is a positional numeral system that uses 16 distinct symbols to represent values. It extends the familiar decimal system by adding six additional symbols (A, B, C, D, E, F) to represent values from 10 to 15. This system has become indispensable in computer science, digital electronics, and software development.
Hexadecimal serves as a bridge between human-readable numbers and binary machine code. Since 16 = 2⁴, each hexadecimal digit represents exactly four binary digits (bits), making it incredibly efficient for representing large binary numbers in a compact, readable format. This relationship makes hexadecimal essential for programmers, system administrators, and digital engineers.
Complete mapping: 0-9 remain the same, 10=A, 11=B, 12=C, 13=D, 14=E, 15=F
While hexadecimal notation gained prominence with the rise of digital computers in the 1960s, base-16 concepts have ancient roots. The system became standardized in computing because early computer architectures often used word sizes that were multiples of 4 bits (nibbles). IBM's System/360 in the 1960s popularized hexadecimal notation for representing memory addresses and machine instructions, establishing conventions still used today.
Hexadecimal numbers follow the same positional notation principle as decimal numbers, but with base 16 instead of base 10. Each position represents a power of 16, starting from 16⁰ on the right and increasing as you move left. This structure allows for efficient representation of large numbers and provides a direct correlation with binary representation.
Hexadecimal numbers are commonly prefixed with "0x" in programming contexts (e.g., 0x1A2F) or suffixed with subscript ₁₆ in mathematical notation (e.g., 1A2F₁₆). Case doesn't matter for the letter digits—both "abc" and "ABC" represent the same values. However, uppercase is more common in professional contexts for better readability.
Different programming languages and systems use various conventions for hexadecimal notation. Understanding these conventions is crucial for cross-platform development and system integration.
Hexadecimal addition follows the same principles as decimal addition, but with a base of 16 instead of 10. When the sum of two hex digits exceeds F (15), you carry 1 to the next position and the remainder becomes the digit in the current position. This process requires familiarity with hex-to-decimal conversions for digits A through F.
F+3=18₁₀=12₁₆, carry 1; 7+B+1=19₁₀=13₁₆; A+1=B₁₆
Hexadecimal subtraction uses borrowing when a digit in the minuend is smaller than the corresponding digit in the subtrahend. When borrowing in hex, you borrow 16 (instead of 10 in decimal) from the next higher position. This process requires converting hex digits to their decimal equivalents for accurate calculation.
A<C: borrow 16, 26-12=14₁₆=E; 1<B: borrow 16, 17-11=6; E-5=9
Hexadecimal multiplication requires knowledge of the hex multiplication table or conversion to decimal for complex calculations. For single-digit multiplication, convert each hex digit to decimal, multiply, then convert back to hex. Multi-digit multiplication follows the same algorithm as decimal multiplication.
| × | 2 | 4 | 8 | A | C | F |
|---|---|---|---|---|---|---|
| A | 14 | 28 | 50 | 64 | 78 | 96 |
| C | 18 | 30 | 60 | 78 | 90 | B4 |
| F | 1E | 3C | 78 | 96 | B4 | E1 |
Hexadecimal division follows the same long division algorithm as decimal, but requires hex arithmetic for each step. For complex divisions, it's often easier to convert to decimal, perform the division, then convert the result back to hex. Integer division in hex programming contexts often uses bitwise operations for efficiency.
Verification: 23₁₆ × C₁₆ = (35 × 12)₁₀ = 420₁₀ = 1A4₁₆ ✓
Converting decimal to hexadecimal uses the division method, repeatedly dividing by 16 and collecting remainders. This method is systematic and works for any decimal number. Understanding this process is crucial for manual calculations and helps in understanding how computer systems perform base conversions.
The relationship between hexadecimal and binary is particularly elegant: each hex digit corresponds to exactly four binary digits. This 1:4 relationship makes conversion straightforward and is why hex is so valuable in computing—it provides a compact representation of binary data.
Beyond basic conversion methods, there are several advanced techniques used in computer science and mathematics for efficient base conversions. These methods are particularly useful in programming and digital system design.
For efficient conversion of large hex numbers: A3F₁₆
Hexadecimal can represent fractional values using a decimal point (or "hexadecimal point"). Positions to the right of the point represent negative powers of 16: 16⁻¹, 16⁻², etc. This is particularly useful in IEEE floating-point representations and fixed-point arithmetic.
Bitwise operations form the foundation of computer logic and are particularly elegant when working with hexadecimal numbers. Each hex digit represents 4 bits, making it easy to visualize and perform bitwise operations. These operations are essential in system programming, cryptography, graphics programming, and embedded systems development. For binary calculations, these principles remain fundamental.
Both bits must be 1 for result to be 1
At least one bit must be 1 for result to be 1
Bits must be different for result to be 1
Inverts all bits (0→1, 1→0)
Bit shifting operations move bits left or right within a number, effectively multiplying or dividing by powers of 2. These operations are extremely fast and are commonly used in optimization, hash functions, and low-level system programming. Understanding shifts in hexadecimal helps visualize the movement of entire hex digits.
Shifts bits left, filling with zeros
Shifts bits right, behavior varies by type
Masking operations use bitwise AND, OR, and XOR to selectively manipulate specific bits or groups of bits within a number. This technique is fundamental in embedded programming, graphics processing, and system-level optimization. Hexadecimal representation makes mask patterns particularly clear and easy to understand.
Extract RGB components from color value 3FA5B2₁₆:
RGB(63, 165, 178) - a teal blue color
Hexadecimal is the standard notation for colors in digital design and web development. RGB color values are typically represented as 6-digit hex codes, where each pair of digits represents the red, green, and blue components. This system provides 16.7 million possible colors (256³) and is universally supported across platforms and applications.
Extended to 8-digit hex for transparency:
Alpha: 00 (transparent) to FF (opaque)
In system programming, memory addresses are traditionally displayed in hexadecimal because they align with byte boundaries and word sizes. A 64-bit address fits neatly into 16 hex digits, making it easier to read and debug compared to 64 binary digits or 20 decimal digits. This convention is used across operating systems, debuggers, and assembly language programming.
Each address is aligned to byte boundaries (divisible by powers of 2)
Network interfaces use hexadecimal MAC addresses (Media Access Control) for unique identification. These 48-bit addresses are typically displayed as six pairs of hex digits separated by colons or hyphens. IPv6 addresses also use hexadecimal notation, employing colons to separate 16-bit groups and supporting compression notation for consecutive zeros.
First 3 bytes identify manufacturer
:: represents consecutive zero groups
Cryptographic hash functions like SHA-256, MD5, and others output fixed-length bit strings that are conventionally represented in hexadecimal. This representation makes it easy to compare hashes, verify integrity, and work with cryptographic keys. The hex format is also used in blockchain technologies, digital signatures, and secure communications.
Embedded systems programming heavily relies on hexadecimal for register configurations, memory-mapped I/O, and hardware interfacing. Microcontroller datasheets specify register addresses and bit fields in hex, making it essential for firmware development. IoT devices use hex for sensor data encoding, protocol implementations, and low-level hardware control.
The IEEE 754 standard for floating-point arithmetic uses hexadecimal representation in its binary format. Understanding how floating-point numbers are encoded in hex is crucial for system programming, graphics programming, and numerical analysis. The standard defines precise bit layouts for single and double precision numbers.
Hexadecimal representation reveals interesting mathematical properties and patterns. Modular arithmetic, greatest common divisors, and prime testing algorithms often work more efficiently with hex representation. The base-16 system also provides insights into divisibility rules and algebraic structures.
These exercises focus on basic hex operations and conversions. Work through them systematically to build confidence with hexadecimal arithmetic and develop intuition for hex-decimal relationships. Use the calculator above to check your work.
These problems combine multiple concepts and require deeper understanding of hexadecimal properties. They introduce bitwise operations, multi-digit calculations, and practical applications commonly encountered in programming and system administration.
A program allocates memory starting at address 0x10000000. If each object is 0x2C bytes, what's the address of the 100th object?
Given color #FF6A33, increase the green component by 0x20 and decrease the blue by 0x10. What's the new color?
Common questions about hexadecimal calculations and our calculator
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