Hex Calculator
Perform addition, subtraction, multiplication, and division in Base-16 hexadecimal notation. Seamlessly convert between hex, 4-bit grouped binary, decimal integers, and octal with two's complement register support.
Mathematical Theory of Hexadecimal Systems
A comprehensive technical analysis of base-16 positional representation, nibble symmetry, and two's complement binary math.
Radix-16 Positional Notation
Hexadecimal represents numbers using base 16. Each place value represents an increasing integer power of 16. For an n-digit hex string d_(n-1)...d_1 d_0:
Digits 0–9 map to decimal 0–9, while letters A, B, C, D, E, F represent values 10, 11, 12, 13, 14, and 15 respectively.
4-Bit Nibble Symmetry
Because 16 = 2⁴, exactly four binary bits constitute one hexadecimal character. This 4-bit unit is called a nibble:
This exact power-of-two alignment eliminates fractional carryovers when converting between binary machine code and human-readable hex logs.
Two's Complement Negation
In fixed-width registers (16, 32, or 64 bits), negative integers are represented using two's complement math:
For example, in a 16-bit register, -1 is encoded as 0xFFFF and -42 is encoded as 0xFFD6. The high bit (MSB) acts as the sign flag.
Hexadecimal Conversion & Architecture Matrix
Complete reference mappings across hexadecimal digits, 4-bit binary nibbles, decimal integers, octal values, and standard register memory limits.
Complete 16-Digit Base Mapping Table
| Hex (Base-16) | Binary (Base-2) | Decimal (Base-10) | Octal (Base-8) | Bitweight Summation |
|---|---|---|---|---|
| 0x0 | 0000₂ | 0₁₀ | 00₈ | 0 |
| 0x1 | 0001₂ | 1₁₀ | 01₈ | 2⁰ = 1 |
| 0x2 | 0010₂ | 2₁₀ | 02₈ | 2¹ = 2 |
| 0x3 | 0011₂ | 3₁₀ | 03₈ | 2¹ + 2⁰ = 3 |
| 0x4 | 0100₂ | 4₁₀ | 04₈ | 2² = 4 |
| 0x5 | 0101₂ | 5₁₀ | 05₈ | 2² + 2⁰ = 5 |
| 0x6 | 0110₂ | 6₁₀ | 06₈ | 2² + 2¹ = 6 |
| 0x7 | 0111₂ | 7₁₀ | 07₈ | 2² + 2¹ + 2⁰ = 7 |
| 0x8 | 1000₂ | 8₁₀ | 10₈ | 2³ = 8 |
| 0x9 | 1001₂ | 9₁₀ | 11₈ | 2³ + 2⁰ = 9 |
| 0xA | 1010₂ | 10₁₀ | 12₈ | 2³ + 2¹ = 10 |
| 0xB | 1011₂ | 11₁₀ | 13₈ | 2³ + 2¹ + 2⁰ = 11 |
| 0xC | 1100₂ | 12₁₀ | 14₈ | 2³ + 2² = 12 |
| 0xD | 1101₂ | 13₁₀ | 15₈ | 2³ + 2² + 2⁰ = 13 |
| 0xE | 1110₂ | 14₁₀ | 16₈ | 2³ + 2² + 2¹ = 14 |
| 0xF | 1111₂ | 15₁₀ | 17₈ | 2³ + 2² + 2¹ + 2⁰ = 15 |
CPU Register Architecture & Boundaries
| Register Type | Bit Width | Hex Digits | Max Unsigned Value | Signed Range (Two's Complement) |
|---|---|---|---|---|
| Byte | 8 Bits | 2 Digits | 0xFF (255) | -128 to +127 (0x80 to 0x7F) |
| Word | 16 Bits | 4 Digits | 0xFFFF (65,535) | -32,768 to +32,767 (0x8000 to 0x7FFF) |
| Doubleword (Dword) | 32 Bits | 8 Digits | 0xFFFFFFFF (4,294,967,295) | -2,147,483,648 to +2,147,483,647 |
| Quadword (Qword) | 64 Bits | 16 Digits | 0xFFFFFFFFFFFFFFFF (18.4 Quintillion) | -9,223,372,036,854,775,808 to +9.22 × 10¹⁸ |
Frequently Asked Questions
Clear technical explanations regarding hexadecimal notation, binary bit patterns, and memory representation.
Why does hexadecimal use the letters A through F?
Hexadecimal is a base-16 positional numeral system. While decimal uses ten symbols (0 through 9), base-16 requires sixteen unique digits. The Latin letters A, B, C, D, E, and F were universally standardized to represent values 10, 11, 12, 13, 14, and 15 respectively without requiring multi-digit notation.
How does 1 hex digit relate directly to 4 binary bits (a nibble)?
Because 16 = 2^4, exactly four binary bits can represent sixteen distinct values (0000 to 1111 in binary). Therefore, each hexadecimal character maps one-to-one to exactly one 4-bit nibble. A standard 8-bit byte is expressed concisely by exactly two hex digits (e.g., 1111 1111 in binary = FF in hex = 255 in decimal).
How does two's complement represent negative hex numbers in 16-bit and 32-bit registers?
In signed binary and hexadecimal representations, the most significant bit (MSB) acts as the sign flag. If the high bit is 1 (in hex, if the leading digit is 8 through F), the number is negative. To find the two's complement, invert all bits (one's complement) and add 1. For example, -1 in a 32-bit register is represented as 0xFFFFFFFF, and -256 is 0xFFFFFF00.
What is the difference between uppercase and lowercase hex characters?
In computational mathematics, programming compilers, and memory addressing, uppercase (A-F) and lowercase (a-f) hex characters are completely identical in value and interchangeable. For example, 0x7F2A and 0x7f2a evaluate to the exact same decimal quantity (32,554). Our calculator accepts both and standardizes outputs to uppercase for clean reading.
How are hex values used in 24-bit RGB web colors and memory offsets?
A 24-bit TrueColor hex code (like #3B82F6) bundles three 8-bit color channels: Red (#3B = 59), Green (#82 = 130), and Blue (#F6 = 246). In computer memory architecture, pointers and register offsets are formatted in hex (such as 0x7FFF0100) because grouping 32-bit or 64-bit addresses into 8 or 16 hex characters is vastly more readable than 32 or 64 binary digits.
Related Computational & Mathematical Tools
Explore our complete suite of low-level computer science, binary, and statistical calculators.
Binary Calculator
Calculate base-2 arithmetic, bitwise shifts, and two's complement.
Log Calculator
Solve custom base logarithms, natural log ln, and base-10 log10.
Scientific Notation Calculator
Convert standard decimal numbers to exponential scientific form.
Basic Math Calculator
Arithmetic solver with strict PEMDAS/BODMAS order of operations.