Add, subtract, multiply, and divide hexadecimal numbers (Base 16), or convert between Hexadecimal and Decimal with step-by-step mathematical breakdowns, Two's Complement signed notation, and cross-base equivalents (Binary & Octal).
• Module 1: Perform base-16 arithmetic (+, -, ×, ÷) on arbitrary length hex values with carry, borrow, quotient & remainder calculation.
• Module 2: Convert any hexadecimal string to decimal with full positional weight expansion ($16^n$).
• Module 3: Convert base-10 decimal numbers into hexadecimal using successive division-by-16.
Perform base-16 arithmetic with automatic Decimal, Binary, and Octal cross-conversions.
Convert base-16 hexadecimal values into base-10 decimal numbers with positional power breakdowns.
Convert base-10 decimal numbers to base-16 hexadecimal strings using successive division by 16.
An in-depth technical reference for software engineers, systems architects, and computer science students.
The hexadecimal number system (often abbreviated as hex) is a base-16 positional numeral system. While the standard decimal system uses 10 symbols (0 through 9) and binary uses 2 (0 and 1), hexadecimal requires 16 distinct digits. It achieves this by combining standard decimal digits 0–9 with the English letters A–F to represent values from 10 to 15:
Why Computer Science Uses Hexadecimal: Computers fundamentally process binary bits (0 and 1). However, long streams of binary digits (like 110110101101) are difficult for humans to read, debug, and communicate. Because 16 = 2⁴, exactly 4 binary bits (known as a nibble) map directly to 1 hexadecimal digit. An 8-bit computer byte is concisely represented by exactly 2 hex digits (ranging from 00₁₆ to FF₁₆).
Use this standard reference table to quickly translate between 4-bit binary nibbles, hexadecimal characters, decimal values, and octal codes:
| Hex (Base 16) | Decimal (Base 10) | Binary 4-Bit Nibble (Base 2) | Octal (Base 8) |
|---|---|---|---|
| 0 | 0 | 0000 | 0 |
| 1 | 1 | 0001 | 1 |
| 2 | 2 | 0010 | 2 |
| 3 | 3 | 0011 | 3 |
| 4 | 4 | 0100 | 4 |
| 5 | 5 | 0101 | 5 |
| 6 | 6 | 0110 | 6 |
| 7 | 7 | 0111 | 7 |
| 8 | 8 | 1000 | 10 |
| 9 | 9 | 1001 | 11 |
| A | 10 | 1010 | 12 |
| B | 11 | 1011 | 13 |
| C | 12 | 1100 | 14 |
| D | 13 | 1101 | 15 |
| E | 14 | 1110 | 16 |
| F | 15 | 1111 | 17 |
When adding two hex digits, if their decimal sum reaches or exceeds 16, subtract 16, write the remainder in that column, and carry 1 to the next higher place value.
1423₁₆.When the top hex digit is smaller than the bottom digit, borrow 1 from the next higher column. In base 16, borrowing 1 adds exactly 16 (10₁₆) to the current column.
2CD₁₆.Multiply single digits using base-10 products, divide the product by 16 to determine column quotient (carry) and remainder (digit), then sum partial products using base-16 addition.
Perform standard long division by estimating base-16 multiples of the divisor, subtracting, and carrying down subsequent digits to determine quotient and remainder.
To convert any hex number to decimal, multiply each digit by 16ⁿ, where n represents the digit's position index starting at 0 from the rightmost integer digit:
To convert a decimal integer to hex, divide the number by 16 continuously, record each integer remainder, map remainders 10–15 to A–F, and read the remainders backwards from bottom to top:
Because 16 is a power of 2 (2⁴), conversion between binary and hex is a simple 1-to-1 replacement without any complex division:
Decimal is base 10, using 10 symbols (0 through 9) with column weights increasing by powers of 10 (1, 10, 100, 1000). Hexadecimal is base 16, using 16 symbols (0 through 9, then A through F for 10-15) with column weights increasing by powers of 16 (1, 16, 256, 4096).
In digital logic, signed numbers are represented using Two's Complement notation. In this system, negative values invert all binary bits and add 1. For example, in an 8-bit byte, -1 is represented as 0xFF (1111 1111₂), -2 is 0xFE (1111 1110₂), and -128 is 0x80 (1000 0000₂).
Digital displays use the RGB (Red, Green, Blue) color model, where each color channel is assigned an 8-bit intensity value from 0 to 255. Because 255 in decimal is exactly FF in hex, any 24-bit truecolor shade can be written as a clean 6-character hex string (#RRGGBB).
Computer memory pointers, CPU registers (like x86 EAX/RAX), and network MAC addresses are written in hex because 2 hex digits correspond to 1 memory byte (8 bits). This eliminates long strings of binary while maintaining exact byte alignment.
Yes. The calculation engine utilizes native JavaScript BigInt arithmetic, which supports arbitrary-length precision integers without scientific notation rounding, truncation, or 64-bit IEEE 754 float limits.