Number Base Converter
Convert numbers across Binary, Decimal, Hexadecimal, and Octal with live simultaneous conversion, step-by-step mathematical workings, bit length.
Active Number Input
1101 0101
0xD5
213
325
Step-by-Step Mathematical Workings
Successive division algorithm (Decimal to Binary):
213 = (1×2⁷) + (1×2⁶) + (0×2⁵) + (1×2⁴) + (0×2³) + (1×2²) + (0×2¹) + (1×2⁰)
⚖️ Decision Verdict: 8-Bit Unsigned Byte
Value 213 fits perfectly in a single 8-bit unsigned byte (0 to 255). In hexadecimal, it is represented as 0xD5.
Bitwise Architecture & Memory Telemetry
About this tool
How to use it
- Select your starting base (Decimal base 10, Binary base 2, Hexadecimal base 16, or Octal base 8).
- Enter any integer value in the input field (characters are validated live against the selected radix).
- View instant simultaneous conversion across Binary (formatted in 4-bit nibbles), Hexadecimal (0x prefix), Decimal, and Octal.
- Click the Copy button next to any representation for rapid clipboard pasting into your code editor.
- Study the step-by-step successive division workings and positional powers-of-two expansion formula below.
Why it matters
Modern computer architecture operates entirely on binary logic circuits (0s and 1s), which software engineers abstract into hexadecimal (base 16) and octal (base 8) for readable memory addresses, color codes, file permissions (chmod 755), and network protocols. Being able to effortlessly convert between these bases and inspect bitwise telemetry is essential for systems programming, cryptography, and computer science education.
When to use it
Use this converter when writing bitwise masks, decoding network packet headers, configuring Linux file permissions, inspecting binary memory dumps, or completing computer science homework.
Good to know
Nibble Formatting: Binary values are grouped into 4-bit nibbles (e.g., 1101 0101) because each 4-bit nibble maps directly to exactly one hexadecimal digit (1101 = D, 0101 = 5 -> 0xD5).
Common questions
Why do computers use binary instead of decimal?
Transistors in computer hardware operate as physical electrical switches with two reliable states: ON (high voltage, 1) and OFF (low voltage, 0). Designing circuits that distinguish between 10 discrete voltage levels for decimal would be vastly more error-prone, noisy, and expensive.
How does hexadecimal relate to binary?
Because 16 is equal to 2⁴, exactly four binary bits (one 'nibble') correspond to one hexadecimal character (0 through F). For example, binary 1111 1111 is simply 0xFF in hexadecimal, making long binary strings concise and human-readable.
What is Two's Complement?
Two's complement is the mathematical method modern CPUs use to represent signed negative integers in binary. The most significant bit (MSB) acts as the sign bit: if it is 1, the number is negative, calculated by inverting all bits and adding 1.
What is the difference between Big-Endian and Little-Endian?
Endianness describes byte order in memory. Big-endian stores the most significant byte first (standard in TCP/IP network protocols). Little-endian stores the least significant byte first (standard in x86 and ARM processor architectures).
Where is octal (base 8) used today?
Octal is most famously used in Unix and POSIX file system permissions (`chmod 755`), where each octal digit represents a 3-bit group of read (4), write (2), and execute (1) permissions for user, group, and others.