Number Base Converter

A number base (or radix) defines how many unique digits are used to represent numbers in a positional numeral system. The four most common bases in computing are: decimal (base 10, digits 0-9), binary (base 2, digits 0-1), octal (base 8, digits 0-7), and hexadecimal (base 16, digits 0-9 and A-F). Each position in a number represents a power of the base. For example, the decimal number 255 means 2x10^2 + 5x10^1 + 5x10^0 = 255, while the binary number 11111111 means 1x2^7 + 1x2^6 + ... + 1x2^0 = 255.

Binary is the fundamental language of digital computers because transistors operate in two states (on/off, 1/0). Every piece of data -- text, images, videos, programs -- is ultimately stored and processed as sequences of binary digits (bits). However, reading long binary strings is error-prone for humans, so octal and hexadecimal serve as compact shorthand. Each octal digit represents exactly 3 binary digits (bits), and each hexadecimal digit represents exactly 4 bits. This is why 255 in decimal = 11111111 in binary = 377 in octal = FF in hexadecimal.

The conversion algorithm works in two steps. First, the input is parsed from its source base into an intermediate decimal value by summing digit x base^position for each digit. Second, the decimal value is repeatedly divided by the target base, collecting remainders in reverse order to form the result. For example, converting decimal 255 to hexadecimal: 255 / 16 = 15 remainder 15 (F), 15 / 16 = 0 remainder 15 (F), reading remainders bottom-up gives FF. The prefixes 0b (binary), 0o (octal), and 0x (hexadecimal) are standard conventions for indicating the base of a written number.

Step-by-step instructions:

1. Select the input base by clicking one of the four buttons: Binary, Decimal, Hexadecimal, or Octal.
2. Type your number in the input field. The placeholder shows an example value for the selected base.
3. If you enter an invalid character for the selected base (e.g., "2" in binary or "G" in hexadecimal), a validation error appears immediately.
4. All four base representations are displayed simultaneously in the results grid below, with the source base highlighted in blue.
5. Click the copy button next to any result to copy that specific representation to your clipboard.
6. Switching the input base clears the input field so you can enter a new value in the selected base.

Hexadecimal input is case-insensitive: both "ff" and "FF" are accepted. Results are displayed in uppercase for consistency.

• Programming and debugging: Developers frequently need to convert between hex and binary when working with memory addresses, bitwise operations, file headers, and low-level protocols.
• Web development (CSS colors): CSS hex color codes like #FF5733 are hexadecimal representations of RGB values. Converting FF to decimal gives 255 (full red), 57 gives 87 (green), 33 gives 51 (blue).
• Networking (IP addresses and subnets): Subnet masks and IP addresses are often analyzed in binary to understand network/host bit boundaries. For example, /24 means 11111111.11111111.11111111.00000000 in binary.
• File format analysis: Hex editors display file contents in hexadecimal. Understanding hex-to-binary conversion is essential for analyzing file headers, magic bytes, and binary protocols.
• Digital electronics: Hardware engineers work with binary and hexadecimal when designing logic circuits, programming FPGAs, and reading datasheets that specify register values in hex.
• Assembly language programming: Machine code instructions and memory addresses are conventionally written in hexadecimal for readability while maintaining a direct mapping to binary.
• Permissions (Unix/Linux): Unix file permissions use octal notation: chmod 755 means rwxr-xr-x in binary (111 101 101), where each octal digit maps to 3 permission bits.