Number Base Converter

Number systems are the foundation of how we represent quantities. The decimal system (base 10) is what we use daily, inherited from counting on ten fingers. But computers don't have fingers — they have transistors with two states, making binary (base 2) the natural language of digital hardware.

Octal (base 8) and hexadecimal (base 16) exist as convenient shorthand for binary. Since 8 is 2³ and 16 is 2⁴, each octal digit represents exactly 3 bits and each hex digit represents exactly 4 bits. This clean mapping makes them invaluable for developers working with low-level data.

Binary is essential for understanding bitwise operations, flags, permissions, and hardware registers. In Unix file permissions, 755 (octal) means rwxr-xr-x, with each digit encoding a 3-bit read/write/execute mask. Network subnet masks like 255.255.255.0 are clearer in binary: 11111111.11111111.11111111.00000000.

Hexadecimal dominates in memory addresses, color codes (CSS #RRGGBB), Unicode code points (U+0041 for 'A'), and byte-level debugging. Most hex editors, debuggers, and packet analyzers display data in hexadecimal. Understanding hex is essential for web development, systems programming, and cybersecurity.

Converting between bases involves two steps: parsing the input value into an internal representation, then formatting it in the target base. To convert from any base to decimal, multiply each digit by its positional value (the base raised to the power of its position) and sum the results.

To convert from decimal to any base, repeatedly divide by the target base and collect the remainders. The remainders, read in reverse order, form the result. For example, converting decimal 42 to binary: 42 ÷ 2 = 21 R0, 21 ÷ 2 = 10 R1, 10 ÷ 2 = 5 R0, 5 ÷ 2 = 2 R1, 2 ÷ 2 = 1 R0, 1 ÷ 2 = 0 R1 → 101010.

Base conversion appears throughout computing. IP addresses are 32-bit numbers displayed as four decimal octets. MAC addresses use six pairs of hex digits. Assembly language programmers routinely convert between hex and binary to understand instruction encodings and memory layouts.

In web development, CSS colors use hex (#FF5733) or RGB decimals (rgb(255, 87, 51)). Data encoding schemes like Base64 convert binary data to 64-character ASCII strings for safe transmission over text-only protocols. Cryptographic hashes, UUIDs, and memory addresses are all displayed in hexadecimal for compactness.