Programmer Base Converter with ASCII

A number base (or radix) is the count of unique digits used to represent numbers. Base 10 (decimal) uses 0-9; base 2 (binary) uses 0-1; base 8 (octal) uses 0-7; base 16 (hexadecimal) uses 0-9 plus A-F. Bases beyond 16 extend by using additional letters (G-Z) - base 36 covers the entire alphabet plus digits.

Decimal → other base: repeatedly divide by the target base, collecting remainders bottom-up. To convert 100 to base 7: 100/7 = 14 r2, 14/7 = 2 r0, 2/7 = 0 r2 → 202₇. Other base → decimal: multiply each digit by base^position (rightmost position is 0). 1011₂ = 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8+0+2+1 = 11.

Hexadecimal is base 16, which equals 2⁴. So each hex digit perfectly represents 4 binary digits (bits). The byte 11110000 in binary is F0 in hex (1111=F, 0000=0). This is why memory addresses, MAC addresses, and color codes (#FFAA00) use hex - it's the most compact human-readable representation of binary data.

Unsigned numbers can only be positive (or zero): an 8-bit unsigned value holds 0 to 255. Signed numbers use the leftmost bit to indicate sign: in two's complement (the universal standard), the leftmost bit being 1 means negative. An 8-bit signed value ranges from -128 to +127. Two's complement makes arithmetic work seamlessly without special "subtraction" hardware.

Base 64 is widely used in computing for encoding binary data as ASCII text (email attachments, JWT tokens, data URLs). It uses A-Z, a-z, 0-9, +, / for digits. Beyond base 64, alphabets run out of conventional characters. Base 256 is used internally as raw bytes but isn't typed by humans. For everyday math, base 16 is the highest practical limit.