Binary Base Converter

A number base (or radix) is the number of unique digits used to represent numbers in a positional numeral system. Binary uses base 2 (digits 0-1), octal uses base 8 (digits 0-7), decimal uses base 10 (digits 0-9), and hexadecimal uses base 16 (digits 0-F). Each position in a number represents a power of the base.

The general method: first convert the source number to decimal (base 10) by multiplying each digit by its positional power of the source base, then convert the decimal value to the target base by repeatedly dividing by the target base and recording remainders. Some pairs (like binary↔hex) have shortcut grouping methods.

Binary (base 2) is used internally by all computers. Decimal (base 10) is human-readable output. Hexadecimal (base 16) is used for memory addresses, color codes, and debugging. Octal (base 8) is used in Unix permissions. ASCII maps numbers to text characters.

Computers use binary because electronic circuits have two stable states: on (1) and off (0). Distinguishing between 2 voltage levels is far simpler and more reliable than distinguishing between 10. Binary arithmetic is also simpler to implement in hardware, requiring fewer transistors.

The easiest method is to convert via binary as an intermediate step. Convert hex to binary (expand each hex digit to 4 bits), then re-group the binary bits into 3-bit groups for octal. Direct hex-to-octal conversion without an intermediate step is possible but more complex.

All three are power-of-2 bases: binary is 2¹, octal is 2³, and hex is 2⁴. This means octal groups 3 binary bits and hex groups 4 binary bits. This power-of-2 relationship makes conversions between them simple grouping operations rather than complex arithmetic.