octals

An octal number system, also known as base‑8, uses eight digits: 0 through 7. Each position in an octal numeral represents a power of eight, similar to how each digit in base‑10 represents a power of ten.

An octal digit is written using the characters 0 to 7. For example, the octal value 157 represents the decimal number one hundred and fifty‑seven because 1×8² + 5×8¹ + 7×8⁰ equals 64 + 40 + 7. Unlike base‑10, no other symbols are required, which keeps octal representations compact.

The octal system became common in early computing because a single octal digit maps directly onto a group of three binary bits. Early digital systems organized data into 12‑, 24‑, or 36‑bit words, which aligned neatly with multiples of three, allowing engineers to express machine instructions in a readable octal form.

In modern computing, octal notation appears mainly in contexts that benefit from its concise binary correspondence. Unix file permissions are traditionally displayed as three octal digits, where each digit encodes read, write, and execute rights for owner, group, and others. Octal is also used in hexadecimal escape sequences in programming languages.

Conversion between octal and decimal is performed by multiplying each digit by its corresponding power of eight and summing the results. Converting octal to binary simply replaces each octal digit with its three‑bit binary equivalent, while converting binary to octal groups the binary string into sets of three bits from the right and translates each group into the corresponding octal digit.