Octal Converter

The octal number system is a base-8 positional numeral system that uses eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. Each digit position represents a successive power of eight, exactly as decimal uses powers of ten. The value of an octal number is the sum of each digit multiplied by its positional power.

Octal gained prominence with the PDP minicomputers of the 1960s and early 1970s, where 18-bit and 36-bit word sizes divided neatly into six or twelve 3-bit groups. Unix inherited this legacy and still uses octal to represent the nine permission bits of a file mode: three bits for the owner, three for the group, and three for everyone else. When you type chmod 755 script.sh, those three digits are octal.

In programming languages, an octal literal is written with a leading zero in C and older Go code (0755) or with the explicit 0o prefix in modern Go and Python (0o755). Forgetting the prefix is a common source of permission bugs: os.Chmod(path, 755) passes the decimal number 755 instead of the octal value, resulting in completely wrong permissions.

Multiply each octal digit by 8 raised to the power of its position, counting positions from right to left starting at zero. Add all the products together.

Worked example: convert 755 (octal) to decimal.

So 755 (octal) = 493 (decimal). To convert in the opposite direction — decimal to octal — divide the decimal number repeatedly by 8, recording each remainder. Reading the remainders from bottom to top gives the octal result. For example: 493 ÷ 8 = 61 remainder 5 → 61 ÷ 8 = 7 remainder 5 → 7 ÷ 8 = 0 remainder 7. Reading upward: 755.

Octal and binary have an especially clean relationship: each octal digit maps to exactly three binary bits. This is because 8 = 2³, so one octal digit spans exactly one 3-bit group. You can convert any octal number to binary digit by digit without any arithmetic.

The eight possible octal digits and their 3-bit binary equivalents:

To convert 755 (octal) to binary, substitute each digit: 7 → 111, 5 → 101, 5 → 101. Concatenate the triplets: 111 101 101. This visual grouping explains why octal is the natural representation for Unix permissions — each triplet maps directly to one set of rwx bits.

To convert binary back to octal, group the binary digits into triplets from right to left (pad with leading zeros if needed), then replace each triplet with its octal digit.

There is no direct one-to-one digit mapping between octal and hexadecimal because 8 and 16 are not powers of each other. The reliable two-step method is to convert through binary: octal → binary → hex.

Alternatively you can convert octal to decimal first and then from decimal to hex, though the binary route is usually faster for mental arithmetic.

The hex digit F (15 decimal) requires four bits (1111), which is why two hex digits always represent exactly one byte, while three octal digits represent only 9 bits — not a full byte boundary. This mismatch is one reason hex has largely replaced octal in modern computing contexts like color codes and memory addresses.

Every file and directory on a Unix or Linux system has a 9-bit permission mask stored as three groups of three bits. The first triplet controls what the file owner can do, the second controls the owning group, and the third controls all other users. Each triplet encodes read, write, and execute rights: read = 4, write = 2, execute = 1. Add the values of the permissions you want to get the digit for each group.

Special permission bits — setuid (4), setgid (2), and sticky (1) — add a fourth leading octal digit. For example, 4755 makes an executable run as its owner regardless of who launches it (setuid). This tool covers the standard 3-digit octal modes. For 4-digit modes refer to man chmod.

The table below covers the first 16 values in octal (0 through 17 octal = 0 through 15 decimal). These are the values within a single hex digit, making this a useful cross-reference for base conversions.