Octal to Binary

An octal digit is a shorthand for three on-off switches. Once you know that, any octal number can be opened up into its bits and read like a row of lights. This page expands octal numbers to binary and shows the lookup it used, which suits students checking an exercise as well as administrators decoding a permission mode.

The lookup

The digit 0 is 000 and the digit 1 is 001. Then 2 is 010, 3 is 011, 4 is 100, 5 is 101, 6 is 110 and 7 is 111. Because each pattern has a fixed length, converting means replacing digits one for one and joining the results, with no carrying or multiplying. That is also why the reverse conversion is just as simple, and why programmers have always liked power-of-two bases.

Reading permissions

Unix file modes are the most common place octal survives. The three digits stand for the owner, the group and everyone else. Within a digit, the bit worth 4 is read permission, the bit worth 2 is write and the bit worth 1 is execute. Expand 755 and you see 111 101 101: the owner may read, write and execute, while the group and the others may read and execute. Expand 644 and you see 110 100 100, where only the owner can write.

Input rules

Type or paste an octal number, with or without a 0o prefix. Spaces and underscores are ignored. A leading zero does not make a number octal on this page, because you have already chosen the base; in the C family of languages 010 is eight, which has been the cause of many bugs, and Python 3 insists on 0o10 for that reason. A digit 8 or 9 stops that line and the message names it. Several numbers can be pasted, one per line.

Edge cases

Zeros at the start of the result are dropped: octal 7 is binary 111, and octal 07 is also 111. Fractions are allowed and convert digit by digit after the point, so octal 0.4 is binary 0.100, which is 0.5 in decimal. Negative numbers keep the minus sign. Octal escapes in old C strings or tar headers can be pasted here to see the byte value in bits, though the page treats them as a number and not as a character code.

A worked example

The Sample button loads 755. The digit 7 becomes 111, the digit 5 becomes 101 and the last 5 becomes 101, so the binary number is 111101101 once joined. The extra box lists the same quantity as 493 in decimal and 1ED in hexadecimal. A check: 256 + 128 + 64 + 32 + 8 + 4 + 1 equals 493, which matches the decimal figure.

Limits

The page converts a number and does not interpret it. It does not translate bits into rwx letters, flag unusual combinations or add the special leading digit that setuid, setgid and sticky bits need. For permissions, the chmod calculator is better; for pure number work this page is enough. The lookup shown covers the digits that appear in your input, not necessarily all eight.

Related tools

• Binary to Octal groups bits back into octal digits.
• Octal to Hex regroups the same bits into fours.
• Octal to Decimal shows the powers of eight behind a number.
• Chmod Calculator builds permission modes from checkboxes.