Octal Converter

Octal is base-8: every position in an octal number is a power of 8 instead of a power of 10. The number 755 in octal means 7 x 64 + 5 x 8 + 5 x 1, which is 493 in decimal. The converter reverses this place-value expansion for any input base, so one entry gives you the decimal, octal, binary, and hexadecimal forms at once.

Octal survives today mainly in computing. Each octal digit maps to exactly three binary bits, so octal was the compact shorthand for machine code before hexadecimal took over, and it still appears wherever Unix file permissions are written as three digits like 755.

To convert octal to decimal by hand, multiply each digit by 8 raised to its position (counting from 0 on the right) and add the results. Octal 157 becomes 1 x 64 + 5 x 8 + 7 x 1 = 64 + 40 + 7 = 111.

To go the other way, divide the decimal number by 8 repeatedly and read the remainders from bottom to top. Converting 111: 111 / 8 = 13 remainder 7, 13 / 8 = 1 remainder 5, 1 / 8 = 0 remainder 1, giving 157. This is exactly the working the calculator shows.

The cleanest octal trick is the binary triad: replace each octal digit with its three binary bits, and you have the binary number with no arithmetic. Octal 7 is 111, octal 5 is 101, so octal 75 is binary 111101. Group binary digits in threes from the right to go back.

This one-to-three mapping is why octal digits never exceed 7: three bits can only count 0 to 7. The same idea works for hexadecimal with four-bit groups, which is why the calculator also shows the hex form.

The most common place you will meet octal is chmod 755. Each of the three digits is a permission group: owner, group, others. Each digit is the sum of read (4), write (2), and execute (1), so 7 means full access and 5 means read plus execute.

Umask arithmetic is octal arithmetic: the classic default 777 minus a umask of 022 gives 755, computed digit by digit with no borrows because each digit is an independent 3-bit group. Convert 755 above and you will see it is 493 in decimal, which is not a number anyone uses, but the triads 111 101 101 are exactly rwxr-xr-x.

Take the octal number 157 and convert it by hand. From the right: 7 x 1 = 7, 5 x 8 = 40, 1 x 64 = 64. Add them: 7 + 40 + 64 = 111 decimal.

The binary form comes from the triads: 1 becomes 001, 5 becomes 101, 7 becomes 111, giving 001101111, or 1101111 after dropping the leading zeros. The hexadecimal form groups binary in fours: 110 1111 becomes 6F. Enter 157 with the octal base selected to confirm all four results.

The number one octal mistake is typing an 8 or a 9. Those digits do not exist in base-8, and they almost always mean a decimal number was pasted into an octal field by accident. The calculator rejects them with the exact position named, instead of silently misreading the number.

Leading zeros are harmless but meaningful: a leading 0 is the traditional marker for octal in C, Java, and Python 2, so 0755 and 755 are the same octal value but very different decimal ones. For related base conversions, the binary converter handles the bit-level side of the same ideas.