Convert Binary to Octal

Why binary groups into threes

Octal is base 8, and 8 is 2 to the third power. That single fact is the whole method. Because three binary digits produce exactly eight possible patterns — 000 through 111 — and octal has exactly eight digits, every 3-bit group maps to one octal digit with nothing left over and no arithmetic required.

This is why converting binary to octal never involves division, unlike converting binary to decimal. Ten is not a power of two, so decimal conversion means real calculation. Eight is, so octal conversion is substitution: chop the bits into threes, look each group up, write the digit down. The same relationship gives hexadecimal its four-bit groups, since 16 is 2 to the fourth.

How to convert binary to octal, step by step

1. Start from the rightmost bit — the one furthest from the point, not the leftmost.
2. Split the binary digits into groups of three, working right to left.
3. If the leftmost group ends up short, pad it on the left with zeros until it has three digits. Leading zeros never change a number's value.
4. Convert each 3-bit group to its octal digit using the table above.
5. Write the digits in the same left-to-right order as their groups. That is your octal number.

The direction matters. Grouping from the left instead of the right is the single most common error, and it produces a wrong answer whenever the bit count is not already a multiple of three. Binary 1111 grouped from the left gives 111 and 1, which reads as 71 — but grouped correctly from the right it gives 001 and 111, which is 17. The correct answer is 17, and you can check it: 1111 in binary is 15, and 17 in octal is 8 + 7 = 15.

Worked examples

Converting octal back to binary

The reverse is easier, because there is no padding decision to make. Replace each octal digit with its three binary digits, keeping the order, then drop any leading zeros from the final result. Octal 25 becomes 010 and 101, which reads 010101, which is 10101 once the leading zero is dropped.

The one rule people forget is that every digit expands to exactly three bits, including small ones. Octal 5 is 101, but in the middle of a number it must be written as the full 101, and octal 1 must be written 001, not just 1. Octal 41 is 100001, not 1001. Use the swap button on the converter above to go this direction without retyping anything.

Binary fractions and the point

Bits after a binary point convert to octal just as cleanly, but the padding direction flips. Before the point you pad on the left; after it you pad on the right. The reason is that place values radiate outward from the point in both directions, so the group nearest the point is always the one that must stay complete.

Take 0.11. There are two fractional bits, so pad on the right to 0.110, which is one group: 110, or octal 6. So 0.11 binary is 0.6 octal. Check it: 0.11 binary is a half plus a quarter, or 0.75, and 6/8 is also 0.75. Now take 101.101 — the integer part 101 is one complete group giving 5, and the fractional part 101 is one complete group giving 5, so the answer is 5.5 octal. Not one of the seven pages currently ranking for this term handles fractional input at all.

Where octal actually gets used: Unix file permissions

If you are not a computer science student, there is a good chance the only octal you will ever meet is a chmod command — and it is octal for precisely the reason this page is about. Unix permissions come in three groups of three bits: read, write and execute, for the owner, the group and everyone else. Nine bits, three groups of three, one octal digit per group. chmod 755 is not an arbitrary code; it is binary 111101101 written the short way.

Read that number through the grouping method and it decodes itself. 111 is 7, meaning read, write and execute all on. 101 is 5, meaning read and execute on, write off. So 755 is rwx for the owner, r-x for the group, r-x for everyone — the standard permission set for a directory or an executable script. Every chmod number you have ever copied off a forum post is a binary-to-octal conversion someone else already did.

The same three-bit logic shows up in a few other corners. File permission masks in umask are octal for the same reason, just inverted. Older Unix system calls and PDP-11 assembly used octal throughout because those machines had word sizes that divided neatly by three. And in C, Python and several other languages, a number written with a leading zero is still interpreted as octal — which is why 0755 and 755 mean different things in code, and why a stray leading zero in a config file can produce a genuinely baffling bug.

Who uses this and why