# Octal

Octal is a counting system that uses eight digits.

Instead of using only 0's and 1's like binary, or the characters '0' to '9' of the decimal number system; octal uses the characters '0' to '7', so does not need what would normally be classed as alphabetic characters to represent digits as Hexadecimal does.

## Uses

The octal number system was used widely in the Electronics and Computer Industry, as although digital electronics is based on gates with only two states and is therefore fundamentally binary, binary numbers can quickly become long and hard to transcribe without errors. Their octal equivalents are much shorter and easier to remember, and have a straight-forward way of conversion to/from binary.

The PDP-11 computer made by the Digital Equipment Corporation used the octal numeric system exclusively for displaying memory addresses and content.

Unix file system permissions have three sets (user, group, others) of three bit permissions (read, write, execute), which is naturally represented in octal.

The use of octal numbers has declined as most modern computers no longer base their word length on multiples of three bits, (they are based on multiples of four bits, so hexadecimal is more widely used).

## Comparing counts from zero in different number systems

C.f. Common number base formatting and Common number base parsing

Binary Octal Decimal 0 0 0 1 1 1 10 2 2 11 3 3 100 4 4 101 5 5 110 6 6 111 7 7 1000 10 8 1001 11 9 1010 12 10 1011 13 11 1100 14 12 1101 15 13 1110 16 14 1111 17 15 10000 20 16 10001 21 17 10010 22 18 10011 23 19 10100 24 20 10101 25 21

## Converting binary to octal

- Split a binary number into groups of three digits, counting from right to left.
- Pad the leftmost group of binary digits with zeros on their left if their are less than three digits.
- Use the following table to translate each group of three binary digits, in order, to its octal equivalent.

Binary digits Octal equivalent digit 000 0 001 1 010 2 011 3 100 4 101 5 110 6 111 7

### An example conversion

Binary Number: 1011010111 Split: 1 011 010 111 Pad:001 011 010 111 Translate groups: 1 3 2 7 Octal answer: 1327