Octal

The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. This is to say that 10_octal represents eight and 100_octal represents sixty-four. However, English, like most languages, uses a base-10 number system, hence a true octal system might use different vocabulary.

In the decimal system, each place is a power of ten. For example:

${\displaystyle \mathbf {74} _{10}=\mathbf {7} \times 10^{1}+\mathbf {4} \times 10^{0}}$

In the octal system, each place is a power of eight. For example:

${\displaystyle \mathbf {112} _{8}=\mathbf {1} \times 8^{2}+\mathbf {1} \times 8^{1}+\mathbf {2} \times 8^{0}}$

By performing the calculation above in the familiar decimal system, we see why 112 in octal is equal to ${\displaystyle 64+8+2=74}$ in decimal.

Octal numerals can be easily converted from binary representations (similar to a quaternary numeral system) by grouping consecutive binary digits into groups of three (starting from the right, for integers). For example, the binary representation for decimal 74 is 1001010. Two zeroes can be added at the left: (00)1 001 010, corresponding to the octal digits 1 1 2, yielding the octal representation 112.