What “10” means in octal
In a positional number system, a digit’s value depends on both the digit and its place. Each step to the left represents a larger unit: in decimal, places are ones, tens, hundreds; in octal, they are ones, eights, sixty-fours.
So octal 10 means one group of eight and zero leftover ones. It represents the same quantity as decimal 8—not decimal ten. The symbols have not changed their values; the place-value rules have.
Why the count rolls over
Octal has no single-digit symbol for eight. After using its eight available digits, from 0 through 7, there is no next digit to put in the ones place. The count therefore carries into the next place, just as decimal carries after 9:
… 6, 7, 10, 11 …
That carry is a bookkeeping change, not a jump over a quantity. Octal 7 is followed by octal 10 because the latter means eight ones in total. This is also why octal can be handy for reading binary: each octal digit corresponds neatly to a group of three binary digits, since three binary places can represent values from zero to seven.
