Positional number representation

Positional number representation is the convention that a digit’s contribution to a number depends on both its value and its position. The same digit “1” means one in the units place, ten in the tens place, and a hundred in the hundreds place — its weight is the base raised to its position index.

A number in base $k$ with $n$ integer digits and $m$ fractional digits is

$\text{Value}={\sum }_{i=-m}^{n-1}{b}_i\cdot k^i$

where $b_i$ is the digit at position $i$. The integer positions go $0,1,2,\dots$ to the left of the radix point; the fractional positions go $-1,-2,\dots$ to the right.

The base $k$ tells you two things: how many distinct digit symbols you have ($0$ through $k-1$), and how much each position weighs more than the one to its right. Decimal uses base $10$ because we have ten fingers; computers use base 2 because a switch has two states.

Example. The binary fraction $(1010.01{)}_2$ is

$1\cdot 2^3+0\cdot 2^2+1\cdot 2^1+0\cdot 2^0+0\cdot 2^{-1}+1\cdot 2^{-2}=8+2+0.25=10.25$

Hexadecimal works the same way with $16$ as the base: $\text{0xFF}=15\cdot 16^1+15\cdot 16^0=255$. The "$\text{0x}$" prefix is just a convention to flag “the digits that follow are base-$16$” — the value is independent of how it’s written.

For going between bases mechanically, see Base Conversion.