Place Value
Place value is the value of a digit based on its position within a number, such as ones, tens, hundreds, and thousands.
Definition
Place value means the value a digit has because of where it sits in a number; the same digit can mean different amounts depending on its position. In the base-$10$ positional system, each position represents a power of $10$: from right to left, ones ($10^0$), tens ($10^1$), hundreds ($10^2$), thousands ($10^3$), and so on, so the value of a digit equals $\text{digit} \times 10^{\text{position}}$. In base $b$ generally, a digit $d$ at position $k$ (counting from $0$ at the right) has value $d \cdot b^k$, and any positive integer can be uniquely written this way using the division algorithm repeatedly, directly mapping to polynomial evaluation.
Example
In $352$, the $3$ is in the hundreds place (worth $300$), the $5$ is in the tens place (worth $50$), and the $2$ is in the ones place (worth $2$); the digit $5$ in $500$ is worth ten times more than the $5$ in $50$. In $4{,}709$: $4$ is in the thousands place ($4{,}000$), $7$ is in the hundreds place ($700$), $0$ contributes nothing, and $9$ is in the ones place. Horner's method exploits this structure, evaluating a base-$10$ numeral $d_n \ldots d_1 d_0$ as $(\ldots((d_n \cdot 10 + d_{n-1}) \cdot 10 + d_{n-2})\ldots) \cdot 10 + d_0$, reducing the work to just $n$ multiplications.
Key Insight
Position is everything: place value makes arithmetic algorithms like addition with carrying and long division work, and it explains why shifting a decimal point one place multiplies or divides by $10$. The concept extends to negative and fractional positions too: a digit $d$ at position $-k$ contributes $d \cdot b^{-k}$, giving decimal (or binary) fractions, and floating-point computer arithmetic is built entirely on this extension.