Place Value
ArithmeticPlace 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.
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 $2$).
Key Insight
The digit $5$ in $500$ is worth ten times more than the $5$ in $50$, and a hundred times more than the $5$ in $5$. Position is everything.
Definition
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. The value of a digit equals $\text{digit} \times 10^{\text{position}}$.
Example
In $4{,}709$: $4$ is in the thousands place ($4 \times 1000 = 4{,}000$), $7$ is in the hundreds place ($7 \times 100 = 700$), $0$ is in the tens place ($0 \times 10 = 0$), and $9$ is in the ones place ($9 \times 1 = 9$).
Key Insight
Place value makes arithmetic algorithms (addition with carrying, long division) work. Understanding it also explains why shifting a decimal point one place multiplies or divides by $10$.
Definition
Place value in base $b$ assigns value $d \cdot b^k$ to a digit $d$ at position $k$ (counting from $0$ at the right). Any positive integer $n$ can be uniquely written in base $b$ using the division algorithm repeatedly. This representation is the basis for all positional arithmetic and directly maps to polynomial evaluation: the number equals $p(b)$ for polynomial $p$ with digit coefficients.
Example
Horner's method evaluates 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 $n$ multiplications to just $n$, which is computationally efficient.
Key Insight
The concept extends to negative and fractional positions: $d$ at position $-k$ contributes $d \cdot b^{-k}$, giving decimal (or binary) fractions. Floating-point computer arithmetic is built entirely on this extension.