Positional Notation — Definition, Formula & Examples

Positional notation is a way of writing numbers where each digit's value depends on its position. For example, the 3 in 300 means something different from the 3 in 30 because it sits in a different place.

Positional notation is a numeral system in which each digit is multiplied by a power of the base determined by its position relative to a fixed reference point (typically the ones place). In base $b$ , a digit $d$ at position $n$ (counting from 0 on the right) contributes $d \times b^n$ to the total value of the number.

Key Formula

\text{value} = d_n \times b^n + d_{n-1} \times b^{n-1} + \cdots + d_1 \times b^1 + d_0 \times b^0

Where:

$d$ = A digit in the number (0 through b−1)
$b$ = The base of the number system (10 for decimal)
$n$ = The position of the leftmost digit, counting from 0 on the right

How It Works

In our everyday base-10 system, positions from right to left represent ones, tens, hundreds, thousands, and so on. Each position is worth 10 times more than the position to its right. To find the value a digit contributes, multiply the digit by the value of its position. Add up all these contributions to get the total number.

Worked Example

Problem: Break down the number 4,725 using positional notation in base 10.

Identify positions: From right to left, the digits 5, 2, 7, and 4 sit in positions 0, 1, 2, and 3.

Multiply each digit by its place value: Multiply each digit by the corresponding power of 10.

4 \times 10^3 + 7 \times 10^2 + 2 \times 10^1 + 5 \times 10^0

Calculate: Evaluate each term and add them together.

4000 + 700 + 20 + 5 = 4725

Answer: 4,725 = 4 × 1000 + 7 × 100 + 2 × 10 + 5 × 1

Why It Matters

Positional notation makes arithmetic operations like addition with carrying and long division possible. It also forms the foundation for understanding binary (base 2), which is how every computer stores and processes data.

Common Mistakes

Mistake: Confusing a digit with its value — thinking the 5 in 503 is just worth 5.

Correction: The 5 is in the hundreds place, so its value is 5 × 100 = 500. Always check which position a digit occupies before stating its value.