Place Value Formula

The Formula

d_n \times 10^n + d_{n-1} \times 10^{n-1} + \cdots + d_1 \times 10^1 + d_0 \times 10^0

When to use: In 352, the 3 is worth 300 because it's in the hundreds place.

Quick Example

In 47, the 4 is in the tens place (worth 40) and 7 is in the ones place.

Notation

Each digit d_k in a number has value d_k \times 10^k, where k is its position counting from the right starting at 0

What This Formula Means

The value a digit represents based on its position in a number; the same digit means different amounts in different places.

In 352, the 3 is worth 300 because it's in the hundreds place.

Formal View

In base 10, an n-digit number d_{n-1}d_{n-2}\ldots d_1 d_0 represents \sum_{k=0}^{n-1} d_k \cdot 10^k, where each digit d_k \in \{0,1,\ldots,9\}. The positional system generalizes to any base b: \sum_{k=0}^{n-1} d_k \cdot b^k.

Worked Examples

Example 1

easy
Write the number 4{,}072 in expanded form.

Solution

  1. 1
    Identify each digit and its place: 4 is in the thousands place, 0 is in the hundreds place, 7 is in the tens place, 2 is in the ones place.
  2. 2
    Expanded form: 4 \times 1000 + 0 \times 100 + 7 \times 10 + 2 \times 1.
  3. 3
    Simplify: 4000 + 0 + 70 + 2 = 4{,}072.

Answer

4 \times 1000 + 7 \times 10 + 2 \times 1
Expanded form reveals the place value of every digit. The zero in the hundreds place is a placeholder — it contributes 0 \times 100 = 0 but is essential for correct positional meaning of the other digits.

Example 2

medium
In the number 3{,}456{,}789, what is the value of the digit 4?

Example 3

medium
What is the value of the digit 7 in the number 473{,}218?

Common Mistakes

  • Confusing the digit with its value — in 352, the digit 3 represents 300, not 3
  • Writing numbers in wrong order — putting the ones digit first and hundreds digit last
  • Forgetting that zero holds a place — writing 35 instead of 305 because the tens digit is zero

Why This Formula Matters

Enables us to write any number, no matter how large, using just 10 digits (0-9) by using position to encode value.

Frequently Asked Questions

What is the Place Value formula?

The value a digit represents based on its position in a number; the same digit means different amounts in different places.

How do you use the Place Value formula?

In 352, the 3 is worth 300 because it's in the hundreds place.

What do the symbols mean in the Place Value formula?

Each digit d_k in a number has value d_k \times 10^k, where k is its position counting from the right starting at 0

Why is the Place Value formula important in Math?

Enables us to write any number, no matter how large, using just 10 digits (0-9) by using position to encode value.

What do students get wrong about Place Value?

Confusing the digit with its place value: in 352, the digit 3 has value 300, not 3.

What should I learn before the Place Value formula?

Before studying the Place Value formula, you should understand: counting, number sense.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Place Value and Measurement: Number Sense Foundations →
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