Place Value Formula

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

The Formula

dn×10n+dn−1×10n−1+⋯+d1×101+d0×100d_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 dkd_k in a number has value dk×10kd_k \times 10^k, where kk 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 nn-digit number dn−1dn−2…d1d0d_{n-1}d_{n-2}\ldots d_1 d_0 represents ∑k=0n−1dk⋅10k\sum_{k=0}^{n-1} d_k \cdot 10^k, where each digit dk∈{0,1,…,9}d_k \in \{0,1,\ldots,9\}. The positional system generalizes to any base bb: ∑k=0n−1dk⋅bk\sum_{k=0}^{n-1} d_k \cdot b^k.

Worked Examples

Example 1

easy
Write the number 4,0724{,}072 in expanded form.

Answer

4×1000+7×10+2×14 \times 1000 + 7 \times 10 + 2 \times 1

First step

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.

Full solution

  1. 2
    Expanded form: 4×1000+0×100+7×10+2×14 \times 1000 + 0 \times 100 + 7 \times 10 + 2 \times 1.
  2. 3
    Simplify: 4000+0+70+2=4,0724000 + 0 + 70 + 2 = 4{,}072.
Expanded form reveals the place value of every digit. The zero in the hundreds place is a placeholder — it contributes 0×100=00 \times 100 = 0 but is essential for correct positional meaning of the other digits.

Example 2

medium
In the number 3,456,7893{,}456{,}789, what is the value of the digit 44?

Example 3

medium
What is the value of the digit 7 in the number 473,218473{,}218?

Common Mistakes

  • Adding the digits instead of weighting them by place - multiply each digit by its column value before combining.
  • Forgetting zero as a placeholder so 305 collapses to 35 - a zero holds an empty column open.
  • Lining up multi-digit numbers carelessly so ones sit under tens - align by place value, ones under ones.

Why This Formula Matters

Place value is the engine that makes ten symbols (0-9) write every number. Carrying, borrowing, decimals, and rounding are all just place-value bookkeeping, so a shaky grasp here quietly breaks all of arithmetic. Recognizing it by "Does the worth of this digit depend on which column it sits in?" — rather than by familiar numbers — is what lets a student tell it apart from face value and base-ten system and number sense in a mixed problem set.

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 dkd_k in a number has value dk×10kd_k \times 10^k, where kk is its position counting from the right starting at 0

Why is the Place Value formula important in Math?

Place value is the engine that makes ten symbols (0-9) write every number. Carrying, borrowing, decimals, and rounding are all just place-value bookkeeping, so a shaky grasp here quietly breaks all of arithmetic. Recognizing it by "Does the worth of this digit depend on which column it sits in?" — rather than by familiar numbers — is what lets a student tell it apart from face value and base-ten system and number sense in a mixed problem set.

What do students get wrong about Place Value?

The procedure for place value is the easy part; the trap is adding the digits instead of weighting them by place. Asking "Does the worth of this digit depend on which column it sits in?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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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