Place Value

Arithmetic
definition

Also known as: positional notation, digit position

Grade K-2

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The value a digit represents based on its position in a number; the same digit means different amounts in different places. Enables us to write any number, no matter how large, using just 10 digits (0-9) by using position to encode value.

This concept is covered in depth in our place value and measurement foundations guide, with worked examples, practice problems, and common mistakes.

Definition

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

💡 Intuition

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

🎯 Core Idea

Position determines value - the same digit means different amounts in different places.

Example

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

Formula

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

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

🌟 Why It Matters

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

💭 Hint When Stuck

Write out the expanded form: break 352 into 300 + 50 + 2 to see what each digit is really worth.

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.

See Also

subtraction

🚧 Common Stuck Point

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

⚠️ 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

Frequently Asked Questions

What is Place Value in Math?

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

What is the Place Value 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 do you use Place Value?

Write out the expanded form: break 352 into 300 + 50 + 2 to see what each digit is really worth.

Prerequisites

countingnumber sense

Next Steps

additiondecimals

How Place Value Connects to Other Ideas

To understand place value, you should first be comfortable with counting and number sense. Once you have a solid grasp of place value, you can move on to addition and decimals.

Want the Full Guide?

This concept is explained step by step in our complete guide:

Place Value and Measurement: Number Sense Foundations →
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Interactive Playground

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