Decimal Place Value Formula

The Formula

\text{digit in } n\text{th decimal place} = \text{digit} \times \frac{1}{10^n}

When to use: Just as moving left of the decimal point makes each place 10 times bigger (ones, tens, hundreds), moving right makes each place 10 times smaller (tenths, hundredths, thousandths). It's like zooming in—each step splits things into 10 equal pieces.

Quick Example

3.847: \quad 3 = \text{ones}, \; 8 = \text{tenths} \left(\frac{8}{10}\right), \; 4 = \text{hundredths} \left(\frac{4}{100}\right), \; 7 = \text{thousandths} \left(\frac{7}{1000}\right)

Notation

Positions right of the decimal point: tenths (\frac{1}{10}), hundredths (\frac{1}{100}), thousandths (\frac{1}{1000}), etc.

What This Formula Means

The value assigned to each digit's position to the right of the decimal point: the first position is tenths (\frac{1}{10}), the second is hundredths (\frac{1}{100}), the third is thousandths (\frac{1}{1000}), and so on.

Just as moving left of the decimal point makes each place 10 times bigger (ones, tens, hundreds), moving right makes each place 10 times smaller (tenths, hundredths, thousandths). It's like zooming in—each step splits things into 10 equal pieces.

Formal View

\text{For a decimal } d_0.d_1 d_2 d_3 \ldots: \; \text{value} = d_0 \times 10^0 + d_1 \times 10^{-1} + d_2 \times 10^{-2} + d_3 \times 10^{-3} + \cdots = \sum_{k=0}^{n} d_k \times 10^{-k}

Worked Examples

Example 1

easy
In the number 4.73, identify the value of each digit.

Solution

  1. 1
    4 is in the ones place: value = 4.
  2. 2
    7 is in the tenths place: value = \(\frac{7}{10} = 0.7\).
  3. 3
    3 is in the hundredths place: value = \(\frac{3}{100} = 0.03\).
  4. 4
    Total: \(4 + 0.7 + 0.03 = 4.73\).

Answer

4 ones, 7 tenths, 3 hundredths
Place value tells us what a digit is worth. Moving right of the decimal point: tenths (÷10), hundredths (÷100), thousandths (÷1000), etc.

Example 2

medium
Write 3 hundredths, 5 tenths, and 2 ones as a single decimal number.

Example 3

medium
In the number 45.0372, what is the place value of the digit 3?

Common Mistakes

  • Confusing tenths with tens (tenths are \frac{1}{10}, tens are 10)
  • Thinking 0.30 is greater than 0.3 (they are equal—trailing zeros don't change value)
  • Reading 0.025 as 'twenty-five hundredths' instead of 'twenty-five thousandths'

Why This Formula Matters

Understanding decimal place value is essential for comparing decimals, rounding, measurement, and all decimal arithmetic. It underpins money calculations (dollars and cents), scientific measurement precision, and converting between fractions and decimals.

Frequently Asked Questions

What is the Decimal Place Value formula?

The value assigned to each digit's position to the right of the decimal point: the first position is tenths (\frac{1}{10}), the second is hundredths (\frac{1}{100}), the third is thousandths (\frac{1}{1000}), and so on.

How do you use the Decimal Place Value formula?

Just as moving left of the decimal point makes each place 10 times bigger (ones, tens, hundreds), moving right makes each place 10 times smaller (tenths, hundredths, thousandths). It's like zooming in—each step splits things into 10 equal pieces.

What do the symbols mean in the Decimal Place Value formula?

Positions right of the decimal point: tenths (\frac{1}{10}), hundredths (\frac{1}{100}), thousandths (\frac{1}{1000}), etc.

Why is the Decimal Place Value formula important in Math?

Understanding decimal place value is essential for comparing decimals, rounding, measurement, and all decimal arithmetic. It underpins money calculations (dollars and cents), scientific measurement precision, and converting between fractions and decimals.

What do students get wrong about Decimal Place Value?

Thinking that more decimal digits always means a larger number (0.45 vs 0.9: students may think 0.45 > 0.9 because 45 > 9).

What should I learn before the Decimal Place Value formula?

Before studying the Decimal Place Value formula, you should understand: place value.

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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