# Place value

Here you will learn about place value, including the names and values of the different positions.

Students will first learn about place value as part of their work with numbers and operations in base ten in elementary school.

## What is place value?

Place value is the value of each digit within a number.

A number is made up of digits. For example, the number 54 has two digits, 5 and 4. The value of each digit can help us understand the size of the number.

To determine the value of a digit within a number you use a place value chart.

For example, for 54, the 5 is in the tens column and the 4 is in the ones column.

This means that the number 54 is equivalent to 5 tens and 4 ones.

5 tens is the same as 5\times{10}=50 and 4 ones is the same as 4\times{1}=4.

Adding 50+4 gives us the number 54.

### Whole number place value

Here are the digits to the left of the decimal point.

Let’s write the number nine million, four hundred and fifty thousand, three hundred and eighty two in the place value chart:

Notice you use a comma after the millions digit (9) and the thousands digit (0).

The comma is used to the left of the decimal point to separate 3 digits at a time.

If the value within a column is 0, you must write a zero for that place value. Here, if you missed the 0 in the thousands position, the number would read 945,382 which is a different number from the number above.

The place value chart above shows positions up to the millions. However, you can use place value to represent very large numbers.

Place Value

Ordinary Number

Power of \bf{10}

Thousands

1,000

10^3

Millions

1,000,000

10^6

Billions

1,000,000,000

10^9

Trillions

1,000,000,000,000

10^{12}

1,000,000,000,000,000

10^{15}

Googol

10,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,
000,000,000, 000,000,000,000

10^{100}

### Decimal place value

Here are the digits to the right of the decimal point.

Let’s write the number two and thirty four thousandths in the place value chart:

There is a period after the ones digit (2). The period tells us that the next part of the number is a decimal or less than 1.

If the value within a column is 0, you must write a zero for that place value. Here, if you missed the 0 in the tenths position, the number would read 2.34 which is a different number from the number above.

The place value chart above shows numbers down to the thousandths. However, you can use place value to represent very small numbers.

Place Value

Ordinary Number

Power of \bf{10}

Thousandths

0.001

10^{-3}

Millionths

0.000001

10^{-6}

Billionths

0.000000001

10^{-9}

Trillionths

0.000000000001

10^{-12}

## Common Core State Standards

• Grade 2 – Numbers and Operations in Base 10 (2.NBT.A.3)
Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

• Grade 4 – Numbers and Operations in Base 10 (4.NBT.A.2)
Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form.

• Grade 5 – Numbers and Operations in Base 10 (5.NBT.A.3)
Read, write, and compare decimals to thousandths.

## How to write the value of a digit in a number

In order to write the value of a digit in a number:

1. Find the digit within the number.
2. Use the place value chart to find the position.
3. Write the value of the digit.

## Place value examples

### Example 1: whole number place value

What is the value of the digit 3 in 4,163?

1. Find the digit within the number.

2Use the place value chart to find the position.

The 3 is in the ones column.

3Write the value of the digit.

The value of the digit 3 in the number 4,163 is 3 ones, or 3.

### Example 2: whole number place value

What is the value of the digit 4 in 385,431?

Find the digit within the number.

Use the place value chart to find the position.

Write the value of the digit.

### Example 3: whole number place value

What is the value of the digit 2 in 1,025,634?

Find the digit within the number.

Use the place value chart to find the position.

Write the value of the digit.

### Example 4: decimal place value

What is the value of the digit 2 in 3.264?

Find the digit within the number.

Use the place value chart to find the position.

Write the value of the digit.

### Example 5: decimal place value

What is the value of the digit 7 in 0.078?

Find the digit within the number.

Recall the place value of that column.

Write the value of the digit.

### Example 6: decimal place value

What is the value of the digit 4 in 10.054?

Locate the digit within the number.

Use the place value chart to find the position.

Write the value of the digit.

### Teaching tips for place value

• When teaching place value, base ten blocks and other manipulatives can be used to help students visualize place value and see how each position relates to the others.

• Place value is based on patterns in our number system, which many students will pick up on. Asking questions like “What do you notice about these numbers?” or “How does the value of a number change if you change the place value of the digits?” can encourage students to think more broadly about the underlying structure of our place value system.

• Regularly exposing students to place value ideas helps them develop a deeper understanding of our number system. Instead of just using worksheets, incorporate it into math each day with a place value game. Something as simple as “I am thinking of a number that has more than 4 hundreds, what could it be?” can help students remember and think flexibly about place value.

• Reinforce the connection between standard form, expanded form and word form by asking students to provide all three for any given number.

### Easy mistakes to make

• Thousands or thousandths
Numbers to the right of the decimal point are parts of a whole and so they each end in ths. Numbers to the left of the decimal point are whole numbers and do not end in ths.

• Including too many digits
All of the digits after a place value position are sometimes wrongly included.
For example, What is the value of 4 in the number 243? \quad 43 {\color{red}✘} \quad 40 {\color{green}✔}

• Including a oneths column
The first column to the right of the decimal point is the tenths column – there is no oneths column.
For example, What is the value of the digit 2 in the number 3.524?
\hspace{0.5cm} Two tenths or 0.02 {\color{red}✘} \hspace{3cm} Two hundredths, or 0.02 {\color{green}✔}
(Thinking that the first decimal \hspace{1.7cm} (The first decimal position is tenths
position is oneths and the second \hspace{1.7cm} and the second is hundredths)
is tenths)

### Practice place value questions

1. What is the value of 2 in 475,321?

21

20

2

Two hundreds

The digit 2 is in the tens column, so the value is 2 tens or 20.

2. What is the value of 1 in 1,000,253?

1,000

100,000

1,000,000

One hundred thousand

The digit 1 is in the millions column, so the value is 1 million or 1,000,000.

3. What is the value of 3 in 483,201?

3,000

300

10,000

Ten thousand

The digit 3 is in the thousands column, so the value is 3 thousands or 3,000.

4. What is the value of 7 in 6.072?

Seven hundreds

Seven tenths

Seven tens

Seven hundredths

The digit 7 is in the hundredths column, so the value is 7 hundredths or 0.07.

5. What is the value of 8 in 0.008?

Eight thousandths

Eight tenths

Eight thousands

Eight ones

The digit 8 is in the thousandths column, so the value is 8 thousandths or 0.008.

6. What is the value of 4 in 4.825?

4000

4

Four thousandths

0.4

The digit 4 is in the ones column, so the value is 4 ones or 4.

## Place value FAQ

Why is \bf{10} important in place value?

Our number system is called Base Ten, because ten is the value used to create the place value groupings. That is why all of the place value positions (like the tens place or hundreds place) are multiples of 10. This is also why younger students practice skip counting by tens, which ultimately helps them recognize patterns in our number system.

Why is place value important?

Place value helps you understand the size of a number and gives meaning to the digits in each position. An understanding of place value can help you when comparing numbers and ordering numbers and is also essential when operating with numbers.

What decimal place value positions are used in 5th and 6th grade?

Per the Common Core, in fifth grade students work with the tenths place, the hundredths place and the thousandths place. Sixth graders should extend this to smaller positions. It is also important to be mindful of standard expectations, as they may vary from state to state.

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