Place Value: Digits, Words and Expanded Form | Grade 5 Math | LevelKite

A digit is one of the ten symbols $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ . You can write any whole number ( $0, 1, 2, 3$ and so on) with just these ten digits. So how can ten symbols make millions of different numbers? The secret is where each digit sits.

The spot where a digit sits is called its place. Each place has a name, such as ones, tens or hundreds. The value of a digit is how much that digit is worth in its place.

Look at $2{,}405{,}318$ . It has seven digits in seven places. The table shows what each digit means and how much it is worth.

Place-value chart for 2,405,318

Digit	Place	What the digit means	Value
$2$	millions	2 millions	$2{,}000{,}000$
$4$	hundred thousands	4 hundred thousands	$400{,}000$
$0$	ten thousands	0 ten thousands	$0$
$5$	thousands	5 thousands	$5{,}000$
$3$	hundreds	3 hundreds	$300$
$1$	tens	1 ten	$10$
$8$	ones	8 ones	$8$

Key concept

A digit's place gives it its value

Every place is worth a set amount. The ones place is worth $1$ , the tens place $10$ , the hundreds place $100$ , and so on up to the millions place, $1{,}000{,}000$ . Each place is worth 10 times the place on its right.

The value of a digit is the digit times the worth of its place. In $2{,}405{,}318$ , the 4 sits in the hundred thousands place, so its value is $4 \times 100{,}000 = 400{,}000$ .

That is why the same digit can stand for different values. A 4 in the tens place is worth $40$ . Move it one place to the left, into the hundreds place, and it is worth $400$ .

\text{value of a digit} = \text{digit} \times \text{worth of its place}

Worked example

Same digit, different values

The digit 5 appears twice in $6{,}255{,}410$ . Name the place and the value of each 5. Why are the two values different?

Name the places. Count them from the right: ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions. The 5 on the right is the fourth digit from the right, so it sits in the thousands place. The 5 on the left is the fifth digit from the right, so it sits in the ten thousands place.
Find the value of the 5 on the right. It sits in the thousands place, so it means 5 thousands. Multiply 5 by 1,000.
$5 \times 1{,}000 = 5{,}000$
Find the value of the 5 on the left. It sits in the ten thousands place, so it means 5 ten thousands. Multiply 5 by 10,000.
$5 \times 10{,}000 = 50{,}000$
Explain the difference. The ten thousands place is one place to the left of the thousands place, so it is worth 10 times as much. That makes the left 5 worth 10 times the right 5.
$50{,}000 = 10 \times 5{,}000$

Answer

The 5 on the right is in the thousands place and is worth $5{,}000$ . The 5 on the left is in the ten thousands place and is worth $50{,}000$ . The values differ because the two digits sit in different places, and each place is worth 10 times the place on its right.

A long number is easier to read in chunks. A period is a group of three places. The commas in a number separate its periods. Starting from the right:

The ones period holds the ones, tens and hundreds.
The thousands period holds the thousands, ten thousands and hundred thousands.
The millions period begins with the millions.

Read a number from left to right, one period at a time. Read the digits in a period the way you read any number up to $999$ . Then say the period's name, million or thousand. Say no name for the ones period.

In words, $2{,}405{,}318$ is two million, four hundred five thousand, three hundred eighteen. Each comma in the words matches a comma in the number. The table shows where each part comes from.

Reading 2,405,318 one period at a time

Period	Digits in the period	Say and write
millions	$2$	two million
thousands	$405$	four hundred five thousand
ones	$318$	three hundred eighteen

Worked example

Write a number in words

Write $6{,}040{,}217$ in words.

Split the number into periods at the commas, because the commas mark where one period ends and the next begins. You will read the periods from left to right: millions, thousands, ones.
$\underbrace{6}_{\text{millions}}\;,\;\underbrace{040}_{\text{thousands}}\;,\;\underbrace{217}_{\text{ones}}$
Read the millions period. Its only digit is 6, so write six and then the period's name, million.
$6 \rightarrow \text{six million}$
Read the thousands period. Its digits are 040. That means 0 hundreds, 4 tens and 0 ones, which is forty. Then write the period's name, thousand.
$040 \rightarrow \text{forty thousand}$
Read the ones period. Its digits are 217, which is two hundred seventeen. The ones period gets no name.
$217 \rightarrow \text{two hundred seventeen}$
Join the three parts in the order you read them. Put a comma between periods, in the same spots as the commas in the number.

Answer

In words, $6{,}040{,}217$ is six million, forty thousand, two hundred seventeen.

You can now write a number with digits and with words. Writing it with digits, like $2{,}405{,}318$ , is called standard form. There is one more way. Expanded form shows a number as a sum: the values of its digits added together.

To write expanded form, find the value of each digit. Start with the digit on the left, which sits in the greatest place. Then join the values with plus signs.

Look back at the first table. The ten thousands place of $2{,}405{,}318$ holds 0, so the value there is $0$ . Adding 0 changes nothing, so you can leave it out of the sum. Here is the expanded form:

2{,}405{,}318 = 2{,}000{,}000 + 400{,}000 + 5{,}000 + 300 + 10 + 8

Worked example

From expanded form back to standard form

Write $700{,}000 + 3{,}000 + 60 + 2$ in standard form.

Read each value as a digit in a place. 700,000 is 7 hundred thousands. 3,000 is 3 thousands. 60 is 6 tens. 2 is 2 ones.
List every place from the greatest place in the sum down to the ones: hundred thousands, ten thousands, thousands, hundreds, tens, ones. The number needs a digit in all six places.
Find the places the sum skips. Nothing fills the ten thousands place or the hundreds place, so each of them holds 0. Writing those zeros into the sum does not change it, because adding 0 changes nothing.
$700{,}000 + 0 + 3{,}000 + 0 + 60 + 2$
Write one digit for each place, in order: 7, 0, 3, 0, 6, 2. Put a comma between the thousands period and the ones period.
$700{,}000 + 3{,}000 + 60 + 2 = 703{,}062$

Answer

In standard form, $700{,}000 + 3{,}000 + 60 + 2$ is $703{,}062$ .

Takeaway: A digit's value comes from its place, and each place is worth 10 times the place on its right. Periods let you read a long number three digits at a time and write it in words. Expanded form adds up the value of every digit, and a 0 holds a place open in standard form.

Next, you will use place value to compare, order and round whole numbers.