Place Value and the Number System

Learning goals

Read a digit's value as the digit times its place value
Explain why each place is worth ten times the place to its right
Write a number in expanded form, and say what a zero digit holds open
Group large numbers into periods of three to read and write them
Compare two whole numbers left to right, stopping at the first place that differs

The base-10 system

Our number system is base 10, also called the decimal system. Base ten means that we bundle in groups of ten. Ten ones make one ten, ten tens make one hundred, ten hundreds make one thousand, and the bundling carries on without ever stopping. People chose that bundle size, and everything else about the system follows from it. There are exactly ten digits, $0$ through $9$ , because ten single units are the point at which a new column has to open. And each place is worth ten times the place immediately to its right. One unit of a place is one whole bundle of ten units of the place to its right.

Reading a whole number from right to left, the places are the ones, tens, hundreds, thousands, and so on:

\ldots,\quad 10{,}000,\quad 1{,}000,\quad 100,\quad 10,\quad 1.

Digit value equals digit times place value

A digit by itself is just a symbol. Its value inside a number is the digit multiplied by the value of the place it sits in. Take the number $4{,}572$ and read off each column:

Digit	Place	Place value	Digit value
$4$	thousands	$1{,}000$	$4{,}000$
$5$	hundreds	$100$	$500$
$7$	tens	$10$	$70$
$2$	ones	$1$	$2$

Keep two easily confused words apart. The place of the $5$ is the hundreds; the value of the $5$ is $5 \times 100 = 500$ . The place is a name for the column, and the value is the number the digit contributes.

The same digit 5 is worth 500 here because it sits in the hundreds column. Each column is worth ten times the column immediately to its right.

Now take a single digit and move it. In the chart below there is only one digit, and you choose which column it sits in. Each step to the left multiplies what it is worth by ten, and each step to the right divides it by ten. The symbol never changes; only its address does.

Each step to the left multiplies a digit's value by ten

Digit Column

The 5 sits in the hundreds place. So it is worth 5 times 100, which is 500.

One digit in a place-value chart. Choose the digit, and choose the column it sits in.

Send the $5$ from the ones column to the thousands column, one step at a time, and read its value at each stop. The values are $5$ , then $50$ , then $500$ , then $5{,}000$ . Nothing about the $5$ changed. Then send it back the other way and the same values come back in reverse, because moving right undoes moving left. This is the single fact the rest of the lesson is built on.

Expanded form

Adding up the digit values rewrites a number in expanded form, which lays bare exactly what each digit contributes:

4{,}572 = 4{,}000 + 500 + 70 + 2.

We can make each place value explicit:

4{,}572 = (4 \times 1000) + (5 \times 100) + (7 \times 10) + (2 \times 1).

A zero digit contributes nothing to the sum, but it is not optional. In $3{,}205$ the $0$ says that there are no tens, and it still has to be written:

3{,}205 = (3 \times 1000) + (2 \times 100) + (0 \times 10) + (5 \times 1) = 3{,}000 + 200 + 5.

Remove that $0$ and you are left with $325$ , which is a different and far smaller number. The table below compares what each digit is worth in the two numerals:

Digit	Its value in $3{,}205$	Its value in $325$
$3$	$3{,}000$	$300$
$2$	$200$	$20$
$5$	$5$	$5$

The $3$ and the $2$ each ended up in a column worth a tenth of the one they had. Zero is a placeholder: it earns its column by keeping the other digits in the columns they belong to. The modern number system depends on it.

Check your understanding

In the number $6{,}308$ , what is the value of the digit $3$ ?

3

30

300

3{,}000

Why each place is exactly ten times the next

You do not have to memorise the ten-times rule one column at a time. It follows from the way the columns are built out of bundles of ten.

Each place is exactly $10\times$ the place to its right#

Look at what one unit of a column is made of. Ten ones are bundled into one ten, ten tens are bundled into one hundred, and ten hundreds are bundled into one thousand:

10 = 10 \times 1, \qquad 100 = 10 \times 10, \qquad 1{,}000 = 10 \times 100.

Every column is built the same way. One unit of it is one bundle of ten units of the column to its right, so it is worth ten of those units. That holds for every pair of neighbouring columns, not only for the three pairs written out above.

Bundles also answer a question the digits on their own do not. Ask how many hundreds there are in $52{,}000$ . The hundreds digit is $0$ , but the question is about groups of $100$ . Each thousand holds ten hundreds, so $52$ thousands hold $52 \times 10 = 520$ hundreds:

52{,}000 = 520 \times 100.

So a whole number can be counted in any bundle size you like, not only in the digit sitting in that column. The hundreds digit counts only the hundreds left over once the thousands are taken. The number of hundreds counts every group of $100$ inside the whole number.

Reading and writing large numbers

For numbers with many digits we group the digits into periods of three, separated by commas, working from the right. The periods are ones, then thousands, then millions, then billions.

52{,}408{,}163 \;=\; \underbrace{52}_{\text{millions}}\,,\;\underbrace{408}_{\text{thousands}}\,,\;\underbrace{163}_{\text{ones}}

To read it, say each three-digit group followed by its period name: “fifty-two million, four hundred eight thousand, one hundred sixty-three.” The commas do not create the values, because the positions of the digits already do that. What the commas do is show you where the periods start and stop. Grouping from the right is what keeps you from misreading the size of the number.

Writing a number from words runs the same steps backwards. Take “four hundred six million, ninety thousand, twenty-five.” Write one three-digit block for each period, filling every block after the leading one out to three columns with zeros:

\underbrace{406}_{\text{millions}}\,,\;\underbrace{090}_{\text{thousands}}\,,\;\underbrace{025}_{\text{ones}} \;=\; 406{,}090{,}025.

Ninety becomes $090$ and twenty-five becomes $025$ because each period after the leading one owns three columns, whether or not its digits reach that far.

Worked example 1 Find the value of the digit $7$ in $4{,}709{,}152$

Start by grouping $4{,}709{,}152$ into periods from the right: the last three digits $152$ are the ones period. The next three $709$ are the thousands period, and the leading $4$ is in the millions period.

The $7$ lives inside the thousands period, in the hundreds slot of $709$ . The hundreds slot of the thousands period is the hundred-thousands place, so its place value is $100{,}000$ :

7 \times 100{,}000 = 700{,}000.

The digit on the page is only a $7$ , but in that position it represents seven hundred thousand.

Worked example 2 Write $20{,}605$ in expanded form

Name the place of each digit, working from left to right:

The $2$ is in the ten-thousands place:

2 \times 10{,}000 = 20{,}000.

The $0$ is in the thousands place, and the $6$ is in the hundreds place ( $6 \times 100 = 600$ ). The next $0$ is in the tens place, and the $5$ is in the ones place. Adding only the nonzero contributions,

20{,}605 = 20{,}000 + 600 + 5.

The two zeros contribute nothing to the sum, and both columns are still essential. They are what hold the $2$ in the ten-thousands column and the $6$ in the hundreds.

Comparing and ordering numbers

To compare two whole numbers, lean on place value rather than on which number “looks” bigger:

The number with more digits is larger, assuming neither has leading zeros. The smallest four-digit number is $1{,}000$ and the largest three-digit number is $999$ , so $1{,}000 > 999$ .
If they have the same number of digits, compare digit by digit from the left. The first place where they differ decides the whole comparison.

Compare $3{,}481$ and $3{,}479$ . They tie in the thousands ( $3 = 3$ ) and in the hundreds ( $4 = 4$ ). In the tens place, $8 > 7$ , so $3{,}481 > 3{,}479$ , and we never even look at the ones place. Going left to right works because a single higher place outweighs everything to its right combined. One extra hundred ( $100$ ) is more than the largest the tens and ones together can reach ( $99$ ). The comparison symbols are $<$ (“less than”), $>$ (“greater than”), and $=$ (“equal to”).

Check your understanding

Which symbol makes a true statement: $5{,}092 \;\square\; 5{,}190$ ?

>

<

=

D none of these

Worked example 3 Compare $48{,}300$ and $48{,}295$

Both numbers have five digits, so compare place by place from the left.

The ten-thousands tie ( $4 = 4$ ) and the thousands tie ( $8 = 8$ ), so keep moving right. The hundreds place is the first to differ: $3$ against $2$ , and $3 > 2$ . That single column decides everything:

48{,}300 > 48{,}295.

Once a higher place settles the comparison, the lower places ( $00$ against $95$ ) cannot overturn it. That is because $300$ already beats the most that $295$ can offer.

Common mistakes

That settles what a written number means. The next question is what you are allowed to do with one. Calculations can often be reordered or regrouped to make them easier. The next lesson asks which of those rearrangements are guaranteed to leave the answer alone, and which only look as though they should.