Understand the Place Value System

What Is Place Value?

In Mathematics, place value refers to the value of a number depending on its position. Let’s start by looking at the different places a digit can go using a  place value chart. Here are the first four places to the left of the decimal. 

Depending on where we put a digit, let’s say four, can determine different numbers. If we place the four in the ones, it means four. If we place the four in the tens place, now it means forty! We can place the four in the hundreds to represent four hundred! The further we move the number to the left on the chart, the larger it gets! 

4

40

400

The Importance of Zero

Did you notice that when we moved the digit over, the number had more zeros in it? These zeros are used as place holders to indicate what place value we want the four to be in. Imagine if we didn’t use those zeros! Can you tell the difference between the numbers below without the zeroes? Now you know why zero is very important as a placeholder when determining values.  

.004

.04

.4

Writing Numbers Using Place Values

When we write down numbers we are writing a shortened version that relies on the understanding of place values. 

6,827

The number that has been written in six thousand, eight hundred twenty seven. If we wanted to write this number using the place values we would need to show that the 6 is really 6,000. The 8 is really representing 800, the 2 is really 20. Below the number is written using the place values of each digit. This is also known as the expanded form.

A shortened version of writing a number using place values is:

Both ways show the number six thousand eight hundred twenty seven!

How Do We Show a Zero Using Place Value?

As we mentioned, zeros are very important when it comes to place values. So how do we show that in a number that has zeros? Let’s take a look at an example. 

4,705

First we can put this number in the place value chart.

Now we can write it using expanded form:

The part that is equal to zero is not usually written.

When there is a zero in the number and you are going from expanded form to standard form, make sure there are not any zeros missing! 475 is very different than 4,705!

Organizing Place Values

Did you notice that when a number has a thousands place value there is a comma? Commas are used to help read larger numbers. Starting from the decimal, a  comma is placed every three place values to the left. 

Commas are not used to the right of the decimal point.

The Decimal

Have you ever seen a dot in a number? That is called the decimal. A decimal separates whole numbers from fractional parts. Let’s look at some visual representations to better understand decimals. Base ten blocks are a great way to help visualize decimals.

Notice that the whole is a ten by ten block. There are 100 smaller squares that make up the whole. Each block is one one hundredth, 0.01, or. Each row or column has ten blocks. This means that a full row or column is one tenth, 0.1, or. 

Now let’s practice looking at how different fractions and decimals are represented by the diagram.

What fraction is represented by the following diagram?

First we can count how many full columns there are. There are five full columns. This is equivalent to five tenths, 0.5, or. Now we can count the number of squares in the partial column. There are four, which is equal to four hundredths, 0.04, or. 

We can now combine the tenths and hundredths. To get fifty four hundredths!

0.5 +0.04 =0.54

Now that we have reviewed decimals, let’s revisit the place value chart from earlier and look to the right of the one’s place. 

The first place value to the right of the decimal is the tenth place. Ten tenths makes one whole. That is just like how ten ones makes ten, or ten tens make one hundred! Let’s try writing a number with a decimal in expanded form. 

263.93

We have already written the numbers to the left of the decimal in expanded form. Two hundred sixty three will look like:

Now for the decimal portion. We know that the nine is in tenth place. Tenth written as a fraction is one over ten. The three is in the hundredths place. One hundredth is written as one over one hundred. 

Fantastic!

The Relationship Between Place Values

Consider the place value chart below. Each digit is 10 times greater than the digit to its right. Let’s look at the ones and tens. There are ten ones in one ten, or one times ten is ten. This pattern continues as we move to the left. There are ten tens in one hundred, or ten times ten is one hundred.

We can use what we know about place value to quickly multiply and divide decimals by 10.

Likewise, each digit isthe size of the digit to its left.

How does this help us? It means that we can follow a pattern when multiplying or dividing by 10.

Multiplying by 10

We can use the place value chart to multiply by 10. Multiplying by 10 makes a number bigger by shifting the place value to the left. Consider the equation below.

Step 1: Determine the current place value of each digit using the place value chart.

Step 2: Shift the place value.
Since we are multiplying by 10, we know we need to shift the place value to the left.

Step 3: Analyze how the number changed.
We can see in the place value chart that 362.489 is now written as 3,624.89.

The number 3,624.89 is ten times larger than 362.489.

Dividing decimals by 10

The place value chart can be used to divide decimals by 10 as well. Dividing by 10 makes a number smaller by shifting the place value to the right. Remember that dividing by 10 is the same as multiplying by.

Let’s look at another example.

The expression could also be expressed as

.

Step 1: Determine the current place value of each digit.

Step 2: Shift the place value.
Since we are dividing by 10, we know we need to shift the place value to the right.

Step 3: Analyze how the number changed.
We can see in the place value chart that 64.28 is now written as 6.428.

The number 6.428 is ten times smaller, or one tenth of the value of 64.28.

Using Exponents to Write Large and Small Numbers

What are exponents?

Exponents are used to show repeated multiplication. When a number is multiplied by itself, exponents can be used to shorten the expressions. For example:

→

The base is the number that is being multiplied, in this case four. The exponent indicates the number of times the base is being multiplied by itself, five times.

There is a relationship between exponents and place value. We have seen how multiplying and dividing by tens moves a number over in a place value. If we want six to become sixty, we can multiply by ten. If we want six to become sixty we can multiply by one hundred. What if we wanted the six to become six thousand?

We can break this into six times one thousand. We know that ten times one hundred is one thousand. One hundred is ten times ten. We have broken six thousand into six times ten, times ten, times ten.

Now  we have a ten being multiplied by itself three times.

–>

This gives us a base of ten and an exponent, or power of, three. This is read as ten to the third power. The exponent is written at the upper right hand corner of the base. 

How does this relate to place values? By multiplying by ten to the third power we have moved the six from the ones place to the thousands place! The higher the exponent means the more palace values that the number will move.

We have made some really large numbers by multiplying by powers of ten. Now we can look at really small numbers by dividing by ten. When we have four and we divide by ten we get nine tenths. If we divide by 10 we can move the decimal one place over to the left. If we divide by another ten we get four hundredths. One more time dividing by ten and we get four thousandths. four divided by ten divided by ten divided by ten is equal to four thousandths. Let’s write this with exponents. There are three tens so we can write that as ten to the third power.

Each time the four is divided by ten the place value becomes one place smaller.

Exponents are helpful with multiplying and dividing by tens. The exponent tells us how many times ten is being multiplied or divided. We know that each ten represents a place value and we can use a shortcut with exponents to help us. When we multiply we move the decimal to the right the same number as the exponent. When we divide we move the decimal to the left  the same number as the exponent.

Place Value and Face Value

We have been reviewing and learning about place values. The place value gives the value of a digit based on its position or location in a number. Face value is the value of the actual digit. With any digit, the face value cannot be changed, but the palace value can be changed by moving it to a different position. Let’s use the digit 5 to look at how place value and face value compare. 

Notice how the place value of the five changes with it’s position in the number. The face value doesn’t change. There is a limited number of face values since we use ten digits. By combining those ten digits in different orders we can make infinite different place values! 

When writing numbers in expanded form we are taking the face value and assigning a palace value to that digit. 

35,948

Each of the digits starts with its face value and is then multiplied by a number to give it the place value.

Did you know?

• Our place value is base 10 since there are ten digits. There are other numeric systems like base 2. The number 5 is written as “101”.
• The face value of a number is the value of the digit. This is different than the place value. For example with the number 426, the place value of the 4 is 400, but the face value is just 4. 
• The place value of a number is dependent on the numbers around it to give it the position and value of its place. The face value is independent of the position of the numbers surrounding it. 
• Tenth is an ordinal number. An ordinal number indicates the order or position of something. For example, if you are the tenth person to finish the quiz, that means that nine people finished before you and the eleventh person will finish after you. 
• Some countries switch the use of decimals and commas. In the United States a number may be written as:

23,059.92

In other countries it may be written as:

23.059,92

Practice:

Write the following in expanded form:

5,286

Solution:

Let’s place the numbers in the place value chart to help match the digits with the values.

We can write it using multiplication:

Or we can write the number using the place value amounts:
5,000 + 200 + 80 + 1

Practice:
Fill in the blank with the missing value in the expanded form of the given number:

600 + _____ + 4 = 684

Solution:
The missing digit from 684 is 8. Since the 8 is in the tens place it has the value of 80, giving us the missing value! 80

Practice:
What number has 4 thousands, 3 hundreds, 7 tens and 2 ones?

Solution:
Let’s change the words into values:
4 thousands = 4,000
3 hundreds = 300
7 tens = 70
2 ones = 2
We can add these values together!

4,000 + 300 + 70 + 2 = 4,372

Practice:
Write the following number in standard form:

Solution:
First we can look at what values the numbers are being multiplied by. This will indicate the place values of those numbers. We have a three in the thousand place, a four in the hundreds place, and a six in the tenths place.

Fill in the place values without any digits with zeros.

Now we have the standard form:
3,400.6

Practice:
What fraction is represented by the following diagram?

Solution:
Since all of the blocks represent one whole, we want to find the number of tenths and hundredths. To find the tenths, count the number of full rows. To find the hundredths, count the number of blocks in the partial row.

Tenths: 4                                            0.4
Hundredths: 7                                  0.07
0.47

Practice:
What place value will the 7 be in for the following number?

Solution:
Ten to the sixth power is equal to ten being multiplied by itself six times





This gives us one million. Seven times ten to the sixth power is equal to seven million. This places the seven in the millions place.

Practice:
Rearrange the digits so that the three has a place value of 30 and the 7 has the same place value as its face value. 

753

Solution:
We have three digits, seven, five, and three. We have three place values, hundreds, tens, and ones. We want the three to have a value of 30, this means that it must be placed in the tens place. The seven needs to have the same place value as its face value. Face values are the same in the ones place. This means the seven is in the ones place. This leaves the five to be in the hundreds place.

537

Practice:
Which digit always has the same face value and place value?

Solution:
Zero! All numbers will have the same place value and face value when they are in the ones place, but a zero has the same face value and place value no matter where it is in the order. The best way to see this is using expanded form.
602                                                               

5024  

Notice how the face value changes to place value after the multiplication. This is not true with the zero since zero multiplied by anything is zero.

Practice without solutions:

True or False:
The 4 in the following number is in the tens place
6,724
False

What number has 6 thousands, 5 hundreds, 2 tens and 8 tenths?

6,520.8

Write the following number in standard form:

2,080.5

Which number is ten times larger than 295.29?

29.529                  2,952.9                          2.9529                                29,529

2,952.9

Solve:

12.5

What is the value of 5 in 3,502.82 ?

500

The value of 2 in 52.8 is ________ than the value of 6 in 34.6

Ten times more

6 hundreds ÷ 10 = ?

60