
Introduction
to
Decimals 

Unit 12 > Lesson 1 of 12 
Problem: 
Order these numbers from least to greatest:



Analysis: 
We know that
is a
mixed
number: it consists of a whole
number and a
fraction.
Let's use place
value to help us compare these numbers.


57 
= (5 x 10) + (7 x 1) 

= (5 x 10) + ( 7 x 1) + (4 x ) +
(9 x
)

58 
= (5 x 10) + (8 x 1)


Answer: 
Ordering these numbers from least to greatest, we get:



That's a lot of writing! We can use decimals to write
more easily.

Definition: 
A decimal is any number in our baseten number system.
Specifically, we will be using numbers that have one or more
digits to the right of the decimal point in this unit of lessons. The decimal point is used to separate the ones place from
the tenths place in decimals. (It is also used to separate dollars from
cents in money.) As we move to the right of
the decimal point, each number place is divided by 10.

Below we have expressed the number
in expanded
form and in decimal form.

Mixed Number 
 E x p a n d e d F o r m
 
Decimal Form 

= (5 x 10) + ( 7 x 1)

+ (4 x ) + (9 x
)

= 57.49

As you can see, it is easier to write
in decimal form. Let's look at this decimal number in a placevalue chart to
better understand how decimals work.

PLACE
VALUE AND DECIMALS 



















5 
7 
. 
4 
9 




As we move to the right in the place value chart, each number place is divided by
10. For example, thousands divided by 10 gives you hundreds. This is also true for
digits to the right of the decimal point. For example, tenths divided by 10 gives you
hundredths. When reading decimals, the decimal point should be
read as "and." Thus, we read the decimal 57.49 as "fiftyseven and
fortynine hundredths." Note that in daily life it is common to read the decimal
point as "point" instead of "and." Thus, 57.49 would be read as "fiftyseven point four
nine." This usage is not considered mathematically
correct.

Example 1: Write each phrase as a fraction and as a decimal.
phrase 
fraction 
decimal 
six tenths 

.6 
five hundredths 

.05 
thirtytwo hundredths 

.32 
two hundred sixtyseven thousandths 

.267 
So why do we use decimals?
Decimals are used in
situations which require more precision than whole numbers can provide. A good
example of this is money: Three and onefourth dollars is an amount between 3 dollars and 4
dollars. We use decimals to write this amount as $3.25. A decimal may have both a wholenumber part and a fractional
part. The wholenumber part of a decimal are those digits to the left of the
decimal point. The fractional part of a decimal is represented by the digits to
the right of the decimal point. The decimal point is used to separate these
parts. Let's look at some examples of this.

decimal 
wholenumber part 
fractional part 
3.25 
3 
25 
4.172 
4 
172 
25.03 
25 
03 
0.168 
0 
168 
132.7 
132 
7 
Let's examine these decimals in our placevalue chart.
PLACE
VALUE AND DECIMALS 




















3 
. 
2 
5 










4 
. 
1 
7 
2 








2 
5 
. 
0 
3 










0 
. 
1 
6 
8 







1 
3 
2 
. 
7 





Note that 0.168 has the same value as .168. However, the zero
in the ones place helps us remember that 0.168 is a number less than one.
From this point on, when writing a decimal that is less than one, we will always
include a zero in the ones place. Let's look at some more examples of decimals.

Example 2: Write each phrase as a decimal.
Phrase 
Decimal 
fiftysix hundredths 
0.560 
nine tenths 
0.900 
thirteen and four hundredths 
13.040 
twentyfive and eightyone hundredths 
25.810 
nineteen and seventyeight thousandths 
19.078 
Example 3: Write each decimal using words.
Decimal 
Phrase 
0.0050 
five thousandths 
100.6000 
one hundred and six tenths 
2.2800 
two and twentyeight hundredths 
71.0620 
seventyone and sixtytwo thousandths 
3.0589 
three and five hundred eightynine tenthousandths 
It should be noted that five thousandths can also be written as zero and five
thousandths.
Expanded Form
We can write the whole number 159 in expanded
form as follows: 159 = (1 x
100) + (5 x 10) + (9 x 1). Decimals can also be written in expanded form.
Expanded form is a way to write numbers by showing the value of each digit. This is shown in the example below.

Example 4: Write each decimal in expanded form.
Decimal 

Expanded form 
4.1200 
= 
(4 x 1) + (1 x )
+ (2 x ) 
0.9000 
= 
(0 x 1) + (9 x ) 
9.7350 
= 
(9 x 1) + (7 x )
+ (3 x ) + (5 x ) 
1.0827 
= 
(1 x 1) + (0 x )
+ (8 x )
+ (2 x )
+ (7 x ) 
Decimal Digits
In the decimal number 1.0827, the digits 0, 8, 2 and 7 are called decimal digits.
Definition: 
In a decimal number, the digits to the right of the decimal point that name the fractional part of that number, are called decimal digits. 
Example 5: Identify the decimal digits in each decimal number below.
Decimal Number 
Decimal Digits 
1.40000 
1.40000 
359.62000 
359.62000 
54.00170 
54.00170 
0.72900 
0.72900 
63.10148 
63.10148 
Writing whole numbers as decimals
A decimal is any number, including whole numbers, in our baseten number system. The decimal point is usually not written in whole numbers, but it is implied. For example, the whole number 4 is equivalent to the decimals 4. and 4.0. The whole number 326 is equivalent to the decimals 326. and 326.0. This important concept will be used throughout this unit.

Example 6: Write each whole number as a decimal.
Whole Number 
Decimal 
Decimal with 0 
17 
17. 
17.0 
459 
459. 
459.0 
8 
8. 
8.0 
1,024 
1,024. 
1,024.0 
519 
519. 
519.0 
63,836 
63,836. 
63,836.0 
Often, extra zeros are written to the right of the last digit of a decimal
number. These extra zeros are place holders and do not change the value of the decimal. For example:
7.5 = 
7.50 = 
7.500 = 
7.5000 
and so on. 
9 = 
9. = 
9.0 = 
9.00 
and so on. 
Note that the decimals listed above are equivalent
decimals.
So how long can a decimal get?
A decimal can have any number of decimal places to the right of the decimal
point. An example of a decimal number with many decimal places is the numerical value of Pi,
shortened to 50 decimal digits, as shown below: 

3.14159 26535 89793 23846 26433 83279 50288 41971 69399
37510 
Summary: 
A decimal is any number in our base ten number system. In this lesson we
used numbers that have one or more digits to the right of the decimal point.
The decimal point is used to separate the whole number part from the fractional
part; it
is handy separator. Decimal numbers are used in
situations which require more precision than whole numbers can provide. As we move to the right of
the decimal point, each number place is divided by 10.

Exercises
Directions: Read each question below. Select your answer by clicking on its button. Feedback to your answer
is provided in the RESULTS BOX. If you make a mistake, choose a different button.

1. 
Which of the following is equal to ? 



2. 
Which of the following is equal to ? 



3. 
Which decimal represents seven hundred sixtytwo and three tenths? 



4. 
Which of the following is the expanded form of 0.546? 



5. 
Which of the following is equal to
twentyfive and sixtynine thousandths? 



This lesson is by Gisele Glosser. You can find me on Google.
