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Introduction Prepared by Sa’diyya Hendrickson

Name: Date:

Package Summary

• Defining Decimal Numbers

• Things to Remember

• Adding and Subtracting Decimals

• Multiplying Decimals

• Expressing Fractions as Decimals

• Shifting Decimal Points

• Irrational Numbers

• Let’s Play! (Exercises)

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Decimal Numbers Level: Q

1. decimal number:

A decimal number is a rational number expressed in our Base – 10 numeration

system. A decimal point is used to separate the integer portion of the number (to

the left) from the fractional portion (to the right).

• The positive and zero powers of 10 are represented by place values to the left of

the decimal: 100 = ones place, 101 = tens place, 102 = hundreds place, etc.

• The negative powers of 10 are represented by place values to the right of the

decimal: 10−1 = tenths place, 10−2 = hundredths place, etc.

One strategy for remembering the order of the place values is to do a countdown:

. . . 3, 2, 1, 0, −1, −2, −3, . . .

Examples of Decimal Numbers

1. Every integer has a decimal representation. e.g. 38 = 38.0

2. The mixed number 56714

has the decimal representation 567.25, where “567” is the

integer part and 0.25 represents the fractional part “14”.

3. The decimal representation of a number may not be unique. For example, if the

decimal part ends, we can add zeros at the end of the decimal part, and it remains

the same! e.g. 38 = 38.00000 or 567.25 = 567.2500000

4. Consider the number 8.475624. What is the relationship between the number of

places behind the decimal and the absolute value of the exponent for the last place-

value?


Decimal Numbers Level: Q

Let’s explore Example 2 in more detail:

1. Expanded Form:

567.25 = 5(102) + 6(101) + 7(100) + 2(10−1) + 5(10−2)

= 5(102) + 6(10) + 7(1) + 2

(1

10

)+ 5

(1

102

)2. As a Fraction: (Finding an LCD)

567.25 =56, 725

102

Question: Can you use your knowledge of LCDs and properties of exponents to justify

how we arrived at the rational/fractional form?

567.25 = 5(102) + 6(101) + 7(100) + 2(10−1) + 5(10−2)

=5(102)

1+

6(101)

1+

7(1)

1+ 2

(1

101

)+ 5

(1

102

)

LCD=

5(102)(102)

(1)(102)+

6(101)(102)

(1)(102)+

7(1)(102)

(1)(102)+

2(101)

(101)(101)+

5

102

=5(102+2)

102+

6(101+2)

102+

7(102)

102+

2(10)

101+1+

5

102

=5(104)

102+

6(103)

102+

7(102)

102+

2(10)

102+

5

102

=5(104) + 6(103) + 7(102) + 2(10) + 5

102

=56, 725

102


Things to Remember Level: Q

• In the previous example, we noticed that the LCD of our fraction will always be

determined by the place value of the last digit (since it will have the denominator

with the largest power of 10).

• Recall that for integers, the last digit is in the ones place, and when we read the

number, it tells us precisely how many ones we have! For example: 56 means “56

ones.” This is true for decimal numbers, in general. For example: the last digit of

567.25 is 5 which is in the place, and we found that:

567.25 =56, 725

102= (56, 725)

(1

102

)= 56, 725 hundredths!

• Exercise: Express 0.581 as a fraction.

S1 Determine the last non-zero place-value: thousandths(i.e. 10−3 = 1

103

)S2 Write down number with the decimal point removed:

(This tells us how many thousandths we have in all!) thousandths

S3 Express the result in S2 numerically: (581)(

1103

)= 581

103

• Recall that in this number system, it’s all about groups of 10! There are:

– ten hundredths in 10−1 since (10)(10−2) = 10−1 (ten hundredths in one tenth)

– ten tenths in 100 = 1, since (10)(10−1) = 101−1 = 100 = 1

– ten ones in 101 = 10, since (10)(1) = 10

– ten tens in = 100, since (10)(10) = 102 = 100 . . . and so on

• When we exceed ten in any place-value category (e.g. hundredths, tenths, ones, tens,

etc.), each group of ten can “upgrade” or be “carried” to the next place value!

For example: If we have 23 tens, then 20 tens (i.e. two groups) can upgrade to

2 hundreds, leaving behind 3 tens, and no ones. This is why we write 23 tens as

23(10) = 230, which has 2 hundreds, three tens and zero ones.


Adding and Subtracting Level: Q

To create numbers in the base – 10 system, we must know the size of each place-value

category. For instance, to build the number 23.45, we have to know that there are 2 tens,

3 ones, 4 tenths and 5 hundredths.

Strategy When we are adding or subtracting numbers, we should be sure to align the

place values so that we can easily count how many we have in each place-value category!

Adding Decimals (e.g. 876.052 + 14.56)

Subtracting Decimals (e.g. 876.052− 14.56)


Multiplying Decimals Level: Q

Suppose we were asked to calculate: 12 · 46

To do this without a calculator, many of us have learned the following approach:

Let’s use expanded forms and the distributive property to better understand this process.

First we will recall that the distributive property requires the following:

• In words: If we are multiplying numbers such that some are in brackets and involve

sums or differences, then everything in one pair of brackets must be multiplied with

(i.e. distributed to) everything on the outside of those brackets.

• Some generalizations of the property include:

Now consider the product expressed using expanded forms: 12 · 46 = (2 + 10)(6 + 4(10))

Can you use the distributive property to reach the solution of 552? Be sure to keep up

with how many ones, tens and hundreds you have and the need for upgrades!

12 · 46 = (2 + 10)(6 + 40)

= (2 + 10)(6 + 4(10))

=

...



Below is a detailed solution of the exercise on the previous page:

12 · 46 = (2 + 10)(6 + 40) (1)

= (2 + 10)(6 + 4(10)) (2)

= 2(6) + 2(4(10)) + 10(6 + 4(10)) distributing the 2 (3)

= 12 + 8(10) + 10(6 + 4(10)) 12 ones ⇒ needs an upgrade! (4)

= 2 + 10 + 8(10) + 10(6 + 4(10)) (5)

= 2 + 9(10) + 10(6 + 4(10)) (6)

= 2 + 9(10) + (10)6 + 10(4(10)) distributing the 10 (7)

= 2 + 9(10) + 6(10) + 4(102) (8)

= 2 + 15(10) + 4(102) 15 tens ⇒ needs an upgrade! (9)

= 2 + 5(10) + 10(10) + 4(102) (10)

= 2 + 5(10) + 5(102) (11)

= 552 (12)

In the diagram below, fill in the number of the highlighted line (above) that corre-

sponds to the detail that has been pointed out. Two have been completed, so be sure to

understand those answers before you begin!

• Line 5 is upgrading the group of 10 ones in 12 (i.e. we are “carrying” one ten).

• Line 8 shows the result of distributing 10, which produces the second highlighted

row in the diagram equaling 460 = 4(102) + 6(10)! Note: The “6” was written in the

tens place, underneath the 9 because it was created by multiplying a “1” in the tens

place (i.e. one ten) with six ones, which creates six tens.



Calculate: (23.48)(3.24)

Solution: A popular approach to multiplying decimals is to:

S1 Determine the product of the numbers

when the decimals are removed: i.e. 2348 · 324

S2 Determine the number of digits needed be-

hind the decimal point in the solution by adding

up the total number of digits behind each of the

two decimals. There are two digits behind each decimal,

giving a total of 2 + 2 = 4 decimal places needed in the

solution.

7 6 . 0 7 5 2︸︷︷︸4 places

Why Does This Work?

Consider the fractions approach: We know that: 23.48 = 2348102

and 3.24 = 324102

Therefore, the product is given by:

2348

102· 324

102=

2348 · 324

104

1. On page 3, we discovered that the numerators of these fractions will always be the

number with the decimal point removed, which is what S1 suggests that you use.

2. We also know that the power of 10 in the denominator tells us how many places

there are behind the decimal point. By properties of exponents, this will always be

the sum of the powers of 10 in each decimal number’s denominator (i.e. the sum of

the number of places behind each decimal).


Fractions to Decimals Level: Q

Two Kinds of Repeating Decimal Numbers:

1. Repeating, Non-Terminating Decimal Representations:

A decimal representation that has a string of digits (other than zero) that repeats.

(The first string of repeating digits is called the repetend). For example,

0.1234545 . . . = 0.12345 has the repetend 45. The convention is to write a bar over

the repetend instead of writing ellipses.

2. Repeating, Terminating Decimal Representations:

A decimal representation with a finite decimal expansion. The repetend for these

numbers is zero, which is not usually written. These numbers occur for fractions

who have equivalent fractions with powers of 10. e.g. 0.123 = 0.1230.

Celebrity Terminating Decimals:

There are some very popular terminating decimals whose fractions we’d want to know!

1

4= 0.25

1

2= 0.5

3

4= 0.75

1

5= 0.2

Notice that it’s fairly easy to see the smallest power of 10 that we can create from their

denominators! Can you identify them? For example, what is the smallest power of 10

that has 4 as a factor?

Fractions whose Denominators are Powers of 10

If we get a fraction whose denominator is 10, 100, 1000, 10000, etc, our work is already

done! These are all powers of 10. The number of zeros tells us the power!

e.g.47

1000=

7

103

We have 47 thousandths. The denominator tells us that the decimal representation has

three digits behind the decimal point. We also know that the 7 must be the digit in the

thousandths place. So we work backwards, filling in the blanks from right to left, filling

in empty spaces with zeros: . 4 7


Fractions to Decimals Level: Q

The general method for finding the decimal representation of a fraction is long division.

Let’s look at an example and recall our strategies.

Exercise 1: Express 111

as a decimal.

Solution: Notice that 100 has appeared again, meaning that a pattern has begun and our

repetend is “09.” Therefore: 111

= 0.09

Exercise 2: Express 67

as a decimal.

A Few Questions:

1. How many remainders of 7 did you see before the

decimal expansion started to repeat?

2. What is the total number of possible remainders

when dividing by 7?

3. Is it possible for us to perform the long division in-

definitely?


Shifting Decimal Points Level: Q

Consider taking a decimal number and multiplying it by powers of 10. What do you think

might happen to the number? Let’s explore an example using expanded form and the

distributive property:

Multiplying by 102 simply caused our decimal to shift places to the .

Can you make a prediction of what will happen when we divide by powers of 10?

Dividing by 104 caused our decimal to shift places to the .


Power of Fractions Level: Q

The Power of Fractions

Sometimes operations with decimals are more cumbersome and tedious then with frac-

tions. So, working towards being comfortable with both fractions and decimals will help

you move through mathematics with more ease. Consider the following exercise:

0.75÷ 2.25

1. Option 1: Use fractions

0.75÷ 2.25 =3

4÷ 2

1

4rewriting decimals as fractions

=3

4÷ 9

4changing mixed number to improper fraction

=3

4· 4

9by theorem for fraction division

=1

1· 1

3by reducible pairs (3, 9) and (4, 4)

=1

3= 0.3

2. Option 2: Use long division

Notice that using fraction notation, our problem is: 0.752.25

.

If we wanted to create an equivalent fraction so that our denominator is a whole

number, what should we multiply the numerator and denominator by so that our

decimal moves over two places to the right?

0.75

2.25=

(0.75)( )

(2.25)( )=

75

225⇒ 225

)75

This is a case of a question that is much simpler with fractions, simply because you

can avoid having to work with large numbers. In general, it’s a good idea to keep

fractions in mind when you’re working with decimals!


Irrationals Level: Q

• decimal and fraction representations:

Every fraction has a repeating deci-

mal representation and every repeat-

ing decimal representation has a frac-

tion representation.

Q: Can every number on the number

line be expressed as a fraction?

A: No

• There are infinitely many numbers that cannot be expressed in the form of a fraction.

• These numbers are not rational and consequently are called irrational.

• They are also characterized by their non-repeating, non-terminating decimal expan-

sions. In other words, their decimal expansions never repeat making it impossible

for us to express them numerically.

• Symbols are used to represent them since they don’t have a closed numerical repre-

sentation.

Celebrity Irrational Numbers:

Very well known irrational numbers include:

1. π = 3.141592654 . . . the ratio of a circles circumference and diameter.

2.√

2 = 1.41421356237 . . . This number appears as the length of a diagonal in a square

with side lengths equal to 1.

3. e = 2.718281828459045 . . . This number is named after a very well-known math-

ematician named Euler, who contributed greatly to the development of calculus,

among many other things.


Let’s Play! Level: Q

1. Express the following decimals as mixed number or fractions in reduced

form:

(a) 0.05 (b) 12.36 (c) 0.763

(d) 120.5 (e) 45.25 (f) 1.498

2. Express the following in decimal form:

(a) four-hundred fifty-six tenths

(b) two-thousand six-hundred and eight ten-thousandths

(c) 5(103) + 3(102) + 9(10) + 100 + 4(10−1)

(d) 7(106) + 6(104) + 5(103) + 102 + 8(10−2)

3. Calculate:

(a) 45.16 + 47.325 (b) 40.563− 32.981 (c) 6.542 + 0.798

(d) 364.2− 273.16 (e) 53.7 + 42.0513 (f) 2.398− 0.099

4. Calculate the following products using fractions:

(a) 42× 0.5 (b) 4× 0.75 (c) 0.125× 0.16

(d) 0.05× 40 (e) 0.24× 50 (f) 5.6× 0.625

5. Calculate the following products:

(a) 476.2× 52.1 (b) 36.516× 0.21 (c) 45.63× 0.59 (d) 407.1× 36.54

6. Calculate the following products:

(a) 3.45× 103 (b) 476.3215÷ 102 (c) 542.3876÷ 104 (d) 36.5× 105

7. Express the following fractions as decimals:

(a) 25106

(b) 4563102

(c) 30103

(d)5648348105

(e) 16

(f) 58

(g) 35

(h)139


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