Unit 1: Numeracy Review February 14, 2022

Lecture 7: Rational Numbers

Lecturer: Mr. Lewis Carroll

7.1 Working with Rational Numbers

Definition 7.1 A rational number is a number that can be made by dividing two integers into one another.
Rational numbers may also be defined by decimals or decimals.

Example 7.2 Write three examples of rational numbers below.

Example 7.3 Write three examples of irrational (non-rational) numbers below.

7.1.1 Converting Decimals to Fractions

We are going to consider two types of decimals: non-repeating and repeating.

Example 7.4 Convert the decimal 1.723 into a fraction.

1. Determine what the final decimal place value is (10’s, 100’s, 1000’s, etc.). This will determine your
.
2. The numerator will be the numbers in the original decimal, without the decimal point.

Example 7.5 Convert the decimals 0.9, 0.25, 1.4, 2.005, and -12.0031 into their respective fractional form.

Example 7.6 Convert the decimals 0.3 and 0.577 into their respective fractional form.

1. Like before, look at the final decimal place value. one (1) from this value. This is
now your denominator.
2. Your numerator is still the numbers in your original decimal value without the decimal point.

7-1
7-2 Lecture 7: Rational Numbers

7.1.2 Ordering Rational Numbers

There are many different strategies for ordering rationals. Let’s consider the number line first.

7 7 2 2
Example 7.7 Place the rational numbers , − , , and − ,
12 12 3 3

1. It may sometimes be helpful to the area between two numbers into a number of
equally-spaced ticks equivalent to your .
2. Once your tick marks are placed, it is a matter of counting to each respective tick mark based on the
of each fraction.
3. Lastly, notice that we have pairs of positive/negative fractions. tells us that they
should be the exact same distance away from zero as one another — just on
sides of the zero!

1 4 5
Example 7.8 Place the rational numbers −1 , , and − on the number line below.
2 3 9

−3 −2 −1 0 1 2 3

4 1 2
Whiteboard Exercise #1 — Place the numbers , − , −1 , and 0.7 on the number line below.
5 6 3

−3 −2 −1 0 1 2 3

1 1 2 16
Example 7.9 Put the numbers − , 3 , − , 2.8, in ascending order (least to greatest). Two potential
2 2 5 5
strategies are laid out below.

1. You can use a number line to help organize these.

2. You can convert each number into its respective decimal or fractional form and organize them.
 Lecture 7: Rational Numbers 7-3

7.2 Homework

Evaluate each expression below.

3
i. Suppose you’re given the numbers −0.7, 2, −2 15 , 14
5 , −1 13 . Match each number to its respective letter
on the number line below.

A B C D E

−3 −2 −1 0 1 2 3

ii. Place the numbers -5.6, -4.4, -4.2, and -4.8 on the number line below.

−6 −5 −4

Convert the following numbers into their respective fractional or decimal form.

iii. 0.25 = iv. 0.26 =

7 31
v. 25 = vi. 250 =

−11
vii. −0.03 = viii. − 1000 =

46 2
ix. 9 = x. 9 =

Between each pair of fractions, circle the value that is greater.

1
xi. 3 or − 23 9
xii. − 10 or 7
10

xiii. − 12 or − 35 xiv. −2 18 or −2 41

Between each pair of fractions, circle the value that is smaller.

4 2
xv. 7 or 3 xvi. − 43 or − 35

7
xvii. − 10 or − 35 xviii. −1 43 or −1 54
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7.3 Homework Solutions

−2 15 −1 13 −0.7
3
2
14
5

i. −3 −2 −1 0 1 2 3

−5.6 −4.8 −4.4 −4.2

ii. −6 −5 −4

1 13
iii. 4 iv. 50

v. 0.28 vi. 0.124

3
vii. − 100 viii. 0.011

ix. 5.1 x. 0.2

1 7
xi. 3 xii. 10

xiii. − 12 xiv. −2 18

4
xv. 7 xvi. − 53

7
xvii. − 10 xviii. −1 54