# 1.10 Comparing and ordering rational numbers

Lesson

## The number system

Our real numbers system hasn't been around in its current state forever. It was developed slowly over time. The real number system includes rational numbers, irrational numbers, integers, whole numbers, and natural numbers.

The first numbers we put on the number line are the natural numbers.

Natural numbers

The set of natural numbers are the counting numbers, starting from $1$1:

$1,2,3,4,5,6,7,\ldots$1,2,3,4,5,6,7,

Next, we will add $0$0 to our number line to show the whole numbers.

Whole numbers

The set of whole numbers are the counting numbers, starting from $0$0:

$0,1,2,3,4,5,6,7,\ldots$0,1,2,3,4,5,6,7,

The left side of this line looks pretty empty. If we add all the negatives we now have a set of numbers called the integers.

Integers

Whole numbers together with the negatives of the whole numbers make up the set of integers:

$\ldots,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,\ldots$,7,6,5,4,3,2,1,0,1,2,3,4,5,6,7,

But are there numbers between the ones we already have marked? The answer is yes - an infinite amount of numbers between every little mark!

What sort of numbers are these? Well, rational numbers are all numbers that indicate whole numbers as well as parts of whole numbers. So fractions, decimals, and percentages are added to our number line to create the set of rational numbers.

Rational numbers

Rational numbers are numbers that can be written as the ratio of two integers with a non-zero denominator.

Integers together with all fractions, terminating and repeating decimals, and percents make up the set of rational numbers.

They cannot all be listed, but here are some examples:

$\ldots,-8,-7.4,-7,-6,-5.33387,-4,-2,0,\frac{1}{2},75%,1,2,3,3.5656,\ldots$,8,7.4,7,6,5.33387,4,2,0,12,75%,1,2,3,3.5656,

## Comparing rational numbers

Just as we can compare and order integers on the number line, we can compare rational numbers on the number line.

Numbers on the number line are ordered from smallest to largest. That is, values for numbers get smaller as we move further to the left on the number line.

Values for numbers get larger as we move further to the right on the number line.

Likewise, values for numbers get larger as we move further to the right on the number line.

For example, $-3.12$3.12 is to the left of $\frac{-3}{2}$32, so $-3.12$3.12 is smaller than $\frac{-3}{2}$32. We can write this statement as the following inequalities:

 $-3.12$−3.12 $<$< $\frac{-3}{2}$−32​ $-3.12$−3.12 is less than $\frac{-3}{2}$−32​ because $-3.12$−3.12 is further to the left on the number line $\frac{-3}{2}$−32​ $>$> $-3.12$−3.12 $\frac{-3}{2}$−32​ is greater than $-3.12$−3.12 because $\frac{-3}{2}$−32​ is further to the right on the number line

Sometimes, it's easiest to compare rational numbers if they are written in the same form. For example, by converting decimal values to fractions or vice versa. We might also convert among fractions, decimals, and percents, such as in the following worked example.

#### Worked example

##### QUESTION 1

Compare the numbers $0.53$0.53, $60.3%$60.3%, and $\frac{17}{77}$1777 and put them in order from smallest to largest.

Think: How can we put them all in the same form so we can compare them easily?

Do: Let's convert both $0.53$0.53 and $\frac{17}{77}$1777 into percentages.

 $0.53\times100$0.53×100 $=$= $53$53 $0.53$0.53 $=$= $53%$53% $\frac{17}{77}$1777​ $=$= $\frac{17}{77}\times100$1777​×100 $%$% $=$= $\frac{1700%}{77}$1700%77​ $=$= $22\frac{6}{77}$22677​ $%$%

$22\frac{6}{77}$22677% < $53%$53% < $60.3%$60.3%

Therefore, the list from smallest to largest is: $\frac{17}{77}$1777, $0.53$0.53, $60.3%$60.3%

#### Practice questions

##### QUESTION 2

Consider the values $71%$71% and $0.31$0.31.

1. First convert $0.31$0.31 to a percentage.

2. Select the inequality sign that makes the statement true.

 $71%$71% ? $0.31$0.31

$=$=

A

$>$>

B

$<$<

C

$=$=

A

$>$>

B

$<$<

C

##### QUESTION 3

Consider the statement:

$\frac{67}{50}$6750 > $154%$154%

1. First convert $\frac{67}{50}$6750 to a percentage

2. Hence, is the statement True or False?

True

A

False

B

True

A

False

B

##### QUESTION 4

Consider the following values:

$71%$71%, $\frac{4}{6}$46, $\frac{84}{1000}$841000, $0.7$0.7, $0.99$0.99, $50.8%$50.8%

1. Which has the largest value?

$71%$71%

A

$\frac{84}{1000}$841000

B

$\frac{4}{6}$46

C

$50.8%$50.8%

D

$0.7$0.7

E

$0.99$0.99

F

$71%$71%

A

$\frac{84}{1000}$841000

B

$\frac{4}{6}$46

C

$50.8%$50.8%

D

$0.7$0.7

E

$0.99$0.99

F
2. Which has the smallest value?

$\frac{84}{1000}$841000

A

$50.8%$50.8%

B

$0.99$0.99

C

$71%$71%

D

$0.7$0.7

E

$\frac{4}{6}$46

F

$\frac{84}{1000}$841000

A

$50.8%$50.8%

B

$0.99$0.99

C

$71%$71%

D

$0.7$0.7

E

$\frac{4}{6}$46

F
3. Which has a value closest to $0.5$0.5?

$0.99$0.99

A

$\frac{4}{6}$46

B

$50.8%$50.8%

C

$0.7$0.7

D

$71%$71%

E

$\frac{84}{1000}$841000

F

$0.99$0.99

A

$\frac{4}{6}$46

B

$50.8%$50.8%

C

$0.7$0.7

D

$71%$71%

E

$\frac{84}{1000}$841000

F

### Outcomes

#### 6.NS.C.6c

Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.