# 1.06 Comparing and ordering real numbers

Lesson

Now that we are familiar with the real number system and comparing and ordering rational numbers, percentages, and integers we can compare and order all real numbers.

## Comparing and ordering real numbers

Just like we have compared rational numbers to rational numbers and integers to integers, we can compare all of the different subsets of real numbers to each other. It is always helpful to convert all numbers you are comparing to the same form. Usually decimal form is most appropriate, especially when irrational numbers are involved.

#### Worked example

##### QUESTION 1

Compare the numbers $\frac{1}{3}$13 , $3.2\times10^{-3}$3.2×103, $\pi$π, and $2%$2%and put them in descending order (from smallest to largest).

Think: What form should we put them in so we can compare them easily?

Do: Let's convert them all into decimals

 $\frac{1}{3}$13​ $=$= $0.\overline{3}$0.3 commonly known repeating decimal $3.2\times10^{-3}$3.2×10−3 $=$= $0.0032$0.0032 move the decimal $3$3 places to the left $\pi$π $=$= $3.14$3.14... approximately $3.14$3.14 $2%$2% $=$= $\frac{2}{100}$2100​ divide by $100$100 $=$= $0.02$0.02

$0.0032$0.0032 < $0.02$0.02 < $0.0032$0.0032 < $\pi$π

Therefore, the list from smallest to largest is: $3.2\times10^{-3}$3.2×103, $2%$2%, $\frac{1}{3}$13$\pi$π

#### Practice questions

##### QUESTION 2

For each of the following pairs of numbers, select the number with the smallest value.

1. $4\pi$4π

A

$12$12

B

$4\pi$4π

A

$12$12

B
2. $\pi^2$π2

A

$144$144

B

$\pi^2$π2

A

$144$144

B

##### QUESTION 3

Compare: $0.31$0.31 and $45%$45%

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

2. Which of the two values is greater?

$45%$45%

A

$0.31$0.31

B

$45%$45%

A

$0.31$0.31

B

### Outcomes

#### 8.NS.2

Use rational approximations of irrational numbers to compare the size of irrational numbers, plot them approximately on a number line, and estimate the value of expressions involving irrational numbers