Topics
1. Real Numbers
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United States of America
Grade 8

1.06 Compare and order real numbers

Lesson
Practice

Introduction

Now that we are familiar with the  real number system  , evaluating  square roots of perfect squares  and  cube roots of perfect cubes  , as well as  estimating the values of irrational numbers  , we can now compare and order real numbers.

Compare and order real numbers

Just like we have compared rational numbers with rational numbers, and integers with integers, we can compare all of the different subsets of real numbers too.

It is helpful to convert the numbers you are comparing to be in the same form. Usually decimal form is most appropriate, especially when irrational numbers are involved.

Examples

Example 1

Compare the numbers \dfrac{4}{3}, \dfrac{\pi}{2}, and \sqrt{2} and arrange them in ascending order.

Worked Solution
Create a strategy

Convert the numbers into decimal form to compare and arrange them from least to greatest.

Apply the idea
\displaystyle \dfrac{4}{3}\displaystyle =\displaystyle 1.\overline{3}Convert to decimal

The value of \pi is approximately 3.14, so:

\displaystyle \dfrac{\pi}{2}\displaystyle \approx\displaystyle 3.14\div 2Divide by 2
\displaystyle \approx\displaystyle 1.57Evaluate the division

To estimate \sqrt{2}:

\displaystyle 1 \lt 2\displaystyle <\displaystyle 4Identify closest perfect squares
\displaystyle \sqrt{1} \lt \sqrt{2}\displaystyle <\displaystyle \sqrt{4}Sqaure root the numbers
\displaystyle 1 \lt \sqrt{2}\displaystyle <\displaystyle 2Evaluate the square roots of the perfect squares

Determining whether \sqrt{2} is closer to 1 or 2:

\displaystyle 4-2\displaystyle =\displaystyle 2The difference between the two largest squares
\displaystyle 2-1\displaystyle =\displaystyle 1The difference between the two smallest squares

\sqrt{2} is closer to 1, but still close to the middle. We can start squaring 1.3 and the tenths above it until we determine which tenths \sqrt{2} lies between.

\displaystyle 1.3^2\displaystyle =\displaystyle 1.69
\displaystyle 1.4^2\displaystyle =\displaystyle 1.96
\displaystyle 1.5^2\displaystyle =\displaystyle 2.25
\displaystyle 1.96 \lt 2\displaystyle <\displaystyle 2.25Compare the values
\displaystyle \sqrt{1.96} \lt \sqrt{2}\displaystyle <\displaystyle \sqrt{2.25}Square root the numbers
\displaystyle 1.4\lt\sqrt{2}\displaystyle <\displaystyle 1.5Evaluate the square roots of the perfect squares

Determining whether \sqrt{2} is closer to 1.4 or 1.5:

\displaystyle 2.25-2\displaystyle =\displaystyle 0.25The difference between the two largest squares
\displaystyle 2-1.96\displaystyle =\displaystyle 0.04The difference between the two smallest squares

This shows us \sqrt{2} is closer to 1.4.

1.\overline{3} \lt 1.4 \lt 1.57

The list from smallest to largest is: \dfrac{4}{3},\,\sqrt{2},\,\dfrac{\pi}{2}.

Example 2

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

a
A
12
B
4\pi
Worked Solution
Create a strategy

Find an approximate value for 4\pi and compare this to 12.

Apply the idea
\displaystyle \pi\displaystyle \approx\displaystyle 3.14...Approximate \pi
\displaystyle 4\pi\displaystyle \approx\displaystyle 4\times 3.14 ...Multiply by 4
\displaystyle \approx\displaystyle 12.56...Evaluate
\displaystyle 12.56 ...\displaystyle >\displaystyle 12Compare values

12 has a smaller value, so the correct answer is option A.

b
A
\pi^2
B
3\sqrt{8}
Worked Solution
Create a strategy

Approximate the value of \pi^2 and compare it to an approximation of the square root of 8 multiplied by 3.

Apply the idea

We know that \pi\approx 3.14 so

\displaystyle \pi^2\displaystyle \approx\displaystyle 3.14^2
\displaystyle \approx\displaystyle 9.8596

To estimate the value of \sqrt{8}:

\displaystyle \sqrt{4}\lt \sqrt{8}\displaystyle <\displaystyle \sqrt{9}Identify closest perfect squares
\displaystyle 2\lt \sqrt{8}\displaystyle <\displaystyle 3Evaluate square roots

This means \sqrt{8} will be less than 3. Multiplying a number less than 3 by 3 will result in a number less than 9.

The correct answer is Option B.

Reflect and check

We can check our answer with a calculator:

\displaystyle \pi^2\displaystyle \approx\displaystyle 9.869604401089359\ldots
\displaystyle 3\sqrt{8}\displaystyle \approx\displaystyle 8.48528137423857\ldots
c
A
\sqrt{75}
B
\sqrt[3]{150}
Worked Solution
Create a strategy

Estimate the value of irrational numbers by identifying the closest perfect cubes and perfect squares and evaluating their roots.

Apply the idea

To estimate the value of \sqrt{75}:

\displaystyle \sqrt{64}\lt \sqrt{75}\displaystyle <\displaystyle \sqrt{81}Identify closest perfect squares
\displaystyle 8\lt \sqrt{75}\displaystyle <\displaystyle 9Evaluate square roots

To estimate \sqrt[3]{150}:

\displaystyle \sqrt[3]{125}\lt \sqrt[3]{150}\displaystyle <\displaystyle \sqrt[3]{216}Identify closest perfect cubes
\displaystyle 5\lt \sqrt[3]{150}\displaystyle <\displaystyle 6Evaluate cube roots

Since \sqrt{75} is between 8 and 9 but \sqrt[3]{150} is between 5 and 6, \sqrt[3]{150}\lt \sqrt{75}.

The correct answer is Option B.

Idea summary

In comparing and ordering real numbers, 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.

Outcomes

8.NS.A.2

Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g. π^2).

8.EE.A.2

Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

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