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# 7.2: Rational and Irrational Numbers

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Circular Garden

In the front of Kenneth Graham Middle School there is a flag with a circular garden beneath it. The students in Mr. Kennedy’s homeroom decided that this circular garden would be their community service project. The students elected Candice the leader of the project and she got right to work organizing the decorating. She asked for a group of students to plant flowers and rake the leaves left from last autumn. It was a perfect spring project.

“We need more dirt,” Sam said soon after the clean-up had begun.

“I think so too,” said Kyle.

Candice went out to assess the situation. The rain and snow of the winter and early spring had left the ground sparse. There definitely was not enough dirt to plant in. Candice began to figure out the area of the circular garden.

She knew that the formula for area is \begin{align*}A = \pi r^2\end{align*}. The diameter of the garden is 16 feet.

That is as far as Candice got. She couldn’t remember the next step. This is where you come in. Using irrational numbers is necessary to solve this problem. But first, you should understand what we mean when we say “irrational number”.

What You Will Learn

In this lesson you will learn how to complete the following skills.

• Classify real numbers as whole numbers, integers, rational numbers (including terminating and repeating decimals) or irrational numbers.
• Compare and order real numbers on a number line.
• Approximate solutions to equations involving irrational numbers.
• Solve real – world problems involving rational and irrational numbers.

Teaching Time

I. Classify Real Numbers as Whole Numbers, Integers, Rational Numbers (Including Terminating and Repeating Decimals) or Irrational Numbers

There are many different ways to classify or name numbers. All numbers are considered real numbers. When you were in the lower grades, you worked with whole numbers. Whole numbers are counting numbers. We consider whole numbers as the set of numbers \begin{align*}\{0, 1, 2, 3, 4 \ldots \}\end{align*}. In middle school, you may also have learned about integers. The set of integers includes whole numbers, but also includes their opposites. Therefore, we can say that whole positive and negative numbers are part of the set of integers \begin{align*}\{ \ldots -2, -1, 0, 1, 2, 3 \ldots \}\end{align*}.

We can’t stop classify numbers with whole numbers and integers because sometimes we can measure a part of a whole or a whole with parts. These numbers are called rational numbers. A rational number is any number that can be written as a fraction where the numerator or the denominator is not equal to zero. Let’s think about this. A whole number or an integer could also be a rational number because we can put it over 1. Look at this example.

Example

-4 could be written as \begin{align*}-\frac{4}{1}\end{align*}, therefore it is an integer, but also a rational number.

Exactly. We can also think about decimals too. Many decimals can be written as fractions, so decimals are also rational numbers.

There are two special types of decimals that are considered rational numbers and one kind of decimal that is NOT a rational number. A terminating decimal is a decimal that is considered to be a rational number. A terminating decimal is a decimal that looks like it goes on and on, but at some point has an end. It terminates or ends somewhere.

.3456798

This is a terminating decimal. It goes on for a while, but then ends.

A repeating decimal is also considered a rational number. A repeating decimal has values that repeat forever.

.676767679...

This is a repeating decimal.

Ah ha! This is the last type of number that is a decimal, but is NOT a rational number. It is called an irrational number. An irrational number is a decimal that does not end and has no repetition. It goes on and on and on. Irrational numbers cannot be represented as fractions. The most famous irrational number is pi \begin{align*}(\pi)\end{align*}. We use 3.14 to represent \begin{align*}\pi\end{align*}, but you should know that \begin{align*}pi\end{align*} is an irrational number meaning that it goes on and on and on forever.

How can we determine if a fraction or a decimal is rational or irrational?

If a number can be written in fraction form then it is rational. If a number cannot be written in fraction form then it is irrational. Besides \begin{align*}\pi\end{align*}, roots of many numbers are also examples of irrational numbers. For example, \begin{align*}\sqrt{2}\end{align*} and \begin{align*}\sqrt{3}\end{align*} are both irrational numbers.

Example

Is \begin{align*}\frac{23}{4}\end{align*} rational or irrational?

Because the number is written as a fraction, it must be a rational number.

Example

Is \begin{align*}\sqrt{7}\end{align*} a rational or an irrational number?

To figure this out, we convert this fraction to a decimal on our calculator.

\begin{align*}\sqrt{7}=2.645751311...\end{align*}

This decimal doesn't end or repeat. This is an irrational number.

Write each of these definitions and one example of each in your notebook.

II. Compare and Order Rational Numbers on a Number Line

You have used number lines to compare numbers before. They can be extremely helpful in comparing the values of different numbers, including irrational numbers. The best strategy is to convert each individual value to a decimal. In the case of irrational numbers, you will have to round them to a reasonable place value. Once the numbers are decimals, you can easily compare them on a number line. Remember when you find the solutions to these types of problems that after you order the values, you should convert them back into their original form.

Example

Place the following values on a number line: \begin{align*}-3.2, \sqrt{2}, 2.\overline{3}, \sqrt{9}\end{align*}.

First find the decimal values of each number.

The number -3.2 is already a decimal.

The number \begin{align*}\sqrt{2}\end{align*} is an irrational number. Its decimal, rounded to the nearest thousandth, is 1.414.

The number \begin{align*}2.\overline{3}\end{align*} is a rational number because it is a repeating decimal. It is equivalent to the fraction \begin{align*}\frac{7}{3}\end{align*}.

The number \begin{align*}\sqrt{9}\end{align*} simplifies to 3, since \begin{align*}3^2\end{align*} is equal to 9.

Then you can place these values on a number line.

This may seem tricky, but if you think about the decimal value of each number then it becomes easier.

III. Approximate Solutions to Equations Involving Irrational Numbers

Sometimes, you will need to find estimates of irrational numbers to solve an equation. The easiest way to do this is to find a decimal value on your calculator that is close to the irrational number. Remember that the more decimal points you include, the more accurate your answer will be. For these purposes, it is usually okay to round an irrational number to the nearest hundredth or thousandth. Once you have found the decimal approximate, solve the equation normally. It is crucial to use words or signage to show that your answer is approximate, not exact. The symbol \begin{align*}\approx\end{align*} means approximately equal to, and is more appropriate than an equals sign in these situations.

Example

Solve for \begin{align*}a\end{align*}: \begin{align*}a=4 \pi\end{align*}.

First find a decimal approximation for \begin{align*}\pi\end{align*} using your calculator. The value of \begin{align*}\pi\end{align*} is 3.1415927... This can be rounded to 3.14 for these purposes.

To solve the equation, multiply 3.14 by 4. This will be the approximate value of \begin{align*}a\end{align*}.

\begin{align*}3.14 \times 4=12.56\end{align*}

So the value of \begin{align*}a\end{align*} is approximately 12.56. \begin{align*}a \approx 12.56\end{align*}

Example

Solve for \begin{align*}y\end{align*}: \begin{align*}12-\sqrt{7}=y\end{align*}.

First find a decimal approximation for \begin{align*}\sqrt{7}\end{align*} using your calculator. The value of \begin{align*}\sqrt{7}\end{align*} is 2.64575... This can be rounded to 2.65 for these purposes.

To solve the equation, subtract 2.65 from 12. This will be the approximate value of \begin{align*}y\end{align*}.

\begin{align*}12-2.65=9.35\end{align*}

So, the value of \begin{align*}y\end{align*} is approximately 9.35. \begin{align*}y \approx 9.35\end{align*}

IV. Solve Real – World Problems Involving Rational and Irrational Numbers

Irrational and rational numbers will show up in real-world problems just like any other mathematical concept. Remember to translate the phrases carefully into mathematical expressions and equations. If necessary, you can convert the irrational numbers into approximated decimals to find the missing values. Remember to solve equations carefully and always keep them balanced.

A common place where you will see irrational numbers is when you are working with circles or spheres. Since pi is related to a circle, you will need to work with irrational numbers when solving problems involving circles.

Example

Henrietta knew that to find the circumference of a circle, she needed to multiply the diameter by \begin{align*}\pi\end{align*}. If the diameter of Henrietta’s circle is 6 inches, what is the approximate circumference of the circle?

First, translate the information in the problem into an equation. Let \begin{align*}C\end{align*} equal the circumference.

\begin{align*}C=\pi \times 6\end{align*}

The value of \begin{align*}\pi\end{align*} is 3.1415927... Rounded to the nearest hundredth, this value is 3.14. You can substitute this value back into the equation to find the value of \begin{align*}C\end{align*}. Remember to use the approximately equals sign after you make this estimation.

\begin{align*}C&=\pi \times 6\\ C &\approx 3.14 \times 6\\ C &\approx 18.84\end{align*}

The circumference of Henrietta’s circle is approximately 18.84 inches.

Now let’s use what we have learned on the problem from the introduction.

## Real-Life Example Completed

The Circular Garden

Here is the original problem from the introduction. Reread it and then solve the problem for the area of the circle.

In the front of Kenneth Graham Middle School there is a flag with a circular garden beneath it. The students in Mr. Kennedy’s homeroom decided that this circular garden would be their community service project. The students elected Candice the leader of the project and she got right to work organizing the decorating. She asked for a group of students to plant flowers and rake the leaves left from last autumn. It was a perfect spring project.

“We need more dirt,” Sam said soon after the clean-up had begun.

“I think so too,” said Kyle.

Candice went out to assess the situation. The rain and snow of the winter and early spring had left the ground sparse. There definitely was not enough dirt to plant in. Candice began to figure out the area of the circular garden.

She knew that the formula for area is \begin{align*}A = \pi r^2\end{align*}. The diameter of the garden is 16 feet.

Notice that you have been given the measurement for the diameter and not the radius.

Solution to Real – Life Example

First, let’s take the measurement for the diameter and figure out the measurement of the radius. The radius is one-half of the diameter.

\begin{align*}16 \ feet &= diameter\\ 8 \ feet &= radius\end{align*}

Now we can substitute this into the formula and solve. We will use 3.14 for \begin{align*}\pi\end{align*} because it is an irrational number.

\begin{align*}A &= \pi r^2\\ A &= (3.14)(8^2)\\ A &= 200.96 \ sq. feet\end{align*}

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Whole Numbers
the set of positive counting numbers.
Integers
the set of whole numbers and their opposites.
Rational Numbers
any number that can be written in fraction form including terminating and repeating decimals.
Irrational Numbers
any number that cannot be written in fraction form. Numbers that do not have an end point when written in decimal form – the decimal values continue indefinitely. These numbers do not fit into the set of rational numbers.
Pi
\begin{align*}\pi\end{align*}, the ratio of the diameter to the circumference of a circle. We use 3.14 to represent this irrational number.
Real Numbers
the set of rational and irrational numbers make up the set of real numbers.

## Time to Practice

Directions: Classify each of the following numbers as real, whole, integer, rational or irrational. Some numbers will have more than one classification.

1. 3.45
2. -9
3. 1,270
4. 1.232323
5. \begin{align*}\frac{4}{5}\end{align*}
6. -232,323
7. -98
8. 1.98
9. \begin{align*}\sqrt{16}\end{align*}
10. \begin{align*}\sqrt{2}\end{align*}

Directions: Answer each question as true or false.

1. An irrational number can also be a real number.
2. An irrational number is a real number and an integer.
3. A whole number is also an integer.
4. A decimal is considered a real number and a rational number.
5. A negative decimal can still be considered an integer.
6. An irrational number is a terminating decimal.
7. A radical is always an irrational number.
8. Negative whole numbers are integers and are also rational numbers.
9. Pi is an example of an irrational number.
10. A repeating decimal is also a rational number.

Directions: Approximate the solution for each equation given the irrational numbers.

1. \begin{align*}\sqrt{2}+5=x\end{align*}
2. \begin{align*}8 = \sqrt{2} + x\end{align*}
3. \begin{align*}t=\pi-5.3\end{align*}
4. \begin{align*}\sqrt{h}=\sqrt{6}-\frac{3}{4}\end{align*}
5. Mrs. DeFazio wrote the following equation on the board. \begin{align*}w=\sqrt{11}-2^2\end{align*} What is the value of \begin{align*}w\end{align*} in Mrs. DeFazio’s equation?

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