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# Sets and Symbols

## Whole numbers, integers, rational numbers, and irrational numbers

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Practice Sets and Symbols
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Classify Real Numbers

Have you ever thought about circles? What about types of numbers? Take a look at this dilemma.

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 A=πr2\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”.

### Guidance

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 {0,1,2,3,4}\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 {2,1,0,1,2,3}\begin{align*}\{ \ldots -2, -1, 0, 1, 2, 3 \ldots \}\end{align*}.

We can’t stop classifying 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.

-4 could be written as 41\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 (π)\begin{align*}(\pi)\end{align*}. We use 3.14 to represent π\begin{align*}\pi\end{align*}, but you should know that pi\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 π\begin{align*}\pi\end{align*}, roots of many numbers are also examples of irrational numbers. For example, 2\begin{align*}\sqrt{2}\end{align*} and 3\begin{align*}\sqrt{3}\end{align*} are both irrational numbers.

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

Now that you have had some practice identifying real numbers, you can compare them.

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.

Place the following values on a number line: 3.2,2,2.3¯,9\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 2\begin{align*}\sqrt{2}\end{align*} is an irrational number. Its decimal, rounded to the nearest thousandth, is 1.414.

The number 2.3¯\begin{align*}2.\overline{3}\end{align*} is a rational number because it is a repeating decimal. It is equivalent to the fraction 73\begin{align*}\frac{7}{3}\end{align*}.

The number 9\begin{align*}\sqrt{9}\end{align*} simplifies to 3, since 32\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 approximate decimal value of each number then it becomes easier.

Classify each real number.

#### Example A

7\begin{align*}\sqrt{7}\end{align*}

Solution: Irrational number

#### Example B

19\begin{align*}\frac{1}{9}\end{align*}

Solution: Rational number

#### Example C

98\begin{align*}-98\end{align*}

Solution: Integer and rational number

Now let's go back to the dilemma from the beginning of the Concept.

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.

Now we can substitute this into the formula and solve. We can use 3.14 as an approximation for π\begin{align*}\pi\end{align*} in order to get the approximate area.

AAA=πr2=(3.14)(82)=200.96 sq.feet

### Vocabulary

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. These are numbers that do not have an end point or repetition when written in decimal form – the decimal values continue indefinitely.
Pi
π\begin{align*}\pi\end{align*}, the ratio of the diameter to the circumference of a circle. We use 3.14 to approximate this irrational number.
Real Numbers
the set of rational and irrational numbers make up the set of real numbers.

### Guided Practice

Here is one for you to try on your own.

Is 234\begin{align*}\frac{23}{4}\end{align*} rational or irrational?

Solution

Because the number is written as a fraction, we know it is a rational number.

### 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. 45\begin{align*}\frac{4}{5}\end{align*}
6. -232,323
7. -98
8. 1.98
9. 16\begin{align*}\sqrt{16}\end{align*}
10. 2\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.

### Vocabulary Language: English

inequality

inequality

An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are $<$, $>$, $\le$, $\ge$ and $\ne$.
Integer

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...
Irrational Number

Irrational Number

An irrational number is a number that can not be expressed exactly as the quotient of two integers.
Pi

Pi

$\pi$ (Pi) is the ratio of the circumference of a circle to its diameter. It is an irrational number that is approximately equal to 3.14.
rational number

rational number

A rational number is a number that can be expressed as the quotient of two integers, with the denominator not equal to zero.
Real Number

Real Number

A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.
Whole Numbers

Whole Numbers

The whole numbers are all positive counting numbers and zero. The whole numbers are 0, 1, 2, 3, ...