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Sets and Symbols

Whole numbers, integers, rational numbers, and irrational numbers

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

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 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 @$\begin{align*}-\frac{4}{1}\end{align*}@$ , 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 A

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

Solution: Irrational number

Example B

@$\begin{align*}\frac{1}{9}\end{align*}@$

Solution: Rational number

Example C

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

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.

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

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

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

Vocabulary

Whole Numbers
The whole numbers are all positive counting numbers and zero. The whole numbers are 0, 1, 2, 3, ...
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...
Rational Numbers
A rational number is a number that can be expressed as the quotient of two integers, with the denominator not equal to zero.
Irrational Numbers
An irrational number is a number that can not be expressed exactly as the quotient of two integers.
Pi
@$\begin{align*}\pi\end{align*}@$ (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.
Real Numbers
A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.

Guided Practice

Here is one for you to try on your own.

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

Solution

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

Video Review

Khan Academy Identifying Rational Numbers

Explore More

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.

Vocabulary

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, ...

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