Is -3 Irrational Or Rational

7.1: Rational and Irrational Numbers

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Learning Objectives

Identify rational numbers and irrational numbers
Classify different types of real numbers

be prepared!

Before you get started, accept this readiness quiz.

Write 3.xix as an improper fraction. If you missed this trouble, review Example 5.1.4.
Write $\dfrac{5}{11}$ as a decimal. If yous missed this problem, review Example 5.v.3.
Simplify: $\sqrt{144}$. If yous missed this problem, review Case 5.12.1.

Identify Rational Numbers and Irrational Numbers

Congratulations! You have completed the first six capacity of this book! It'south time to have stock of what you have done so far in this course and think about what is ahead. You take learned how to add, decrease, multiply, and divide whole numbers, fractions, integers, and decimals. Y'all have become familiar with the language and symbols of algebra, and have simplified and evaluated algebraic expressions. Yous accept solved many dissimilar types of applications. You have established a good solid foundation that you need so you can be successful in algebra.

In this affiliate, we'll make sure your skills are firmly fix. We'll take another look at the kinds of numbers nosotros have worked with in all previous capacity. We'll piece of work with properties of numbers that volition help you amend your number sense. And nosotros'll practice using them in means that we'll use when we solve equations and complete other procedures in algebra.

Nosotros take already described numbers equally counting numbers, whole numbers, and integers. Do you remember what the deviation is among these types of numbers?

counting numbers	one, two, 3, 4…
whole numbers	0, 1, 2, three, 4…
integers	…−3, −2, −1, 0, 1, ii, iii, four…

Rational Numbers

What type of numbers would you lot get if you started with all the integers and and so included all the fractions? The numbers you would have form the set of rational numbers. A rational number is a number that can be written equally a ratio of two integers.

Definition: Rational Numbers

A rational number is a number that can be written in the class $\dfrac{p}{q}$, where p and q are integers and q ≠ 0.

All fractions, both positive and negative, are rational numbers. A few examples are

\[\dfrac{4}{v}, - \dfrac{vii}{8}, \dfrac{13}{4},\; and\; - \dfrac{20}{3}\]

Each numerator and each denominator is an integer.

We demand to look at all the numbers nosotros have used so far and verify that they are rational. The definition of rational numbers tells us that all fractions are rational. We will now look at the counting numbers, whole numbers, integers, and decimals to make sure they are rational.

Are integers rational numbers? To decide if an integer is a rational number, nosotros try to write it as a ratio of two integers. An like shooting fish in a barrel way to do this is to write information technology as a fraction with denominator i.

\[3 = \dfrac{three}{1} \quad -8 = \dfrac{-8}{1} \quad 0 = \dfrac{0}{ane}\]

Since whatsoever integer tin can exist written as the ratio of two integers, all integers are rational numbers. Recollect that all the counting numbers and all the whole numbers are also integers, and and so they, too, are rational.

What virtually decimals? Are they rational? Let'due south await at a few to see if we can write each of them equally the ratio of ii integers. We've already seen that integers are rational numbers. The integer −viii could be written as the decimal −8.0. So, clearly, some decimals are rational.

Think almost the decimal seven.3. Tin can we write it as a ratio of two integers? Because 7.3 means $7 \dfrac{iii}{10}$, we tin write information technology every bit an improper fraction, $7 \dfrac{three}{10}$. And then 7.3 is the ratio of the integers 73 and ten. Information technology is a rational number.

In general, whatever decimal that ends later a number of digits (such equally 7.3 or −1.2684) is a rational number. We tin utilise the reciprocal (or multiplicative changed) of the place value of the last digit as the denominator when writing the decimal as a fraction.

Example $\PageIndex{1}$:

Write each equally the ratio of two integers: (a) −15 (b) 6.81 (c) $−3 \dfrac{6}{7}$.

Solution

(a) −15

Write the integer as a fraction with denominator 1.

$$\dfrac{-15}{one}$$

(b) 6.81

Write the decimal as a mixed number.	$$6 \dfrac{81}{100}$$
And then catechumen it to an improper fraction.	$$\dfrac{681}{100}$$

Catechumen the mixed number to an improper fraction.

$$- \dfrac{27}{7}$$

Exercise $\PageIndex{1}$:

Write each as the ratio of ii integers: (a) −24 (b) 3.57.

Answer a: $\frac{-24}{1}$

Answer b: $\frac{357}{100}$

Exercise $\PageIndex{two}$:

Write each every bit the ratio of ii integers: (a) −19 (b) 8.41.

Answer a: $\frac{-19}{1}$

Answer b: $\frac{841}{100}$

Let's look at the decimal form of the numbers we know are rational. We have seen that every integer is a rational number, since a = $\dfrac{a}{one}$ for whatever integer, a. We tin too change any integer to a decimal past calculation a decimal point and a zero.

\[\begin{split} Integer \qquad &-ii,\quad -1,\quad 0,\quad 1,\; \; 2,\; three \\ Decimal \qquad &-2.0, -1.0, 0.0, 1.0, 2.0, 3.0 \cease{split up}\]

These decimal numbers cease.

We have also seen that every fraction is a rational number. Await at the decimal form of the fractions we simply considered.

\[\begin{split} Ratio\; of\; Integers \qquad \dfrac{four}{5},\quad -\dfrac{7}{viii},\quad \dfrac{thirteen}{4},\;&- \dfrac{20}{three} \\ Decimal\; forms \qquad 0.8, -0.875, 3.25, &-half dozen.666 \ldots \\ &-6.\overline{66} \finish{split}\]

These decimals either end or echo.

What do these examples tell y'all? Every rational number tin be written both as a ratio of integers and as a decimal that either stops or repeats. The table below shows the numbers we looked at expressed as a ratio of integers and as a decimal.

Rational Numbers
	Fractions	Integers
Number	$$\dfrac{4}{v}, - \dfrac{vii}{viii}, \dfrac{13}{4}, \dfrac{-20}{3}$$	$$-ii, -1, 0, 1, 2, iii$$
Ratio of Integer	$$\dfrac{4}{five}, \dfrac{-7}{eight}, \dfrac{xiii}{4}, \dfrac{-twenty}{iii}$$	$$\dfrac{-2}{1}, \dfrac{-1}{1}, \dfrac{0}{1}, \dfrac{ane}{1}, \dfrac{2}{one}, \dfrac{3}{1}$$
Decimal number	$$0.eight, -0.875, iii.25, -vi.\overline{6}$$	$$-ii.0, -i.0, 0.0, 1.0, 2.0, 3.0$$

Irrational Numbers

Are at that place any decimals that do non stop or repeat? Yes. The number $\pi$ (the Greek letter pi, pronounced 'pie'), which is very important in describing circles, has a decimal form that does not stop or repeat.

\[\pi = 3.141592654 \ldots \ldots\]

Similarly, the decimal representations of square roots of whole numbers that are not perfect squares never end and never repeat. For case,

\[\sqrt{5} = 2.236067978 \ldots \ldots\]

A decimal that does non stop and does non echo cannot be written as the ratio of integers. Nosotros telephone call this kind of number an irrational number.

Definition: Irrational Number

An irrational number is a number that cannot be written every bit the ratio of ii integers. Its decimal form does not stop and does not repeat.

Let's summarize a method we can apply to determine whether a number is rational or irrational.

If the decimal form of a number

stops or repeats, the number is rational.
does not stop and does not repeat, the number is irrational.

Example $\PageIndex{2}$:

Place each of the following equally rational or irrational: (a) 0.58$\overline{three}$ (b) 0.475 (c) three.605551275…

Solution

(a) 0.58$\overline{3}$

The bar to a higher place the 3 indicates that it repeats. Therefore, 0.583 – is a repeating decimal, and is therefore a rational number.

(b) 0.475

This decimal stops after the 5, and then it is a rational number.

The ellipsis (…) means that this number does non terminate. There is no repeating design of digits. Since the number doesn't cease and doesn't repeat, it is irrational.

Exercise $\PageIndex{3}$:

Identify each of the following every bit rational or irrational: (a) 0.29 (b) 0.81$\overline{vi}$ (c) 2.515115111…

Respond a: rational

Answer b: rational

Answer c: irrational

Exercise $\PageIndex{4}$:

Identify each of the following equally rational or irrational: (a) 0.two$\overline{3}$ (b) 0.125 (c) 0.418302…

Respond a: rational

Answer b: rational

Answer c: irrational

Let's retrieve nearly foursquare roots now. Foursquare roots of perfect squares are always whole numbers, so they are rational. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational.

Example $\PageIndex{iii}$:

Identify each of the following as rational or irrational: (a) 36 (b) 44

Solution

(a) The number 36 is a perfect square, since sixⁱⁱ = 36. So $\sqrt{36}$ = 6. Therefore $\sqrt{36}$ is rational.

(b)Remember that viⁱⁱ = 36 and vii^two = 49, so 44 is not a perfect square. This means $\sqrt{44}$ is irrational.

Exercise $\PageIndex{5}$:

Identify each of the following as rational or irrational: (a) $\sqrt{81}$ (b) $\sqrt{17}$

Reply a: rational

Answer b: irrational

Practice $\PageIndex{half dozen}$:

Identify each of the following equally rational or irrational: (a) $\sqrt{116}$ (b) $\sqrt{121}$

Answer a: irrational

Answer b: rational

Classify Real Numbers

We accept seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. Irrational numbers are a split up category of their own. When we put together the rational numbers and the irrational numbers, nosotros get the set of real numbers. Figure $\PageIndex{1}$ illustrates how the number sets are related.

The image shows a large rectangle labeled

Figure $\PageIndex{1}$ - This diagram illustrates the relationships between the different types of existent numbers.

Definition: Real Numbers

Real numbers are numbers that are either rational or irrational.

Does the term "real numbers" seem foreign to you lot? Are there any numbers that are non "real", and, if so, what could they exist? For centuries, the only numbers people knew almost were what we now call the real numbers. And then mathematicians discovered the set of imaginary numbers. You lot won't encounter imaginary numbers in this course, but you will later on in your studies of algebra.

Instance $\PageIndex{4}$:

Determine whether each of the numbers in the post-obit listing is a (a) whole number, (b) integer, (c) rational number, (d) irrational number, and (e) real number.

\[−vii, \dfrac{14}{5}, 8, \sqrt{5}, 5.9, − \sqrt{64}\]

Solution

The whole numbers are 0, 1, 2, 3,… The number 8 is the merely whole number given.
The integers are the whole numbers, their opposites, and 0. From the given numbers, −7 and 8 are integers. Too, notice that 64 is the square of viii so $− \sqrt{64}$ = −8. So the integers are −7, eight, $− \sqrt{64}$.
Since all integers are rational, the numbers −vii, 8, and $− \sqrt{64}$ are also rational. Rational numbers as well include fractions and decimals that terminate or repeat, so $\dfrac{14}{five}$ and 5.ix are rational.
The number 5 is not a perfect square, so $\sqrt{v}$ is irrational.
All of the numbers listed are real.

Nosotros'll summarize the results in a tabular array.

Number	Whole	Integer	Rational	Irrational	Existent
-7		$\checkmark$	$\checkmark$		$\checkmark$
$\dfrac{xiv}{five}$			$\checkmark$		$\checkmark$
8	$\checkmark$	$\checkmark$	$\checkmark$		$\checkmark$
$\sqrt{5}$				$\checkmark$	$\checkmark$
5.9			$\checkmark$		$\checkmark$
$- \sqrt{64}$		$\checkmark$	$\checkmark$		$\checkmark$

Exercise $\PageIndex{7}$:

Determine whether each number is a (a) whole number,(b) integer,(c) rational number,(d) irrational number, and (e) existent number: −three, $− \sqrt{2}, 0.\overline{three}, \dfrac{ix}{5}$, 4, $\sqrt{49}$.

Answer

Number	Whole	Integer	Rational	Irrational	Real
-3		$\checkmark$	$\checkmark$		$\checkmark$
$-\sqrt{2}$				$\checkmark$	$\checkmark$
$0.\overline{3}$			$\checkmark$		$\checkmark$
$\dfrac{9}{v}$			$\checkmark$		$\checkmark$
$4$	$\checkmark$	$\checkmark$	$\checkmark$		$\checkmark$
$\sqrt{49}$	$\checkmark$	$\checkmark$	$\checkmark$		$\checkmark$

Exercise $\PageIndex{8}$:

Decide whether each number is a (a) whole number,(b) integer,(c) rational number,(d) irrational number, and (eastward) existent number: $− \sqrt{25}, − \dfrac{3}{8}$, −1, vi, $\sqrt{121}$, two.041975…

Reply

Number	Whole	Integer	Rational	Irrational	Existent
$− \sqrt{25}$		$\checkmark$	$\checkmark$		$\checkmark$
$-\dfrac{iii}{8}$			$\checkmark$		$\checkmark$
$-one$		$\checkmark$	$\checkmark$		$\checkmark$
$6$	$\checkmark$	$\checkmark$	$\checkmark$		$\checkmark$
$\sqrt{121}$	$\checkmark$	$\checkmark$	$\checkmark$		$\checkmark$
$two.041975…$				$\checkmark$	$\checkmark$

Practice Makes Perfect

Rational Numbers

In the following exercises, write equally the ratio of ii integers.

(a) v (b) 3.19
(a) 8 (b) −ane.61
(a) −12 (b) 9.279
(a) −sixteen (b) four.399

In the following exercises, determine which of the given numbers are rational and which are irrational.

0.75, 0.22$\overline{iii}$, 1.39174…
0.36, 0.94729…, ii.52$\overline{8}$
0.$\overline{45}$, ane.919293…, three.59
0.1$\overline{3}$, 0.42982…, 1.875

In the following exercises, identify whether each number is rational or irrational.

(a) 25 (b) 30
(a) 44 (b) 49
(a) 164 (b) 169
(a) 225 (b) 216

Classifying Real Numbers

In the following exercises, make up one's mind whether each number is whole, integer, rational, irrational, and real.

−8, 0, 1.95286...., $\dfrac{12}{5}, \sqrt{36}$, 9
−9 , $−3 \dfrac{4}{9}, − \sqrt{ix}, 0.4\overline{09}, \dfrac{11}{6}$, 7
$− \sqrt{100}$, −vii, $− \dfrac{viii}{3}$, −1, 0.77, $iii \dfrac{ane}{4}$

Everyday Math

Field trip All the 5th graders at Lincoln Elementary School will go on a field trip to the science museum. Counting all the children, teachers, and chaperones, at that place will exist 147 people. Each bus holds 44 people.
1. How many buses will be needed?
2. Why must the respond be a whole number?
3. Why shouldn't you round the answer the usual mode?
Child care Serena wants to open a licensed child intendance heart. Her land requires that there be no more than than 12 children for each teacher. She would like her child intendance centre to serve xl children.
1. How many teachers volition be needed?
2. Why must the answer be a whole number?
3. Why shouldn't y'all round the reply the usual fashion?

Writing Exercises

In your ain words, explain the difference between a rational number and an irrational number.
Explicate how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.

Self Bank check

(a) Afterwards completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

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(b) If about of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to apply them. What did yous do to go confident of your ability to practise these things? Be specific.

…with some help. This must exist addressed quickly because topics yous do not master become potholes in your road to success. In math, every topic builds upon previous work. Information technology is important to make certain you take a stiff foundation before yous move on. Who tin can you ask for help? Your fellow classmates and instructor are good resource. Is there a identify on campus where math tutors are available? Can your written report skills be improved?

…no—I don't get it! This is a warning sign and you lot must not ignore it. You should get aid correct away or you will quickly exist overwhelmed. See your teacher as soon as you can to discuss your state of affairs. Together you can come up with a plan to get y'all the help yous demand.

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana Higher). This content is licensed nether Creative Eatables Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@v.191."