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Q4.

Expert-verified

Algebra 1

Found in: Page 10

Book edition Student Edition

Author(s) Carter, Cuevas, Day, Holiday, Luchin

Pages 801 pages

ISBN 9780078884801

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Short Answer

Name the set or sets of numbers to which each real number belongs.
$\frac{56}{7}$

The real number $\frac{56}{7}$ belongs to the sets of natural numbers, whole numbers, integers, and real rational numbers.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step-1. Apply the concept of the real number system

Real numbers include all the numbers except complex numbers and have the following five subsets:

Natural numbers: Includes counting objects and starting from 1.
Whole numbers: Includes the set of natural numbers along with 0.
Integers: Z = Includes numbers that are not fraction (positive and negative whole numbers)
Rational numbers: Includes the numbers which can be written in the form of $\frac{p}{q}$ where p and q are integers, $q \neq 0$ .
Irrational numbers: Includes numbers that cannot be written in the form of $\frac{p}{q}$ where p and q are integers, $q \neq 0$ .

Step-2. Examples of the real number system.

1. Natural numbers: $\{1, 2, 3, 4, . . .\}$

2. Whole numbers: $\{0, 1, 2, 3, 4, . . .\}$

3. Integers: Measurement of debts, temperatures, etc., fall under the set of integers $\{. . ., - 3, - 2, - 1, 0, 1, 2, 3, . . .\}$

4. Rational numbers: If we cut a cake into equal pieces, then we may have a piece that represents a fraction like $\frac{5}{6}, 1.5 (= \frac{3}{2}), \frac{6}{9}, \frac{8}{3}$

5. Irrational numbers: The numbers that are square roots of positive rational numbers, cube roots of rational numbers, etc., such as $\sqrt{2}, \sqrt[3]{5}, - \sqrt{3}$

Step-3. Analyze the given real number.

Consider the given real number $\frac{56}{7}$

We can rewrite it as:

$\frac{56}{7} = \frac{8 \times 7}{7} = 8 = \frac{8}{1}$

Now we will check each subset of real numbers:

Natural numbers: They are positive, countable, and start from 1. So, $\frac{56}{7} = 8$ being positive, countable, and greater than 1 is a natural number.
Whole numbers: They are natural numbers including 0. So, $\frac{56}{7} = 8$ being a natural number is a whole number as well.
Integers: They are whole numbers that are both positive and negative. So, $\frac{56}{7} = 8$ being a whole number is an integer as well.
Rational numbers: They can be written in the form $\frac{p}{q}$ where p and q are integers, $q \neq 0$ . Since, $\frac{56}{7} = \frac{8}{1}$ it is a rational number.
Irrational numbers: They cannot be written in the form $\frac{p}{q}$ where p and q are integers, $q \neq 0$ . Since from the above point, $\frac{56}{7}$ can be written in the form $\frac{p}{q}$ , it is not an irrational number.

Therefore, the real number $\frac{56}{7}$ belongs to the sets natural numbers, whole numbers, integers and real rational numbers.

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