Department of Education

Schools Division Office – Muntinlupa City Student Center for Life Skills Bldg., Centennial Ave., Brgy. Tunasan, Muntinlupa City 1 1 (02) 8805-9935 / (02) 8805-9940

MARINELLA C. SANTILLAN WRITER

ENRICO L. OBSEQUIO EDITOR

EMELITA D. BAUTISTA EdD VALIDATOR

Illustrate the different subsets of real

numbers and arrange real numbers in

increasing or decreasing order .

MATHEMATICS 7

ACCOUNTMATHEMATICS

9 Module 8

Hello everyone! Welcome to Module 8 of Mathematics 7!

You learned in the last module how to estimate square root of a whole number

To the nearest hundredths and plot irrational numbers on a number line.

Now you will be exposed to the different kinds of real numbers. You will learn

how to arrange them in increasing or decreasing order and plot them on a number

line.

You could feel that I am with you while reading this module. The first part of this

module is a Pre-Test. Read the lessons carefully, then do the exercises and activities,

and finally, answer the Post Test.

There are questions inside a box that you need to answer by yourself first. You are guided by this

bitmoji

Then italicized words after this bitmoji will give you the correct answer.

Whenever you will see this bitmoji a lesson is presented.

After going through this module, you are expected to:

• Illustrate the different subsets of real numbers

• Arrange real numbers in increasing or decreasing order

• Plot real numbers on a number line

Happy reading!

PRE-TEST

I. TRUE or FALSE. Write the word TRUE if the statement is true. Write the word FALSE,

if the statement is false.

1. Zero is a whole number.

2. Zero is an element of the set of counting or natural numbers.

3. Every decimal is an irrational number.

4. The rational numbers are subsets of irrational numbers.

5. √5 is an irrational number.

6. Both rational numbers and irrational numbers can be plotted in a number line.

7. Terminating decimals are rational numbers.

8. Repeating non-terminating decimals are rational numbers.

9. √−1 = - 1

10. Natural numbers are integers.

II. Multiple Choice. Write the letter that corresponds to the correct answer

1. Which one is the smallest?

A. 1

2 B.

1

3 C.

1

4 D.

1

5

2. Which statement is TRUE?

A. Some irrational numbers are integers

B. All irrational numbers are real numbers.

C. No rational numbers are whole numbers.

D. All rational numbers are integers.

3. Where is – 2 1

3 on a number line?

.

4. What is the correct location of √7 on a number line?

5. Which list shows the numbers in increading order?

A. -0.5, 1.5, -2, -0.75, √7 C. -2, -0.75, -0.5, 1.5, √7

B. -0.5, -2, -0.75, 1.5, √7 D. √7, 1.5, -0.5, -0.75, -2

6. What is the correct symbol to be used in: - 12 - 20?

A. < B. > C. ≤ D. ≥

7. Which one is arranged from least to greatest?

A. -0.2, 7

10, -1, √2, and -4 C.

7

10, √2, -0.2, -1, -4

B. -4, -1, -0.2, √2, 7

10 D. -4, -1, -0.2,

7

10, √2

8. Which comparison is correct?

A. 51

6 = 5.16̅ C. 5

1

6 < 5.16̅

B. 51

6 > 5.16̅ D. 5

1

6 ≤ 5.16̅

9. Which of these real numbers is the GREATEST?

A. 64

8 B. 7.99 C. -9 D. -1

10. Three (3) of the real numbers below describe √36. Which one DOESN’T?

A. rational B. integer C. irrational D. natural number

Activity 1: LOOK AROUND! There are fifteen (15) different words/partitions of

numbers are hidden in the puzzle below. Copy this in a sheet of paper. Then use any

color to shade the word that you could find. You may look up, down, across, backward,

and diagonally.

N A F R A C T I O N S I

S P B A C C D Z W N E L

T E O F T O G E H E R A

O R H S I U J R O G A M

I C R K I N R O L A T I

L E E L M T N A E T I C

A N A O P I I Q L I O E

R T L R S N T V U V N D

U I N T E G E R E E A A

T I R R A T I O N A L I

A N O N I N T E G E R S

N N U M N U M B E R S S

Words: 1. NUMBERS 2. INTEGER 3. IRRATIONAL

4. FRACTION 5. PERCENT 6. REAL

7. COUNTING 8. ZERO 9. WHOLE

10. NEGATIVE 11. RATIONAL 12. DECIMAL

Activity 2: RECALLING YOUR PAST.

Try to recall your experiences in the past by answering the questions below.

1. What were the first set of numbers you learn during your early stage when you

were just starting to talk?

2. What number represents “ nothing”?

3. You are now grown-ups and started to study the “opposites”. Can you give

examples of numbers that specifically describe debt or below zero temperature?

4. Give examples of numbers that are part of a whole. For example, you cannot eat

the whole cake in one seating. How do you express the part of the cake you

consumed?

5. In your studies, there are three kinds of decimals. They are

1) terminating decimals

2) repeating and non-terminating (repeats and never ends)

3) non-repeating and non-terminating (do not repeat and never ends)

Give 3 examples for each.

Answers may vary. For numbers 3 to 5, some of the possible answers are:

1. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

2. 0

3. -100, -5°, etc.(any negative numbers)

4. 1

4 ,

1

2 ,

2

3, 𝑒𝑡𝑐. (any fractions)

5. terminating decimals: 0.4, 5.14, 8.35, etc

Repeating and non-terminating: 0.2222…, 6.131313… 15.424242…

Non-repeating and non-terminating: 3.141592654… 2.236067978…

. Since you were a child you experience to play and deal with numbers.

• First, when you are just starting to talk you begin uttering numbers

like 1, 2, 3, …up to 10. This means you are already counting. That is why

this first set of numbers you learn are called COUNTING or NATURAL NUMBERS.

• As you grow older, more numbers are added to your list

of numbers. Aside from counting/natural numbers, you learn ZERO to represent “nothing”.

Counting/Natural Numbers + zero = WHOLE NUMBERS.

• Eventually you found out that not all numbers are composed

of zero and counting numbers. You learn that there are numbers BELOW ZERO. These are

the NEGATIVE of whole numbers. So, whole numbers + negative of counting/

natural numbers = INTEGERS.

• Then time comes that you experience taking only a part of a whole,

thus finding that integers could be used as a numerator and denominator forming a fraction. These

types of numbers are called RATIONAL NUMBERS. Take note that not all integers can be used

as a denominator. ZERO cannot be used as a denominator of a fraction. This is not a legal fraction

because its overall value is undefined.

• Fractions have numerators and denominators. If the numerators and

denominators are integers, and you divide numerators by the denominators this results to decimals.

Decimals could be terminating, repeating non-terminating, non-repeating non-terminating.

Terminating decimals and repeating and non-terminating decimals are RATIONAL NUMBERS.

Non-repeatng and non-terminating decimals are IRRATIONAL NUMBERS. Rational numbers

CAN NOT be expressed as a quotient of two integers.

• RATIONAL NUMBERS and IRRATIONAL NUMBERS are both

REAL NUMBERS.

Look at the Venn Diagram

here to show real numbers and their

subsets.

Activity 3: TRUE or FALSE

Read the following sentences carefully. Write the word TRUE if the statement is true;

otherwise, write the word FALSE.

1. All integers are rational.

2. “5” is a whole number, an integer, a rational number and a real number.

3. All integers are whole numbers.

4. Zero is a rational number.

5. √9 is a rational number.

Answers:

1. TRUE 2. TRUE 3. FALSE 4. TRUE 5. TRUE

Activity 4:

Put a check ( ) for each that the number is a part of:

Counting/

Natural

Numbers

Whole

Number Integers

Rational

Numbers

Irrational

Numbers

Real

Numbers

Example:

25

1. - 8

2. 2

5

3. 13

4. √10

5. 0.333…

Answers:

Counting/

Natural

Numbers

Whole

Number Integers

Rational

Number

s

Irrational

Numbers

Real

Numbers

1. - 8

2. 2

5

3. 13

4. √10

5. 2.64575…

The figure on the right is a number line. On a number line, positive numbers are to the right of

zero. Negative numbers are to the left of zero.

Zero is neither positive nor negative.

The arrows at either end of the line indicate that

the number line extends forever in each

Direction. There is NO greatest positive number and there is

NO smallest negative number.

The number line can be used to

compare and order positive and negative numbers.

Going from left to right, numbers increase in

value. Going from right to left, numbers decrease in

value.

Inequality symbols like < (read as “is less than”) and >(read as “is greater than”) are used to show the ordering of real numbers. If a and b are the two real numbers, and a is on the right of b, then we can say that a > b. If a is on the left of b, then a < b.

Example: Arrange the real numbers 21

4, - 3, 0, √8, -5 in increasing order and plot

them in a number line.

Solution: 2 1

4 needs to be converted to decimal (divide 1 by 4) = 2.25

√8 needs to get the approximate value to the nearest hundredths = 2.83

0

Whichever is on the farthest right has the greatest value. We will now plot all of these

numbers on a number line.

Answers:

- 5 < - 3 < 0 < 2 1

4 < √8

Activity 5.

Fill in the blanks with the correct symbol between two given real numbers.

Choose your answer from <, >, or =

1. – 4 _____ -1

2. 0 _____ -3

3. 3 1

4 _____

4. 𝜋 _____ 2 1

10

5. −√12 _____ - 4

Answers:

1. – 4 < -1

2. 0 > -3

3. 3 1

4 <

4. 𝜋 > 2 1

10

5. −√12 < - 4

Activity 6.

Arrange the given real numbers from least to greatest.

1. 3, -2, 0, -1

2. 𝜋, √25, - 4.25, - 6

3. 41

5, - 2

1

2, 0, - 1

1

4

4. 4.13…., 4.75, 5, 3.75

5. −√16, −√9, −√25, −√1

Answers:

1. -2, -1, 0, 3

2. -5, -4.25, 𝜋, √25

3. - 21

2, - 1

1

4, 0, 4

1

5

4. 3.75, 4.13…, 4.75, 5

5. −√25, −√16, −√9, −√1

2 1

4

√8 3 −5

Activity 7

Copy the number line below in your sheet of paper. Graph the given real numbers under

Activity 6.

I. Do you agree with the following statements? Write YES if you do. If you don’t, write NO.

1. -17.5 > −√121

2. -14 could NOT be a natural number.

3. 5

10 is a rational number.

4. 3 is a natural, whole, integer, rational, and real number.

5. -36 is real, rational, whole number.

II. List all the classifications of numbers that apply to the real number. . The choices for your

answers are:

natural/counting numbers,

whole numbers

integers

rational numbers

irrational numbers

real numbers

Study the example given.

Example: Classify what kind of number is 5.

Answer: 5 is a counting number, whole number, integer, rational number, and real number

1. 7

2. 3

5

3. - 17

4. 8.05

5. 7.416198487…

III. Do what is asked.

1. Circle all of the natural numbers in the list: -2.4, 3, 12, 1

3 , √8

2. Circle all of the whole numbers in the list: -7, 2

5 , 14, 0.15, 0

3. Circle all of the integers in the list: - 13, 1.5, 15, 0, -√6

4. Circle all of the rational numbers in the list: -16, 𝜋, −8

15 , 10.9, 13.03840481…

5. Circle all of the irrational numbers in the list: 𝜋, √100, √99, √9

16, 24.123123…

IV. Arrange the following numbers from greatest to least.

1. √7, 2, √8

2

2. √200, - 11, √81, 11,5

3. √8, 2𝜋, 3.5

4. 𝜋, √6, 7

3

5. √18, 16

3,

17

4

V. Graph the real numbers under test IV.

I. Let us see if you could fill in the boxes below correctly. Choose from the

“word pool” below.

WORD POOL

Rational Numbers

Whole Numbers

Irrational Numbers

Natural Numbers

Integers

Negative Integers

Zero

Fractions/Decimals

Real Numbers

II. Define the following subsets of real numbers:

1. natural or counting numbers

2. whole numbers

3. integers

4. rational numbers

5. irrational numbers

III. Answer the following questions:

1. How do you arrange real numbers from least to greatest? From greatest to least?

2. What are the different mathematical symbols that could be used in comparing real numbers?

3. How do you plot real numbers on a number line?

Identify the set of numbers that best describes the situation. Explain your choice.

Example:

• The number of people wearing eyeglasses in a room.

Answer: The set of whole numbers best describes the situation. Once we count,

we start from 1, 2, 3, and so on. So we might think the answer is natural/counting numbers. But what

about if no one wears eyeglasses, the answer could be 0.

1. the amount of water in a glass as it evaporates.

2. the score in a basketball game after the first half (after 2 quarters)

3. change in P100-peso bill after buying goods amounting to P65.75

4. temperature in the North Pole

5. the circumference of your electric fan

POST TEST I. TRUE or FALSE.

Write the word TRUE if the statement is true; otherwise, write the word FALSE.

1. Repeating decimal like 4.17777… is an irrational number.

2. A decimal that stops like 5.43 is a rational number.

3. A non-ending and non-repeating decimal is an irrational number.

4. All rational numbers are integers.

5. Some irrational numbers are integers.

6. No integers are irrational numbers.

7. All whole numbers are integers.

8. No real numbers are irrational numbers.

9. All integers greater than 0 are whole numbers.

10. All real numbers, both rational and irrational numbers can be plotted on a number line.

II. Multiple Choice. Write the letter that corresponds to the correct answer

1. Which of the numbers below is a natural/counting, whole, rational, and real number?

A. 7.8 B. 21 C. 174

5 D. – 12

2. How would you classify √42?

A. rational, integer C. irrational

B. irrational, integer D. whole, integer, rational

3. How do you classify number 321?

A. irrational C. rational, integer, whole

B. natural, whole, integer, rational D. rational, integer

4. What is the order of 2𝜋, √37, 6.75 from GREATEST TO LEAST?

A. 6.75, √37, 2𝜋 C. √37 , 2𝜋, 6.75

B. 2𝜋, √37, 6.75 D. 6.75, 2𝜋 √37

5. Which represents a rational number?

A. √65 B. √80 C. √110 D. √324

6. Which number is the SMALLEST?

A. 1.04 B. 1.4 C. √2 D. 1.4

7. Which of the following is an integer BUT NOT a whole number?

A. – 9 B. – 4 C. 0 D. 4.8

8. Which set of numbers best describes the displayed weights on a digital scale that shows

each weight to the nearest half pound?

A. whole numbers C. real numbers

B. rational numbers D. integers

9. Which is in order from LEAST to GREATEST?

A. 3,3, 10

3, 𝜋,

11

4 C. 𝜋,

10

3,

11

4, 3.3

B. 10

3, 3.3,

11

4, 𝜋 D.

11

4 , 𝜋, 3.3,

10

3

10. Which number line below is set up correctly?

A. C.

B. D.

ANSWER KEY

Pre Test Post Test 1. TRUE I. 1. NO I. 1. FALSE

2. FALSE 2. YES 2. TRUE

3. FALSE 3. YES 3. TRUE

4. FALSE 4. YES 4. FALSE

5. TRUE 5. NO 5. FALSE

6. TRUE II. 6. TRUE

7. TRUE 1. 7 is a natural, whole, rational, 7. TRUE

8. TRUE real number 8. FALSE

9. FALSE 2. 3

5 is a rational, real number 9. TRUE

10. TRUE 3. – 17 is an integer, rational, and 10. TRUE

II. real number II.

1. D 4. 8.05 is a rational, real number 1. B

2. B 5. 7.416198487… is an irrational, 2. C

3. D III. 3. B

4. C 1. -2.4, 3, 12, 1

3 , √8 4. D

5. C 2. -7, 2

5 , 14, 0.15, 0 5. D

6. B 3. - 13, 1.5, 15, 0, -√6 6. A

7. D 4. -16, 𝜋, −8

15 , 10.9, 13.03840481… 7. D

8. A 8. B

9. A 5. 𝜋, √100, √99, √9

16, 24.123123… 9. D

10. C IV. 10. A

1. √7, 2, √8

2

2. √200, 11.5, √81, - 11

3. 2𝜋, 3.5, √8

4. 𝜋, √6, 7

3

5. 16

3,

17

4, √18

THE SET OF REAL NUMBER SYSTEM

ANSWERS:

1. rational numbers

2. natural or counting numbers

3. rational numbers

4. integer, rational numbers

5. irrational numbers

References:

https://flexbooks.ck12.org/cbook/ck-12-elementary-intermediate-college-

algebra/section/1.3/primary/lesson/subsets-of-real-numbers-c-alg

https://www.ck12.org/book/ck-12-algebra-ii-with-trigonometry-concepts/section/1.1/

https://www.slideshare.net/GraceRobledo2/subsets-of-real-numbers

https://algebra1coach.com/1-3-real-numbers-and-the-number-line/

https://nv01912265.schoolwires.net/cms/lib/NV01912265/Centricity/Domain/880/module01%208th%20grade.p

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