BANGA NATIONAL HIGH SCHOOL
SCIENCE, TECHNOLOGY
& ENGINEERING
PROGRAM
ELEMENTARY ALGEBRA
Quarter 1 – Module 1
The Set of Real Numbers and
The Concept Of Opposites
The following are some reminders in using this module:
1. Use the module with care. Do not put unnecessary mark/s on any part of the
module. Use a separate sheet of paper in answering the exercises.
2. Read the instruction carefully before doing each task.
3. Observe honesty and integrity in doing the tasks and checking your answers.
4. Finish the task at hand before proceeding to the next.
5. Return this module to your teacher/facilitator once you are through with it.
If you encounter any difficulty in answering the tasks in this module, do not hesitate
to consult your teacher or facilitator. Always bear in mind that you are not alone.
We hope that through this material, you will experience meaningful learning and
gain deep understanding of the relevant competencies. You can do it!
The module is divided into two lessons, namely:
Lesson 1 – The Set of Real Numbers
Lesson 2 – The Concept of Opposites
After going through this module, you are expected to:
1. define and identify the set of real numbers;
2. graph each set of numbers on the number line, and
3. find the opposite of a given number.
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LESSON
THE SET OF
REAL NUMBERS
1
Discussion:
Real Numbers are the set of numbers that is formed by combining the rational
numbers and the irrational numbers.
1. NATURAL NUMBERS are used for counting.
{1, 2, 3, 4, …..}
2. WHOLE NUMBERS are the set of numbers formed by adding 0 to the set of natural
numbers.
{0, 1, 2, 3, 4, …..}
3. INTEGERS are the set of numbers formed by positive whole numbers, negative
whole numbers, and zero.
{…, -3, -2, -1, 0, 1, 2, 3, …}
4. RATIONAL NUMBERS are the set of all numbers which can be expressed in the
𝑎
form: 𝑏 , where a and b are integers and b≠0. It can be expressed as terminating or
repeating decimals.
5. IRRATIONAL NUMBERS is a number that cannot be written as the ratio of two
integers. Its decimal form does not stop and does not repeat.
The real number line and some of its points are shown below:
5⁄
2
-4
-3
− 1⁄3
−√2
-2
-1
0
3⁄ √4
2
1
2
3
4
The real number line
The real number system is represented below in a tree diagram. It shows how all of the sets
of numbers relate to one another.
2
What’s More
State whether each sentence is TRUE or FALSE.
1. The counting numbers are also called natural numbers.
2. The set of integers consists of positive and negative numbers.
1
3. The mixed number 3 is a rational number.
4
4. A number which can be expressed in the form
𝑎
𝑏
, where a and b are integers
and b ≠ 0, is a rational number.
5. The value of 6 ÷ 10 is in the set of integers.
6. The repeating decimal 0.888… is a rational number.
7. Between every pair of rational numbers, there are infinitely many rational
numbers.
8. The irrational numbers are found on the number line.
9. Fractions and decimals are rational numbers.
10. Irrational numbers cannot be found on the number line.
B. Name the coordinates of the points graphed on each number line.
1.
-4
-3
-2
-1
0
1
2
3
4
-8
-7
-6
-5
-4
-3
-2
-1
0
2.
What I Have Learned
Place each numbers in the most SPECIFIC categories of real numbers they belong.
9
-4
0
𝟒
√𝟏𝟔
1. Natural Numbers
:
2. Whole Numbers
:
3. Integers
:
4. Rational Numbers
:
5. Irrational Numbers
:
𝟓
6. Real Numbers :
3
3
-2.4
4.17
- √𝟑
What I Can Do
A. Determine whether each number is rational or irrational.
1. 0
2.
𝟑
𝟒
3. 15.125
4.
𝟏𝟏
𝟏𝟐
5. √121
6. √18
7. 52
8. 2√5
9. 0.333…
10. 4.121314…
B. Graph each set of numbers.
1. {−𝟖. 𝟒, −𝟕. 𝟐, −𝟔, −𝟒. 𝟖, −𝟐. 𝟓}
𝟐
𝟏
𝟏
𝟒
2. {−𝟑 𝟑 , −𝟐, 𝟑 , −𝟏, 𝟑 , − 𝟑 , 𝟎 }
4
LESSON
THE CONCEPT OF
OPPOSITES
2
DISCUSSION:
The concept of opposites is commonly demonstrated in real life. In terms of
direction, going south is the opposite of going north; in terms of length, short is the opposite
of long; in terms of altitude, low is the opposite of high; in terms of quantity, few is the
opposite of many.
In mathematics, opposites are denoted by the signed numbers called
integers. If a direction going to the right is represented by a positive (+) sign, then going to
the left is represented by a negative (-) sign. If going up is +, then coming down is - .
Study the following number lines.
negative (-) integers
-4
-3
-2
positive (+) integers
-1
0
+1
+2
+3
+4
opposites
+4
+3
positive (+) integers
+2
+1
0
-1
negative (+) integers
-2
-3
-4
5
Examples of numbers that denote opposite signs.
1. an increase of P5 denotes +5, while a decrease of P5 denotes -5
2. a profit of P100 denotes +100, while a loss of P100 denotes -100
3. a direction of 2 blocks east denotes +2, while 2 blocks west denotes -2
4. a rise of 8 degrees in temperature denotes +8, while a drop of 8 degrees denotes -8
5. a deposit of P200 denotes +200, while a withdrawal of P200 denotes -200
What’s More
Complete the following table.
Then –x equals:
If x equals:
1. 5
2. 35
3. 0
4. -5
5. -15
6. -50
What I Can Do
Represent the following with integers. And state the opposite of the quantities
described.
1. a weight increase of 3 kilograms
2. going up the stairs by 8 steps
3. walking 5 blocks north
4. pushing a crate 6 meters to the right
5. raising the flag 10 meters high
6. winning by 10 points in a game
7. marching 15 steps to the right
8. traveling 10 kilometers south
9. an increase of 510 in weekly allowance
10. a decrease of 4 kilograms in weight
11. a drop of 10°C in temperature
12. climbing a mountain 8200 meters high
13. 875 meters below sea level
14. accelerating by 2 meters per second
15. moving a chair 3 meters forward
6
7
What I Can Do
integers
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
+3
+8
+5
+6
+10
+10
+15
-10
+510
-4
-10
+8200
+875
+2
+3
opposite
-3
-8
-5
-6
-10
-10
-15
+10
-510
+4
+10
-8200
-875
-2
-3
WHAT’S MORE
1.
2.
3.
4.
5.
6.
-5
-35
0
5
15
50
THE CONCEPT OF OPPOSITES
WHAT I CAN DO
A.
1.
2.
3.
4.
5.
6.
Rational
Rational
Rational
Irrational
Rational
6. Irrational
7. Rational
8. Irrational
9. Rational
10. Irrational
B.
1.
2.
WHAT’S MORE
WHAT I HAVE LEARNED
A.
1.
2.
3.
4.
5.
9 , -4 , 0 , √𝟏𝟔 , 𝟓 , 3 , -2.4 , 4.17 , - √𝟑
6.
3 , - √𝟑
5.
9 , √𝟏𝟔 , 0 , -4 , 𝟓 , -2.4 , 4.17
4.
9 , √𝟏𝟔 , 0 , -4
3.
9 , √𝟏𝟔 , 0
2.
9 , √𝟏𝟔
1.
𝟒
TRUE
TRUE
TRUE
TRUE
FALSE
6. FALSE
7. TRUE
8. FALSE
9. TRUE
10. FALSE
B.
1. {-4, -2, 0, 2, 4}
2. {-7, -6, -5, -3, -2}
𝟒
THE SET OF REAL NUMBERS
Answer Key
References
Bernabe, J., 2009. Elementary Algebra. Revised Ed. 1251 Gregorio Araneta
Avenue, Quezon City: SD Publication, Inc., pp.26-28.
Oronce, O. and Mendoza, M., 2007. E-Math. 1st ed. 856 Nicanor Reyes Sr.,
St.,Sampaloc, Manila: Rex Book Store, pp.84-98.
https://cnx.org/contents/CImQfPDv@9.1:dgB2SHi0@20/1-8-The-Real-Numbers
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