Advanced Algebra

and Trigonometry 9
Quarter 1
Self Learning Module 1
Imaginary Numbers
Enriched Mathematics – Grade 9
Quarter 1 – Self Learning Module 1: Imaginary Numbers
First Edition, 2020

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Published by the Department of Education - Schools Division of Pasig City

Development Team of the Self-Learning Module

Writer: Daniel G. Reyes
Editor: Revie G. Santos
Reviewers: Revie G. Santos; Ma. Victoria L. Peñalosa
Illustrator: Name
Layout Artist: Name
Management Team: Ma. Evalou Concepcion A. Agustin
OIC-Schools Division Superintendent
Aurelio G. Alfonso EdD
OIC-Assistant Schools Division Superintendent
Victor M. Javeña EdD
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OIC-Chief, Curriculum Implementation Division

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Librada L. Agon EdD (EPP/TLE/TVL/TVE)

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Joselito E. Calios (English/SPFL/GAS)
Norlyn D. Conde EdD (MAPEH/SPA/SPS/HOPE/A&D/Sports)
Wilma Q. Del Rosario (LRMS/ADM)
Ma. Teresita E. Herrera EdD (Filipino/GAS/Piling Larang)
Perlita M. Ignacio PhD (EsP)
Dulce O. Santos PhD (Kindergarten/MTB-MLE)
Teresita P. Tagulao EdD (Mathematics/ABM)

Printed in the Philippines by Department of Education – Schools Division of

Pasig City
Advanced Algebra
and Trigonometry 9
Quarter 1
Self – Learning Module 1
Imaginary Numbers
Introductory Message
For the Facilitator:

Welcome to the Enriched Mathematics Grade 9 Self-Learning Module on

Imaginary Numbers!

This Self-Learning Module was collaboratively designed, developed and

reviewed by educators from the Schools Division Office of Pasig City headed by its
Officer-in-Charge Schools Division Superintendent, Ma. Evalou Concepcion A.
Agustin, in partnership with the City Government of Pasig through its mayor,
Honorable Victor Ma. Regis N. Sotto. The writers utilized the standards set by the K
to 12 Curriculum using the Most Essential Learning Competencies (MELC) in
developing this instructional resource.

This learning material hopes to engage the learners in guided and

independent learning activities at their own pace and time. Further, this also aims
to help learners acquire the needed 21st century skills especially the 5 Cs, namely:
Communication, Collaboration, Creativity, Critical Thinking, and Character while
taking into consideration their needs and circumstances.

In addition to the material in the main text, you will also see this box in the
body of the module:

Notes to the Teacher

This contains helpful tips or strategies
that will help you in guiding the learners.

As a facilitator you are expected to orient the learners on how to use this
module. You also need to keep track of the learners' progress while allowing them
to manage their own learning. Moreover, you are expected to encourage and assist
the learners as they do the tasks included in the module.
 For the Learner:

Welcome to the Enriched Mathematics Self-Learning Module on Imaginary

Numbers!

This module was designed to provide you with fun and meaningful
opportunities for guided and independent learning at your own pace and time. You
will be enabled to process the contents of the learning material while being an
active learner.

This module has the following parts and corresponding icons:

Expectations - This points to the set of knowledge and skills

that you will learn after completing the module.

Pretest - This measures your prior knowledge about the lesson

at hand.

Recap - This part of the module provides a review of concepts

and skills that you already know about a previous lesson.

Lesson - This section discusses the topic in the module. 

Activities - This is a set of activities that you need to perform.

Wrap-Up - This section summarizes the concepts and

application of the lesson.

Valuing - This part integrates a desirable moral value in the

lesson.

Posttest - This measures how much you have learned from the
entire module.
 EXPECTATION

1. Define and describe imaginary numbers.

2. Derive pattern in finding the value of the powers of i.
3. Find the value using the powers of i.
4. Show accuracy and neatness in doing one’s work.

PRETEST

Multiple Choice.
1. Which classification describes the number - 25 ? √
A. Integer C. Irrational
B. Rational D. Imaginary

2. The number √−25 belongs to which of these sets?

A. Integer C. Irrational
B. Rational D. Imaginary

3. Which of these sets of numbers contains not real number?

1 7
4
A. 12, 2 , - 13 C. - 6, - √ 225 , 8

B. - 3.5041, √ 99 , 0.1436 D. √−81 , 0.75, 0

4. What is the value i574?

A. -1 C. i
B. 1 D. -i

5. Evaluate 9i44 + 4i24

A. -5 C. 5
B. -13 D. 13
 RECAP

Put check to which the following numbers belong.

Natural Whole Integers Rational Irrational Real
1. 0
2. -25
3. 175
4. ¾

5. 3 √5

LESSON

Now consider the equation x2 + 1 = 0. Since the square of any real number
cannot be negative, this equation has no solution in the set of real numbers.
So the set of real numbers is once more extended so that x 2 + 1 = 0, will
have a solution.
x2 + 1 = 0 then x2 = -1
x = ± √ −1 .
We shall call these new numbers imaginary numbers.
Let -1 be i2,
since -1 = i2,

hence √−1 = √ i2 = ±i

Examples: Simplify:

1.) √−4 = √−1(4)

2i
=

2.) √−8 = √−1(4)(2)

= √ i2 (4)(2)
= 2i √ 2
 3.) 2 √−8 = 2 √−1( 4)(2)
= 2 √i2( 4)(2)
= 2(2i) √2
= 4i √2

4.) 4 √−72 = 4 √−1(36)(2)

= 4 √i2 (36)(2)
= 2(6i) √2
= 12i √2

5.) 15 - 5 √−100 = 15 - 5 √−1(100)

= 15 - 5
√ i2 (100)
= 15 – 5(10i)
= 15 – 50i

Consider the following powers of i.

i1 = √ −1 = i √2
i2 = -1
i3 = i2 i = -1(i) = -i
i4 = i2(i2) = 1
i5 = i4(i) = i
i6 = i4(i2) = -1
i7 = i4(i3) = -i
i8 = i4(i4) = 1
The powers of i will cycle through 1, i , −1, - i , this repeating pattern
of four terms can be used to simplify in. Because the powers of i cycle
through 1, i , −1, - i these types of problems can always be simplified when
dividing the power by 4 ,the remainder will always be either 0, 1, 2, or 3.

Examples:
Evaluate the following using the powers of i.
1. i15 Since 15 ¿ 4 has a remainder of 3
Therefore, i15= -i
 So, i15 = -i

2. -3i105 Since 105 ¿ 4 = 26 has a remainder of 1

Therefore, i105 = i
So, -3 i105= -3i

3. i 37
- i43 + i25 Since i 37 = 1,
i43 = -i ,
i25 = i
So, i 37 - i43 + i25 = 1 –(- i ) + i
=1 + i + i
=1+2i

4. 6i77 + 5i28 Since, i77 = i

i28 = 1
So, 6i77 + 5i28
= 6(i) +5(1)
=6i+5

5. 7i25 + 4i(i43) – 2(-5i39) Since, i25 = i

i43 = - i
i39 = - i
So, 7i25 + 4i(i43) – 2(-5i39)
= 7(i) + 4 i(-i) – 2(-i)
= 7 i - 4 i2 + 2 i
But i2 = -1
=7i+4+2i
= 9 i +4

ACTIVITIES

ACTIVITY #1: Let’s Practice

Simplify:

1.
√−36 6. i56

2.
√−32 7. i33

3.
√−20 8. i42
 4
4. √ −
9 9. -i23
5. 3 √−125 10. 5i88

ACTIVITY #2: Keep Practicing

Evaluate each of the following:
1. √−9⋅√−9⋅√−9
2. 5 √−16+3 √−9−25 √−100
3. 5 √−25⋅2 √−36 +4 √−4
4. 6 √ −11−7 √−11+20 √−11
5. 10 √−32+3 √−50−2 √−18
6. i + i26 – 5i35
24

7. 4i24 + 5i38 – 10i47

8. 6i24 + 34i(i26) – 5i35
9. 7i47 + i128 – 8i35
10. -4i(i14) + 5i(i76) – 12i(i65)

ACTIVITY #3: Test Yourself

1. i345
2. i3264
3. i74 + 4 i2 - 10 i2
4. For ix where x is an even number greater than 0, develop a rule to
determine if ix simplified is 1 or -1.
5. Find the error in Aaron’s work
23 - √ 36
= 23 + √ 36
= 23 + 6
= 29

WRAP–UP
An imaginary number is a mathematical term for a number whose square is
a negative real number. Imaginary numbers are represented with the letter i
which stands for the square root of -1.
Let -1 be i2,
since -1 = i2,

hence √ −1 = √2i = ±i
With imaginary numbers, when you square them, the answer is negative.
They are written like a real number, but with the letter i after them.
Powers of i.
The imaginary unit i is defined as the square root of -1.
So i1 = i ,
i2 = -1,
i3 = - i,
i4 = 1.
Therefore, the cycle repeats every four powers. Because the powers of i
cycle through 1, i , −1, - i these types of problems can always be simplified
when dividing the power by 4 ,the remainder will always be either 0, 1, 2, or
3.

VALUING

Imaginary numbers were introduced to solve equations that involve

square root of negative numbers. Same way with our situation today To continue
education in this pandemic time. Distance learning was introduced so that
education will not stopped. Can you cite any other situation wherein somebody
introduce something to solve a particular problem?

POSTTEST

Multiple Choice.
1. What is (4i)2?
A. -8 C. 8
B. -16 D. 16
2. The number √−49 belongs to which of these sets?
A. Rational C. Real
 B. Irrational D. Imaginary
3. What is the value i393?
A. -1 C. i
B. 1 D. -i
4. Simplify 8 √−7+6 √−7
A. 14i √7 C. 14 √ 7i
B. - 14i √ 7 D. 7i √ 14
5. Evaluate 10i27 - 9i31
A. -10i + 9 C. -i
B. -10i - 9 D. i

KEY TO CORRECTION

Pre Test
1. B 2. D 3. D 4. A 5. D

RECAP
Natural Whole Integers Rational Irrational Real
1. 0 / / / /
2. -25 / / /
3. 175 / / / / /
4. ¾ / /
5. / /
3 √5

PRACTICE
2
i
1. 6I 2. 4I √2 3. 2I √5 4. 3 5. 15I √3
6. 1 7. I 8. -1 9. –I 10. 5

KEEP PRACTING
1. -27i 2. -221i 3. -300 + 8i 4. 19i √ 11 5. 49i
√2
TEST YOURSELF
1. i 2. 1 3. 5
x x
4. If 2 is an even number, then ix = 1. If is an odd number,2
then i = -1.
x

5. A negative inside a radical cannot be affected by the sign outside

 REFERENCES

Chua, Simon, et al. “21st Century Mathematics.” Phoenix Publishing House,

Inc., 1997

Coronel, Illuminada, et al. “Mathematics An Integrated Approach.”

Bookmark, Inc., 1992

Milo, Gracia, et al. “Integrated Secondary Mathematics IV.” JMC Press, Inc.,
1993