emagazine

May 2018 issue 

e-Math 

EXPERIENCE BETTER LEARNING 

Easy Steps in Solving Abstract Problems 

Facts and Trivias 

Basic Concepts 

Math Comics

M 

e - ath 

agazine

EDITIon 1 

FIRST ISSUE 

All Rights Reserved.

Research Ad 

About the 

The “e-Math Magazine 

striving goals in e 

stract Algebra. ThT 

way of succeeding 

academic compe 

improvement a 

Mr. Ronel

Cover 

” represents excellence in 

ducation especially in Abhe 

e manner of thinking is a 

beyond phenomenon. An 

tence highers educational 

nd empowers the mind of 

every student. 

Editorial Board: 

Alday 

viser 

Mrs. Teodora A. Dator 

Abstract Algebra Adviser 

Garry D. Babao 

Researcher

Pre 

This is the firs 

zine for Filipino c 

lowing highlights 

first term : Basic 

the Groups. 

Each topic in tht 

development of c 

students to build 

in understanding 

gebra. This empo 

topics which som 

difficult ideas.

face 

t issue of “e-Math” in electronic magaollege 

students. The series have the fol- 

of the topics in Abstract Algebra for the 

Concepts and the Binary Operations with 

he “e-Math” provides knowledge in the 

oncepts in e-math magazine. It helps 

and improve higher order thinking skills 

and evaluating concepts in Abstract Al- 

wers the clarity and accurateness of the 

etimes confuse students in evaluating

Contents 

1 Basic Concepts 

3 Real Numbers 

4 

Properties of Integers 

13 

MODULAR 

ARITHMETIC 

20 

PERMUTATION GROUP 

25 

CONGRUENCE 

CLASSES 

27 

TABLE OF FOUR GROUP 

28 

SPECIAL TABLE FOR 

COMMUTATIVE FOR PERMUTATIONS

Properties of Matrices 

29 

Cancellation Law 

39 

Mathematical Induction 

40 

Chapter Test 

47 

Study Tips 51 

Rebus Puzzle 

Comic Strip 

Divisibility Rules 

53 

59 

62 

Literary Pieces

Abstract Algebra 

Abstract Algebra is a 

challeng- ing course 

in college. This e- 

magazine tackles the 

basic concepts of Abstract 

Algebra with the 

following topics: the real 

numbers, properties 

of real numbers, permutations, 

congruence 

classes, matrices, special 

table for commutative 

of permutation, inverse 

of a matrices, and 

mathematical induction. 

It has also few mathematics 

poems, facts 

and trivia, comic strips, 

and jokes. 

~GDB

P R A Y E R 

M 

A 

Lord, teach me to number my days 

And graph then according to your ways 

Trusting you to base me in my plan 

To complement your perfect diagram. 

T 

H 

E 

M 

A 

T 

I 

C 

S 

Subtract the points you do not want from me, 

But add the values you have set for me 

Divide the dividends I possess accordingly 

So I can multiply them systematically. 

Draw the lines I have to follow 

Guide me properly with your arrow 

Because I tend to be irrational 

Yet all the while you want me to be rational. 

Well I learn that life is slope 

With its ascends and descends that I must cope 

Going through such a wonderful formula 

Is just like solving problem in Algebra. 

Life is indeed an infinite equation 

Perfected by your eternal computation 

And only by a minuscule yet projection 

Give thanks and praise you Almighty creation. Amen. 

~SLSU—CTE (MMS)

T 

1 

e-Math M 

This section ha 

will be tested b 

properties of R 

such as CLOSURE 

IDENTITY, INVER 

TIVE and DISTRI 

numbers as wel 

numbers and ot 

permutations. It 

the test of thes 

other mathema 

like MODULAR 

CONGRUENCE 

other sets in abst

agazine 2 

s examples and 

the following 

EAL NUMBERS 

, ASSOCIATIVE, 

SE, COMMUTA- 

BUTIVE on real 

l as imaginary 

her forms like 

also extends 

properties to 

tical structures 

ARITHMETIC, 

CLASSES, and 

ract forms. 

BASIC CONCEPTS

REAL NUM 

3 

e-Math M 

Lesson 

Objectives: 

At the end of this subject matter, 

students are able to: 

1. Determine the properties of real numbers 

using examples. 

2. Evaluate and solve the following sets 

under four operations. 

3. Show enthusiasm in the discussion and 

apply into real world applications. 

R eal 

Numbers involving rational 

value that represents a quanti 

There are six properties of real nu 

sist of closure, associative, identit 

distributive. It also have their exam

agazine 4 

REAL NUMBERS 

Real application of Real Numbers are everywhere. It can be found 

at the store, can be press on your calculator, counting money value, 

slicing a pie, thermometer label, and even in gambling. 

ARE EVERYWHERE! 

and irrational numbers. It is the 

ty along a line. 

mbers in this section which cony, 

inverse, commutative, and 

ples which may help you grow! 

BERS AND ITS PROPERTIES!

Properties of 

5 

e-Math M 

Real Numbers 

Closure Property 

R is closed under addition 

and multiplication, i.e. ∀ a, b ∈ R 

⇒ a + b ∈ R. 

1 

Associative Property 

Addition is associated under R, 

i.e. (a + b) + c = a + (b +c). 

Multiplication is associated under 

R, i.e. (a b) c = a (b c). 

2 

3 6 

Identity Property 

The identity element in R is 0 under addition 

i.e. ∀ a ∈ R ⇒ a + 0 = 0 +a = a. 

The identity element in R under multiplication 

is 1, i.e. ∀ a ∈ R ⇒ a · 1 = 1 · a = a. 

REMEMBER: The identity element must commute 

with every element of R under the operation.

agazine 6 

Inverse Property 

5 

4 

The inverse of an element 

which is also in R if it exists. 

Additive inverse of a is (-a) : a + 

(-a) = (-a) + a = 0, the identity in 

addition. 

Multiplicative inverse of a is 

(a -1 ) : a · a -1 = a -1 · a = 1, the identity 

in multiplication. 

REMEMBER: The inverse of an element 

must always commute 

with the element. 

Commutative 

Property 

Addition is commutative 

in R, i.e. a + b = b + a. 

Multiplication is commutative 

in R, i.e. b · a = a · b 

Distributive Property 

The multiplication is leftdistributive 

and right-distributive 

over addition under the set of real 

numbers, i.e. 

a (b + c) = ab + ac, left– distributive 

of multiplication over addition, 

(c +b) a = ca + ba, right-distributive 

of multiplication over addition. 

Take 

Note! 

R = Set of Real 

Numbers 

∈ = Element of 

∀ = For All 

⇒ = Implies 

a -1 = Multiplicative 

Inverse of 

-a = Additive Inverse 

of 

S = Set 

Z = Set of Integer 

Q = Set of Rational 

Numbers 

N = Set of Natural 

Numbers 

= Group 

| = Such that

REAL LIFE APPLICATION 

7 

e-Math M 

Tying of shoe lace is an example of COMMUTATIVE PROPERTY. 


Leaf + leaf = Leaves. (a + b = 

c such that a, b, and c ∈ leaves. 


(denoted by parenthesis) 

(Rose + Daisy) + Gumamela = 

Rose + (Daisy + Gumamela) = 

Flowers. 


Look at the mirror and you’ll 

see yourself. 


The inverse of male is female 

and black is white . 

Commutative Property 

If we have two things, 

put equal sign at the middle 

and just flip the place of 

both. 


Money is the one we often distribute. 

Example: 

Determine if the set S = {-1, 0, 1} is closed under 

a) addition, b) subtraction, c) multiplication, and 

d) division. 

R 

EMEMBER: To disprove a 

statement or to sure that a 

property is not satisfied, one 

counter example is enough. 

The answer must be in the set, if not, it’s 

not closed under the operation. 

Solutions: 

The set S is not closed under addition since -1 + (-1) 

= -2 where -2 ∉ S. 

-1 + 1 = 0 where 0 ∈ S (satisfied). 

-1 + 0 = -1 where -1 ∈ S (satisfied). 

-1 + (-1) = -2 where -2 ∉ S (not satisfied). 

The set S is not closed under subtraction since -1 

– (+1) = -2 where -2 ∉ S. 

1 - 1 = 0 where 0 ∈ S (satisfied). 

1 - 0 = 1 where 1 ∈ S (satisfied). 

1 - (-1) = 2 where 2 ∉ S (not satisfied). 

The set S is closed under multiplication since for 

every two element of the set S, the product is also 

in S.

agazine 8 

1 · 1 = 1 where 1 ∈ S (satisfied), 

-1 · -1 = 1 where 1 ∈ S (satisfied). 

0 · -1 = 0 where 0 ∈ S (satisfied). 

0 · 0 = 0 where 0 ∈ S (satisfied). 

-1 · 1 = -1 where -1 ∈ S (satisfied). 

1 · -1 = -1 where -1 ∈ S (satisfied). 

The set S is not closed under division since a 

number divided by zero is undefined which is 

not in the set S. 

1 ÷ 1 = 1 where 1 ∈ S. 

1 ÷ 0 = undefined where undefined ∈ S. 

-1 ÷ 1 = -1 where -1 ∈ S. 

0 ÷ 1 = 0 where 0 ∈ S. 

Example: 

a) Is 2Z closed under subtraction? Explain. 

b) Is 1 + 2Z closed under subtraction? Explain? 

Solutions: 

a) The set 2Z is closed under subtraction, 

since if 2a and 2b ∈ 2Z such that a and b are 

integers, then 2a – 2b = 2 (a-b) is in 2Z. 

If 2Z = {…, -4, -2, 0, 2, 4, …) 

Then, substitute the values in 2Z under 

subtraction, therefore, 

2 (-1) – 2 (0) = -2 where -2 ∈ 2Z 

2 (0) – 2 (3) = -6 where -6 ∈ 2Z 

2 (1) – 2 (-2) = 6 where 6 ∈ 2Z 

b) The set of integers of the form 1 + 2Z is 

not closed under subtraction, since the 

numbers 1 and 3 are in 1 + 2Z, but the 

difference 3 – 1 = 2 where 2 ∉ 1+2Z. 

If 1 + 2Z = {…, -3, -1, 1, 3, …) 

[1 + 2 (-2)] – [1 + 2 (1)] = (-3) – 3 = 0 

where 0 ∉ 1 + 2Z. 

[1 + 2 (-1)] – [1 + 2 (0)] = (-1) – 1 = -2 

where -2 ∉ 1 + 2Z. 

Example : 

Show that = {a + b √2| a, b ∈ Q} is 

closed under multiplication. 

Solution: 

Let and ∈ , where α = {a1 + 

b1√2} and β = {a2 + b2√2}. Then, 

α · β = (a1 + b1√2) · (a2 + b2√2) 

= a1 a2 + 2b1b2 + (a1b2 + a2b1) √2 ∈ R, 

since a1 a2 + 2b1b2 ∈ Q and (a1b2 + a2b1) 

∈ Q. 

Mirror is a representation of Identity Property because equal sign represents a mirror.

COMPLEX NUMBERS 

Do you want to meet them? You’ll going to meet 

them after your lessons… must keep going…. 

9 

e-Math M 

A complex number denoted 

by a + b is the extension of 

the real numbers through the addition 

of the term b to the real 

number a. 

Lesson 

After the lesson, 

the students are able 

to: 

1. Convert complex number 

to Polar form to 

Euler form. 

2. Plot points of complex 

numbers using 

rectangular planes. 

3. Apply the complex 

number in a real 

world scenario.

agazine 10 

The complex number a + b can be expressed 

by the point (x, y) and which in turn 

can be plotted in the coordinate plane as 

shown in the Figure 1. The plotting of points 

such as P1 (-2, 1) and P2 (1, -1) are also illustrated 

in the plane. 

Example: Group of Fourth Roots of Unity 

Find the elements of the fourth roots of unity 

and test the properties of real numbers under the 

set. 

Solution: 

We start with the element =√-1. Then, 

2= · = -1 

3= 2 · = - 

4= 2 · 2 = 1 

Thus, we have a group of fourth roots of unity S = {1, 

-1, , - }. Table 1 

POLAR AND EULER FORMS OF 

COMPLEX NUMBER 

The complex number x + y can also be 

expressed in two different forms as follows: x 

+ y = r (cos + sin ) = re Ө 

we call, x + y as the rectangular form, r (cos 

+ sin ) as polar form, where Ө = arctan y/x 

and r = √(x 2 + y 2 ), re Ө as the Euler form. 

Take note! 

= Imaginary numbers 

cos = Cosine Theta 

sin = Sine Theta 

In looking the table , when the operation 

is: (dot/multiplication) = left 

side before the upper part 

• 1 -1 - 

1 1 -1 - 

-1 -1 1 - 

- -1 1 

- - 1 -1 

Table 1. S = {1, -1, , - } 

Check for Properties of Real Number 


The set S is closed based on the table shown 


The operation of multiplication is associative 

under the set of complex number i.e. 

(1 · ) -1 = 1 ( · -1) 

( ) -1 = 1 (- ) 

- = - (satisfied) 


The multiplicative identity is one (1) under the 

set of complex numbers. Take note that in the table, 

you’ll see the identity number which is one (1) 

because the fourth roots and the answer under 

multiplication are identical.

11 

e-Math M 


To get the inverse number, the answer 

should be the identity number. 

1 -1 = 1 where 1 · 1 = 1 

(-1) -1 = -1 where -1 · -1 = 1 

-1= - where · - = 1 

(- ) -1 = where – · = 1 

Commutative Property 

The set S is closed under commutative 

property as shown in the table. 


The set S is closed under distributive 

property. 

Example: 

a) Find the POLAR form of 1 – . 

b) Find the POLAR form of -1 + . 

c) Find 2e - 

d) Plot the points in a), b) and c). 

Solutions: 

a) We express 1 – in POLAR form by 

computing r and Ө as follows: 

r = (√x 2 + y 2 ) = [√1 2 + (-1) 2 ] 

= (√1 +1) = (√2) 

= arctan (-1÷1) = arctan (-1) 

= 315° or 7 /4 

Thus, the 

polar form of 1 

– = √2 (cos 

7 /4 + sin 

7 /4) 

To have the arctan in the calculator, press 

shift tan and input the number, then add or 

minus it to 180°or 360° depending to the 

quadrant it is placed. Then if want to get the 

radian value, multiply the answer to /180°. 

b) We express -1 + in POLAR form by computing 

r and as follows: 

r = [√ (-1) 2 + 1 2 ] = √2 

= arctan (1÷ -1) = arctan (-1) 

= 315° or 7 /4 

c) We note that x + y = r (cos + sin ) = 

re . For 2e - , the = - . Thus, we have 

r (cos + sin ) = re as 2e - = 2 [cos (- 

) + sin (- )] = 2 (- + 0 ) = -2. 

d) With P (x, y) = x + y, then we plot the following 

points on the rectangular plane. 

P1 = 1 - = (1, -1) 

P2 = -1 + = (-1, 1) 

P3 = -2 = (-2, 0) 

P3 (-2, 0) 

-2 

P2 (-1, 1) 

-1 

2 

1 

1 

-1 

-2 

2 

P1 (1, -1) 

Graph 1. P (x, y) = x + y

agazine 12 

Let’s Apply! 

Example: Group of Third Roots of Unity 

a) Find n such that w n = 1, and the elements 

of the set S = {w, w 2 , w 3 , … , w n =1}. 

b) If Ө = arctan (y/x) and r = √(x 2 + y 2 ), 

find r and Ө for each power of w and express 

these in the form x + y = r (cos Ө + 

sin Ө) = re Ө . 

c) Form a multiplication table for the elements 

of S. 

Solutions: 

a) w 2 = (-1/2 + √3/2) 2 = (1/4 - 3/4 - 

√3/2) = -1/2 - √3 /2 

w 3 = w 2 (w) = (-1/2 - √3/2)((-1/2 + 

√3/2)= (1/4 + 3/4) = 1 

Thus, the value of n = 3 such that w 3 

= 1. Thus, we have the set S = {w, w 2 , 1}. 

b) For w : r = √(x 2 + y 2 ) = √((-1/2) 2 + 

(√3/2) 2 ) = 1 

Ө = arctan y/x = arctan (√3/2)/(- 

1/2) = arctan (-√3). 

Thus, w 2 = -1/2 + √3/2 = 1 (cos 2 /3 + 

sin 2 /3) = e (2 /3) 

For w 2 : Ө = arctan y/x = arctan (-√3/2)/ 

(-1/2) = arctan (√3) = 4 /3. 

Thus, w 2 = -1/2 - √3/2 = cos 4 /3 + sin 

4 /3 = e (4 /3) 

For w 3 : Ө = arctan 0/1 = 0 ° . 

c) We have Table 1 for the multiplication 

of elements in S. Notice that w 4 = w 3 • w 

= 1 w = w 

Another example is electromagnetism. Rather 

than trying to describe an electromagnetic 

field by two real quantities (electric field 

strength and magnetic field strength), it is 

best described as a single complex number, of 

which the electric and magnetic components 

are simply the real and imaginary parts. 

• 1 w w 2 

1 1 w w 2 

W W w 2 1 

w 2 w 2 1 w 

Table 2. S = {1, w, w 2 } 

The table indicates that S is closed under 

the operation of multiplication, i.e. ∀ a, b ∈ S implies 

a b ∈ S.

Multiplication 

13 

e-Math M 

Z 2 modulo 2 

Inverse element 

Modu 

Arith 

Typically, modular arithmetic can be seen in 

tion of modular arithmetic. Another exa

agazine 14 

Table of elements 

Addition 

Identity element 

lar 

metic 

the clock. That is the best real life applicample 

is the basis for bar codes in stores.

We form the table for elements of 

the set S with arbitrary elements a 

and b under multiplication modulo 10 

by performing a · b where 

a · b = c, where 0 ≤ c < 10 

8 · 3 =4 mod 10; 

7 · 7 = 9 mod 10; 

5 · 6 = 0 mod 10; 

12 · 2 = 4 mod 10 

Illustrations: 

3 · 4 = 2 mod 10; and 

9 · 3 = 7 mod 10. 

A clock is 

an example of 

modular arithmetic, 

we use 

clock to measure 

time. Our clock 

system uses modulo 

12 arithmetic . 

However, instead 

of a zero we use 

the number 12. 

Example: 

Construct for the table for elements 

of S1, S2, and S3 modulo 10. 

S1 = {2, 4, 6, 8} under multiplication. 


15 

S3 = {1, 3, 5, 7, 9} under multiplication. 

Determine if each set is closed under 

the indicated operation. 

Find the identity for each set. 

Find the inverse of every element 

of each set. 

Solutions: 

S1 = {2, 4, 6, 8} modulo 10 under 

multiplication. Table 3. 


Table 4. 

S3 = {1, 3, 5, 7, 9} under multiplication. 

Table 5. 

· 2 4 6 8 

2 4 8 2 6 

4 8 6 4 2 

6 2 4 6 8 

8 6 2 8 4 

e-Math M 

Table 3. S1 = {2, 4, 6, 8} mod 10

agazine 16 

· 1 3 7 9 

1 1 3 7 9 

3 3 9 1 7 

7 7 1 9 3 

9 9 7 3 1 

Table 4. S2 = {1, 3, 7, 9} mod 10 

· 1 3 5 7 9 

1 1 3 5 7 9 

3 3 9 5 1 7 

5 5 5 5 5 5 

7 7 1 5 9 3 

9 9 7 5 3 1 

Table 5. S3 = {1, 3, 5, 7, 9} mod 10 

The sets S1, S2, and S3 are closed under 

multiplication based on the table. 

For the identity of every element of 

the sets S1, S2, and S3 are the following: 

The identity element of S1 is 6, i.e. a · 6 

= 6 · a = a, ∀ a ∈ S1. 

The identity element of S2 is 1, i.e. a · 1 

= 1 · a = a, ∀ a ∈ S2. 

The identity element of S31 is 1, i.e. a · 

1 = 1 · a = a, ∀ a ∈ S3. 

In S1: Note that the inverse in 

multiplication is denoted by a 

-1. Read first from the left side 

then the upper part of the table. 

The inverse of 2 is 8, since 2 · 8 

= 6 (the identity element). 







In S2: 









In S3: 

Five (5) has no multiplicative inverse, 

since we cannot find an 

element x ∈ S3 such that 5 · x 

= x · 5 = 1. In S3, the inverses 

of the elements 1, 3, 7 and 9 

are 1, 7, 3, and 9, respectively.

Addition and Multiplication 

in Z4 (mod 4) 

The table formed under addition 

and multiplication of the 

elements of S4 = {0, 1, 2, 3} are 

indicated as follows. Table 6 for 

addition and Table 7 for multiplication. 

+ 0 1 2 3 

0 0 1 2 3 

1 1 2 3 0 

2 2 3 0 1 

3 3 0 1 2 

Table 6. S4 = {0, 1, 2, 3} mod 4 

under addition 

· 0 1 2 3 

0 0 0 0 0 

1 0 1 2 3 

2 0 2 0 2 

3 0 3 2 1 

Table 7. S4 = {0, 1, 2, 3} 

mod 4 under multiplication 

Example: 

Test the properties of real numbers 

under Z4. 

Solutions: 

17 

1) Closure Property 

The set is closed under addition 

and multiplication based on the 

table, i.e. 

∀ a, b ∈ Z4, then a + b ∈ Z4 in Table 

5. 

∀ a, b ∈ Z4, then a · b ∈ Z4 in Table 

6. 

We note that when we say 

“based on the table” for the closure 

property it means that performing 

any two elements on the 

set, the result is on the set. 

2) Associative Property 

Addition is associative in Z4. 

Multiplication is associative in 

Z4. 

This is because the elements in 

Z4 are real numbers in addition 

(or in multiplication) is associative 

under R. 

3) Identity Property 

The identity element of Z4 in addition 

is 0. 

The identity element of Z4 in 

multiplication is 1. 

e-Math M

agazine 18 

4) Inverse Property 

Negatives (Additive Inverse) denoted 

as –a. 

-0 = 0, since 0 + 0 = 0; 

-1 = 3, since 1 + 3 = 0; 

-2 = 2, since 2 + 2 = 0; and 

-3 = 1, since 3 + 1 = 0. 

Reciprocal (Multiplicative Inverse) 

denoted as a -1 . 

1 -1 = 1, since 1 · 1 = 1 

3 -1 = 3, since 3 · 3 = 1 

The elements 0 and 2 have no multiplicative 

inverses, i.e. there exists 

no element x ∈ Z4, such that 

0 · x =1 

Solutions: 

The tables for S with respect to 

the operations of addition Table 

8 and multiplication Table 9. 

Based on the tables, S is closed 

with respect to addition and 

multiplication. 

/the tables are in the next page.. 

That is why… 

2 · x =1 

Example: 

If S = {(0, 0), (1, 0), (0, 1), (1, 1)}, 

where addition ⊕ (circled plus/ 

scalar addition) and multiplication 

⊙ (circled dot/scalar multiplication) 

is defined under modulo 2 as 

(a, b) ⊕ (c, d) = (a +c, b + d), and 

(a, b) ⊙ (c, d) = (ac, bd), then 

form the tables for Z2 ⊕ Z2 and Z2 

⊙ Z2, and discuss the closure 

property.

Supermarkets and retail 

stores have a nasty little secret. 

Every time you scan your purchases, 

they’re using modular 

arithmetic on you! 

19 

e-Math M 

P 

⊕ (0, 0) (1, 0) (0, 1) (1, 1) 

(0, 0) (0, 0) (1, 0) (0, 1) (1, 1) 

(1, 0) (1, 0) (0, 0) (1, 1) (0, 1) 

(0, 1) (0, 1) (1, 1) (0, 0) (1, 0) 

(1, 1) (1, 1) (0, 1) (1, 0) (0, 0) 

Table 8. Z2 under scalar addition 

⊙ (0, 0) (1, 0) (0, 1) (1, 1) 

(0, 0) (0, 0) (0, 0) (0, 0) (0, 0) 

(1, 0) (0, 0) (1, 0) (0, 0) (1, 0) 

(0, 1) (0, 0) (0, 0) (0, 1) (0, 1) 

(1, 1) (0, 0) (1, 0) (0, 1) (1, 1) 

Table 9. Z2 under scalar multiplication 

A permutation in Sn 

= {1, 2, 3, …, n} is a one-toone 

mapping of arrangements 

of integers to integers. This 

topic will be discussed extensively 

in the latter chapters. 

Illustrations: In S3 = {1, 2, 

3}, we have the following arrangement 

of integers for the 

mapping of integers 1, 2, and 

3 into integers 1, 2, and 3. 

There are six arrangements of 

1, 2, and 3, and each arrangement 

may be associated to a 

permutation which in turn 

may be denoted by a Greek 

symbol. The six arrangements 

of 1, 2, and 3 are 123, 132, 

213, 231, 312, and 321. Thus, 

the six permutations named 

in 6 (six) Greek symbols are 

associated to the 6 arrangements 

as follows:

agazine 20 

ermutations 

(Alpha) = 1 2 3 

2 1 3 

(Gamma) = 1 2 3 

3 1 2 

(Theta) = 1 2 3 

1 3 2 

(Beta) = 1 2 3 

2 3 1 

(Epsilon) = 1 2 3 

1 2 3 

(Tau) = 1 2 3 

3 2 1 

The multiplication of permutations 

and denoted by ⊙ 

or simply means we perform 

first followed by . Thus, 

in , we have: 

In , 1 goes to 2, but in , 2 goes 

to 1, so in , 1 goes to 1. 





Thus, the product is the permutation 

1 2 3 , i.e. 

1 3 2 

Did you know that...

21 

e-Math M 


1 2 3 ; this in our 

1 3 2 

natation is , i.e. 

To obtain the product 

In 

In 

In 

, we have: 

, 1 goes to 2, and 2 goes to 1, so 1 goes to 

1 


3 


2. 

Thus, 

Notice that the multiplication of permutation 

is not commutative. 

The multiplication of permutation and denoted 

by ⊙ or simply means we perform 

first followed by . Thus, in we have: 

The products of other elements of S 3 are illustrated 

as follows: 

In , 1 goes to 2, but in , 2 goes to 3, so in 

to 3 


to 2 


to 1. 


1 goes 

2 goes 

3 goes 

; this in our notation is , i.e.

agazine 22 

We form the table for S 3 as follows. 

⊙ 

Ө 

Ө 

Ө 

Ө 

Ө 

Ө 

Ө 

Ө 

Table 10: S 3 = { , Ө, , , , } under multiplication 

Closure 

S 3 is closed based on the table. This mean that 

when we multiply any two elements on S 3 , 

the product is on S 3. 

Associative 

Multiplication of permutation is associative. 

This is exhibited in the following examples. 

( ) = ( ) 

• = Ө • 

= 

Ө ( ) = (Ө ) 

Ө • = • 

= 

Identity 

The identity in S 3 is 1. 

Inverses 

-1 = 

-1 = 

-1 = 

-1 = 

Ө -1 = Ө 

-1 = 

Did you know that… 

The Fibonacci Sequence are 

numbers where each following 

number is the sum of the previous 

two: 

0 1 1 2 3 5 8 13 21 

34 55 89 … 

And Fibonacci Sequence 

can be found in sunflower.

23 

e-Math M 

Example: Given the permutation 

which can be written in the form 

(1 2 3 4 5 6). This is called the cycle notation of . 

Find the cycle form of the following. 

If you add up all the numbers 

from 1 to 100 consecutively 

(1 + 2 + 3 + … , + 100), 

it totals 5050. 

Solution: 

Rubiks Cube 

= (1 3) (2 4) (5 6) 

= (1 6 5 4 3 2) 

= (1 3 5) (2 4 6) 

= (1 5) (2 4) 

Illustration: 

Another application of permutation 

group is the Rubiks Cube (3x3). It 

has six different colors and each 

color is repeated exactly nine times, 

so the cube can be considered as an 

ordered list which has 54 elements 

with numbers 1 and 6, each number 

meaning a color being repeated 

9 times. 

Example: If 

and 

, find the product of the following 

permutations and express the product in cycle form. 

a) LR 2 

b) LR 3 

c) LR 4 

d) LR 5

agazine 24 

Solutions: 

Illustrations: 

Example: 

If = (1 2 3 4 5 6) and S = { , 2, … , n = 

(1)}, find the elements of S. form the multiplication 

table of S. 

Solution: 

To form the table, we compute a few products for set 

S: 

3 • 

3 = 

6 = (1) 

4 • 

3 = 

7 = 

6 • = (1) • = 

4 • 

4 = 

8 = 

6 • 2 = (1) • 2 = 2 

5 • 

5 = 

10 = 

6 • 4 = (1) • 4 = 4 

Then, we have the table for S as follows: 

• (1) 2 3 4 5 

(1) (1) 2 3 4 5 

2 3 4 5 (1) 

2 2 3 4 5 (1) 

3 3 4 5 (1) 2 

4 4 5 (1) 2 3 

5 5 (1) 2 3 4 

Table 11: S = {(1), , 2, 3, 4, 

5,} under multiplication

Piano Keys 

The naming of musical 

notes is modulo 7; if you 

start at the note “D” on a 

piano, and count up 7 

white notes, you’ll end up 

back on “D”, which is the 

same note an octave higher. 

This raises the interesting 

fact that there are 7 

different notes in an octave; 

it gets its name because 

if you count the 

notes at the start and end 

“D” in this case, there are 8 

notes). 

25 

Congruence Classes 

e-Math M 

In congruence modulo n, there are n distinct 

congruence classes. Let Z n denote the set 

of n congruence classes, then 

Z n = {[0], [1], [2], … , [n – 1]}. 

Then addition and multiplication are defined 

in Z n as follows: 

Theorem: Addition in Z n 

The addition of congruence classes is defined 

by [a] + [b] = [a + b]. 

Show the properties that hold in addition under 

Z n . 

Proof: 

It is clear that the rule [a] + [b] = [a + b] is 

in Z n, based on the fact that Z is closed 

under addition. 

The associative property follows from 

[a] + ([b] + [c]) = [a] + [b] + [c] 

= [a + (b + c)] 

= [(a + b) + c] 

= [a + b] + [c] 

= ([a] + [b]) + [c]. 

Notice that the key step here is the fact 

that addition is associative in Z; a + (b 

+ c) = (a + b) + c. 

[0] is the additive identity, since [a] + [0] = 

[a + 0] = [a] and [0] + [a] = [0 + a] = [a]. 

[-a] = [n – a] is the additive inverse of [a], 

since [a] + [-a] = [a + (-a)] = [0] and [-a] 

+ [a] = [(-a) + a] = [0]. 

The commutative property follows from [a] 

+ [b] = [a + b] = [b + a] = [b] + [a].

agazine 26 

Example: 

Discuss the commutative property. 

Find the identity of Z 4 and inverses od each 

element in Z 4 under addition and multiplication. 

Solution: 

We define and give notation for the elements 

of multiples of four as the congruence 

classes [0] 4, where [0] 4 = {4k | k ∈ Z} = {…, 

-4, 0, 4, 6, 8, …}. 

Note that any element can represent the 

congruence class. 

The other elements of Z 4 are as follows: 

[1] 4 = {4k + 1 | k ∈ Z} = {…, -3, 1, 5, 9, 13, 

…}. 

[2] 4 = {4k + 2 | k ∈ Z} = {…, -2, 2, 6, 10, 

14, …}. 

[3] 4 = {4k + 3 | k ∈ Z} = {…, -1, 3, 7, 11, 

15, …}. 

The four sets [0] 4 , [1] 4 , [2] 4 , [3] 4 are 

called four (4) congruence classes of Z 4. We 

note that any element of the setcan represent 

the corresponding congruence class, i.e. [0] 4 

= [4] 4 ; [1] 4 = [5] 4 , etc. 

Using the above definition, the addition 

and multiplication in Z n = {[0] 4 , [1] 4 , [2] 4 , 

[3] 4 } are defined as the following Table 12 

and Table 13. 

+ [0] [1] [2] [3] 

[0] [0] [1] [2] [3] 

[1] [1] [2] [3] [0] 

[2] [2] [3] [0] [1] 

[3] [3] [0] [1] [2] 

Table 12: S = {[0], [1], [2], [3]} 

under addition 

• [0] [1] [2] [3] 

[0] [0] [0] [0] [0] 

[1] [0] [1] [2] [3] 

[2] [0] [2] [0] [2] 

[3] [0] [3] [2] [1] 

Table 13: S = {[0], [1], [2], [3]} 

under multiplication

The addition is commutative. Multiplication 

is also commutative. 

The identity for addition is [0], and 

The additive inverse (that is, negatives) 

Additive inverse of [1] 4 = -[1] 4 = [3] 4 , since 

[1] 4 + [3] 4 = [0] 4 


[2] 4 + [2] 4 = [0] 4 


[3] 4 + [1] 4 = [0] 4 

The identity for multiplication is [1]. 

For the multiplicative inverses of the elements 

of Z 4 , we have the following: 

Multiplicative inverse of [1] 4 = [1] 4 -1 = [1] 4 , 

since [1] 4 • [1] 4 = [1] 

Multiplicative inverse of [3] 4 = [3] 4 -1 = [1] 4 , 

since [3] 4 • [3] 4 = [1] 

Note that: 

The equation [1] • [1] = [1] = [3] • [3] 

show that each of [1] and [3] own 

multiplicative inverse (that is, reciprocal), 

i.e. [1] -1 = [3] and [3] -1 = 

[1]. 

Notice that neither [0] 4 nor [4] 4 has a 

multiplicative inversein Z 4 , i.e. 

there exists no element x ∈ Z 4 . 

[0] • [a] ≠ [1] and [2] • [a] ≠ [1] for every 

[a] in Z4 

Neither [0] nor [2] has a multiplicative 

inverse in Z4 since [0] • [a] ≠ [1] and 

[2] • [a] ≠ [1] for every [a] in Z4. 

Distributive Property on Congruence 

Classes 

Example: 

27 

For n > 1, let Z n denote the congruence 

classes on the integers modulo n: 

Z n = {[0], [1], [2], … , [n – 1]}. 

Define binary operations for addition 

and multiplication in Z n as the rules 

[a] + [b] = [a + b] and [a] • [b] = [ab] 

[a] • ([b] + [c]) = [a (b + c)] 

= [ab + ac] 

= [ab] + [ac] 

= [a] • [b] + [a] • [c], 

So that the left distributive law holds in 

Z n . the right distributive law can be verified in 

a similar manner. 

The Table of Four Group 

Consider the set of four elements S = 

{e, a, b, ab} under with the operation of multiplication 

table defined in Table. This group 

is known as the four group. 

We see that based on the table, S is 

closed under multiplication. 

• e a b ab 

e e a b ab 

a a e ab b 

b b ab e a 

ab ab b a e 

Table 14: S = {[0], [1], [2], [3]} 

under multiplication 

e-Math M

Table 15: S = {[0], [1], [2], [3]} 

under multiplication 

agazine 28 

Special Table for Commutative of Permutations 

In general, multiplication of permutation is 

not commutative. However, we give a special table 

of permutations where multiplication is commutative. 

Let T = {T 1 , T 5 , T 7 , T 11 }, be a set of four permutations 

defined by 

The symbol infinity (Ꝏ) 

was used by the Romans 

to represent 1000. 

To form the multiplication table, we have the 

following products as illustrated: 

Thus, we have 

• T 1 T 5 T 7 T 11 

T 1 T 1 T 5 T 7 T 11 

22, 273 is the largest prime 

in the Bible and it’s aptly in 

Number 3:43 

T 5 T 5 T 1 T 11 T 7 

T 7 T 7 T 11 T 1 T 5 

T 11 T 11 T 7 T 5 T 1 

PREPARE FOR THE PROPER- 

TIES OF MATRICES //

29 

e-Math M 

Properties 

of 

Matrices 

Invertible Matrix 

Definition: 

A 2 × 2 matrix is called an IN- 

VERTIBLE MATRIX if its inverse 

denoted by A -1 exists. The definition of 

the invertible matrix applies only to a 

square matrix. 

If the inverse of a matrix A exists, 

then we can find the matrix B such that 

AB = BA = I 2 , the identity matrix. The 

matrix A and B are called invertible or 

nonsingular. If inverse does not exist 

The inverse of 

is given by the formula 

( ). 

) is comput- 

For example, the inverse of ( 

ed as follows: 

First, compute . 

, then the inverse of a is as fol- 

Since 

lows: 

To check, we need to product of A and A -1 as I 2 , 

i.e 

such as 

, then it is called singular. 

The inverse of 

is invertible 

if the determinant . 

The following matrices have no inverses 

based on the computation of 

Method 1 for Finding Inverse of Matrix 

To find the inverse of a 2x2 matrix 

we form 

and perform matrix operation to obtain . 

This means that 2x2 matrix B is the inverse of A. 

; ;

agazine 30 

For example, we want to find the inverse of 

. First, we form 

Solving the 4 sets of equations gives 

x = -3, y = -4, z = 1, and w = 1. 

Thus, we have the inverse 

. To verify this result, 

we can easily check 

Thus, the inverse of 

is 

Fun Time!... 

To check: 

. 

Method 2 for Finding Inverse of Matrix 

Consider 

of a matrix A called 

. We want to find the inverse 

such that 

This can be translated into a system of equations 

= 

= . 

By equating corresponding component, we have the following 

4 equations: 

and

1 

31 

e-Math M 

Method 3 Formula for Inverse of 2x2 Matrix 

The inverse of 2x2 matrix 

For example, the inverse of 

) is 

is 

2 

To check: 

Inver 

The following have no inverse: 

( ),since 

Example: Find the inverse of the matr 

Solution. We begin by adjoining 

and perform matrix operation un 

( ),since 

Use elemtary row operations to 

-R 1 + R 2 

-6R 1 + R 3 

R 3 + R 1 

R 3 + R 2 

So, the matrix A is i

agazine 32 

3 

To check, the result of A -1 , we multiply A and A -1 to obtain , 

I 3 i.e. 

) 

Notations and Restriction of 2x2 Matrix 

A 2x2 matrix denoted by 

is defined as 

Also we define 

as the set of 2x2 matrix 

se of 3x3 Matrix 

If 

Addition of two matrices: ( )+( ) 

ix ( ) 

the identity matrix to A to form the matrix 

Multiplication of two matrices: ( )( ) 

til we arrive at 

, where 

obtain the matrix 

, as follows. 

R 2 + R 1 

-4R 2 + R 3 

nvertible (or inverse exist) and its inverse is

Fiber 

Optics 

For example, if we multiply 

33 

e-Math M 

is closed under ad- 

Example 1.2.2 Show that 

dition and multiplication 

Solution. Let and , i.e. and . Then 

since and implies 

since , implies 

Example: Show that ) is closed under 


Solution. Let and . Then, 

We note that 

may be represented as rational numbers 

say c, since Q closed under addition and multiplication. The 

same for 

numbers say d. 

which can be represented as one rational

agazine 34 

In related application, physical sciences used matrices 

in the study of optics, electrical circuits, and quantum 

mechanics. 

Example 1.2.4 Given a 2x2 matrix 

Find n such that 

, where the identity matrix, Construct the table of 

set S whose elements are formed from the powers of . 

Solutions. Given then we have the multiples of 

as follows:

35 

e-Math M 

Trigonometric Identity 1.2.1 

Example 1.2.5 We prove the trigonometric 

identity 

Let S as the elements formed from the powers of 

a. Then We construct the multiplication 

table as shown in Table 16 with the products 

of the elements of S as follows: 

For all integers n (positive, zero, or negative). 

Proof. We will prove the statement using 


Step 1. Verification for the truth of the statements 

when n=1. But when n=1, 

* 

which is a true statement. 

Step 2: Induction Hypothesis 

Assume that the statement is true in any integer 

n = k, i.e. 

Table 16: {I 2, , 2, 3,} under multiplication 

The preceding statement serves as the induction 

hypothesis. 

Prove it also true for the next integer n = k 

+ 1, i.e. 

The next statement is a trigonometric identity 

which is special form of a square matrix. Moreover, 

such 2x2 matrix 

is a special 

subset of the set of all invertible matrices. We 

will prove the special matrix in trigonometric 

form.

agazine 36 

Step 3: Proof of the Induction 

Multiply the induction hypothesis by 

to obtain: 

Step 4: Conclusion 

Since the statement is true when n = k = 1 (Step 1), then it must be true for 

the next integer n = k + 1 (Step 3). Since it is true for n = k = 2 (Step 2), then it 

must be true for the next integer n = k + 1 = 3 (Step 3), etc. 

Solution: 

Example 1.2.6 

Find the value of the following using the trigonometric identity in 

Example 1.2.5: 

(cos 90 0 + sin 90 0 ) 4 

(cos 60 0 + sin 60 0 ) 10 

By trigonometric identity 1.2.1, we have: 

(cos 90 0 + sin 90 0 ) 4 = 

To check: 

(cos 60 0 + sin 60 0 ) 10 = 

Since, 

, then removing the multiple of 360° from 

Matrices are used 

much more in daily life 

than people would 

have thought. In fact, 

it is in front of us every 

day when going to 

work, at the university 

and even at home. 

Graphic software 

such as Adobe Photoshop 

on your personal 

computer uses 

matrices to process 

linear transformations 

to render images. A 

square matrix can represent 

a linear transformation 

of a geometric 

object. 

600° gives (cos 60° + sin 60°) 10 = cos 240° + sin 240° =

37 

e-Math M 

1 

Example: Find the element of the set 

Solution: 

When k = 0 and Ө = 90 0 , we have: 

k = 0 ; 

k = 1 ; 

k = 2; 

OTHER PROPERTIES OF MATRICES 

Addition of matrices is associative 

Definition of matrix addition 


Associative law for addition of numbers 


“In the context where algebra is identified with the theory of equations 

has traditionally been known as the “Father of Algebra” but in more re 

whether al-Khwarizmi, who founded the discipline of al-jabr,

agazine 38 

DEFINITION OF MATRIX ADDITION 

This means by adding this matrix to any 2 x 2 

(two by two) matrix leaves the element 

or the matrix unchanged, i.e. if A is any 2 x 2 matrix, A + O M = A. 

2 

We define scalar multiplication as 

For example: 

ZERO DIVISORS 

A matrix A and B are zero divisors if 

B ≠ O M 

, the zero matrix, but A ≠ O M and 

Notice that if and , then 

, the zero matrix. 

But and , then A and B are called zero divisors. 

Thus, the set of 2 x 2 matrices M 2 has zero divisors. 

the Greek mathematician Diophantus 

cent times there is much debate over 

deserves that title instead.”

athematical Induction 

39 

e-Math M 

The cancellation 

law on matrices 

will not hold 

in some cases. Thus, 

the property AX = AY 

does not always 

permit us to say 

X = Y. Unlike in 

the case of real 

numbers, 

Cancellation 

Law holds, 

i.e. ax = ay 

permits us 

to say x = 

y. 

4. Cancellation Law 

Thus, we 

have on matrices: 

Cancellation 

Law on Matrices 

may NOT always 

hold AX = AY, then X ≠ 

Y, if A is zero divisor. 

AX = AY, then X = Y, if 

A -1 exists. 

Lesson 

Obje 

At the end of lesso 

1. Enumerate and 

mathemat 

2. Prove proposition 

The theory of numbe 

ods. There are two basic 

tion. The first is the Well 

of Mathematical Inductio 

Well- Ordering Prin 

teger contains a least po 

∈ S, there exists an intege 

This principle provi 

rem. 

Definition. Mathematical 

the validity of mathema 

tions involving a series of 

Actually, mathem 

starts from particular ca 

drawn by inductive pro 

not quite revealing. 

One disadvantage o 

mathematical statement 

gent guess in the formula 

tain rules, which we beli 

examine the following eq

agazine 40 

ctives: 

n, the students are able to: 

illustrate the steps of the 

ical induction; and 

s by mathematical induction. 

r relies for proofs on great many ideas and methprinciples 

to which we withdraw special atten- 

- Ordering Principle and the other is the Principle 

n. 

ciple. Every non-empty set S or a non-negative insitive 

integer. In other words, for every element x 

r e ∈ S, such that e ≤ x. 

des a firm basis for the proof of subsequent theo- 

Induction is a standard procedure for establishing 

tical statements, which we shall call as proposipositive 

integers. 

atical induction is deductive in nature since it 

ses. Although useful, no certain conclusion can be 

cess and so the name mathematical induction is 

f this tool is that it gives no aid in formulating 

s. However, we may come up with some intellition 

of the properties of integers that behave cereve, 

might hold in general. For instance, we try to 

ualities. 

There are several examples of mathematical 

induction in real life: 

1. One standard example of falling dominoes. 

In a line of closely arranged dominoes, 

if the first domino falls , then all 

the dominoes will fall because if any one 

domino falls, it means that the next 

domino will fall, too. 

2. Another example is the solution to the 

Tower of Hanoi and other similar problems. 

Although there are several other 

proofs, induction is the most common 

and elegant of them all. 

3. A true real-life example is the sinking of 

the Titanic. The crew of the Titanic realized 

that the ship was doomed when 

they realized that the bulkhead that 

was being flooded would be completely 

flooded, and that when a given bulkhead 

was completely flooded, and the 

next bulkhead will undergo the same 

fate, thus sinking the whole ship.

41 

e-Math M 

1 = 1 = 1 2 

1 + 3 = 4 = 2 2 

1 + 3 + 5 = 3 2 

1 + 3 + 5 + 7 = 4 2 

“e” in its name. 

The number 2 is the only prime number that does not have an 

This few cases cited above gives a definite pattern. Then construct a rule that 

gives the mathematical relationship of integers on the right side of the equation. 

1 + 3 + 5 + 7 + … + (2n – 1) = n 2 , n ≥ 1 

Also, it is interesting to note of the next example. 

If we write the consecutive integers from 1 to n in one row, and just the same 

integers this time from n down to 1 in another row, then we have the formation of 

integers as follows: 

1 2 3 4 5 n – 1 n 

n n - 1 n - 2 n - 3 n - 4 2 1 

Notice that each vertical column produces the sum of n + 1, and there are n 

such columns. Thus, the sum of all the integers listed above in n (n + 1). It follows 

that 

2 (1 + 2 + … + n ) = n (n + 1) 

And so, the sum of the first n consecutive integers is 

Theorem 1.4.1. Principle of Mathematical Induction 

Let S be a set of positive integers such that 

i. 1 ∈ S, and 

ii. Whenever the integer k ∈ S, then k + 1 ∈ S. 

Then S is the set of positive integers. 

Proof. Let S be the set of all positive integers and S1 be the set of all positive integer 

not in S. assume S 1 ≠ ∅. Then by the Well-Ordering Principle, there exists a least 

element, say e ∈ S 1 . But by i) 1 ∈ S implies that e > 1 and 0 < e – 1 < e. The choice 

of e being the smallest integer in S 1 implies that e – 1 ∉ S, but by ii), whenever e – 

1 ∈ S, then the next integer (e – 1) + 1 = e ∈ S. But this contradicts the choice of 

e ∈ S 1 . The way out of the contradiction is to claim that S 1 = ∅. Thus, S contains 

the set of all positive integers. 

.

agazine 42 

Procedures for Mathematical Induction 

It consists of four steps. 

Step 1: Verification 

We verify the validity of the proposition for a few particular 

cases of n. We start with the smallest value of n for which the 

proposition holds, usually n = 1, unless otherwise stated. 


This s the assumption made in carrying out of the proof of 

the induction. This means that we have to assume that the statement 

is true for some positive integer n = k. 


This is the basis of induction where the hypothesis is utilize 

to carry out the desired result. 


The proposition is true for all positive integral values of n. 

Example: 

Addition of Suitable Terms 

Show by mathematical induction that 

Proof: 


Verify that the proposition is true for the first few integers. 

Let n = 1, then 

Tower of Hanoi 

Tower of Hanoi is one 

of the application of 


S n is the minimum number 

of moves it takes to solve 

towers of Hanoi where n 

is a positive integer. 

S n = 2 n – 1 

Base Case: 

S 1 = 2 1 – 1 

= 1 


Assume that the proposition is true for some positive integers 

n = k, i.e.

43 


This is the basis of induction where the hypothesis is utilize to carry out the desired result. 


The proposition is true for all positive integral values of n. 

e-Math M 

Example: 


ADDITION OF SUITABLE TERMS 

Proof: 


Verify that the proposition is true for the first few integers. Let n = 1, then 


Assume that the proposition is true for some positive integers n = k, i.e. 


We shall show that whenever the hypothesis holds, then the proposition also holds for the next integer 

n = k + 1, i.e. 

The left side of the hypothesis lacks the term ar k compared to the left side of the proposition to 

be proved. So, adding ar k to both sides of the induction hypothesis, we get 

Simplifying the right side of the preceding equation gives

agazine 44 


Hence, if the proposition is true in n = k, we have 

proved it to be true for n = k +1. But the proposition 

holds for n = 1 = k, hence, by Step 3, it also holds for n 

= k + 1 = 2. Being true for n = 2 = k, then by Step 3, it 

must also hold for n = k + 1 = 3, and so on. 

Thus, the proposition is true for all positive integral 

values of n. 

Example. 

Proved by mathematical induction 

Math About Me 

Numbers, numbers all around, 

Everywhere they can be found. 

Numbers tell how old I am, 

And how many people in my fam. 

How much I weigh and just how tall , 

Where I live, and that’s not all! 

Proof: 


Verify that the statement for n = 1, i.e. 

When to wake up and when to eat, 

What size shoes to buy for my feet. 

How much money something costs. 

A number to call if my dog gets lost. 

I don’t know where I would be, 

If numbers weren’t a part of me! 

:https://www.google.com/search? 

q=math+poems 


Assume that the proposition is true for some positive 

integer n = k, i.e, 

Did You Know That! 

Plus (+) and minus (-) signs were 

first used by mathematicians in 

the sixteenth century.


Show that the statement is also true for 

the next positive integer n = k + 1, whenever it is 

true for n = k, i.e. 

45 

Example: 


e-Math M 

Using the hypothesis, we add (k + 1) (k + 

3) on both sides, i.e. 

Proof: 


Let n = 1, then 


Hence, if the proposition is true in n = k, 

we have proved it to be true for n = k + 1. But 

the proposition holds for n = 1 = k, hence, by 

Step 3, it also holds for n = k + 1 = 2. Being true 

for n = 2 = k, then by Step 3, it must also hold 

for n = k + 1 = 3, and so on. 

Thus, the proposition is true for all positive 

integral values of n. 


Assume that the proposition is true for some 

positive integer n = k, i.e, 

A mathematical name 

for the division sign 

(÷) is called Obelus.

agazine 46 Bibliography 


We shall show whenever the hypothesis 

holds, then the proposition also holds for the 

next integer n = k + 1, i.e. 

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Adding the two terms on the right side 

of the preceding equation, we have: 


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mgrc=woNuYgWolTg9cM: 


Hence, if the proposition is true in n = k, 

we have proved it to be true for n = k + 1. But 

the proposition holds for n = 1 = k, hence, by 

Step 3, it also holds for n = k + 1 = 2. Being 

true for n = 2 = k, then by Step 3, it must also 

hold for n = k + 1 = 3, and so on. 

Thus, the proposition is true for all positive 

integral values of n. 


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3D#imgrc=JoGEdJgypNifHM:

Chapter Test: Give your BEST! 

Give It a TRY! 

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e-Math M 

Properties of Real Numbers! 

1) Determine if the set S = {-1, 1} is closed under a) addition, 

b) subtraction, c) multiplication, and d) division. 

2) Let S = {-60, -69, -68, …, -1, 0, 1, …, 58, 59}. 

a) Which integers are in both S and 6Z? 

b) Which integers in S have 1 as the remainder when 

divided by 6? 

c) Which integers in S are also in -1 + 6Z? 

d) Which integers n in S satisfy n ≡ 3 (mod 6)? 

3) Let T = {-70, -69, -68, …, -1, 0, 1, …, 68, 69}. 

a) Which integers are in both and 7Z? 

b) Which integers in T have 2 as the remainder 

c) Which integers in T are also in 3 + 7Z? 

d) Which integers n in T satisfy n ≡ -1 (modulo 7)? 

4) (a) Is 2Z closed under subtraction? Explain. 

(b) is 1 + 2Z closed under subtraction? Explain.

agazine 48 

5) Express the following in POLAR AND EULER form: 

Complex Number Polar Form Euler Form 

a) (-1/2) - (√3/2) 

b) (-√2/2) - (√2/2) 

c) (-√2/2) ÷ (√2/2) 

d) (√2/2) - (√2/2) 

e) √2 - 

f) -√3 - 

6) Plot the following points. 

a) e -2 

b) 3e - 

c) 3e - /2 

d) 4e-5 /6 

e) 2e-5 /3 

f) 2(cos - sin ) 

g) 2(cos /6 - sin /6) 

h) 2(cos 7 /3 - sin 7 /3) 

i) 5(cos 7 /4 - sin 7 /4) 

j) 5(cos 7 /6 - sin 7 /6) 

7) Which of the sets 3Z, 1 + 3Z, and 2 + 3Z is/are closed under subtraction? Explain.

8) Let a be in 1 + 4Z, and let b in 2 + 4Z. 

(a) Must a + b be in 3 + 4Z? 

(b) Must a – b be in 3 + 4Z? 

(9) Let a be in 1 + 4Z, and let b in 2 + 4Z. 

(a) Must b – a always in 1 + 4Z? 

(b) Must b + a always in 1 + 4Z? 

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e-Math M 

10) (a) Why do a and a + 2d have the same parity for any integers 

a and d? 

(b) Why do m – n and m + n have the same parity for all m and 

n in Z? 

11) Let a be in 1 + 2Z + = {3, 5, 7, ….}. Find one solution in positive 

integer x and y to the equation x 2 – y 2 = a. 

12) Whenever possible, find a solution for each of the following 

equations in the given Z n . 

(a) [4] [x] = [2] in Z 6 

(b) [6] [x] = [4] in Z 12 

(c) [6] [x] = [4] in Z 8 

(d) [10] [x] = [6] in Z 12 

(e) [8] [x] = [6] in Z 12 

13) If a = (2 3 4 1) and S = {a, a 2 , …, a n = (1)}, list the elements 

of S and form the multiplicative table for S. 

14) If a = (3 1 2) and S = {a, a 2 , …, a n = (1)}, list the elements of 

S and form the multiplicative table for S. 

15) Find the products of the following and express the result n 

cyclic form. 

(a) 

(b)

agazine 50 

(c) 

(d) 

(e) 

(f) 

(g) 

16) Find the products of the following in cycle 

form. 

(a) (3 4 1 2) (1 2) (3 4) 

(b) (3 4 1 2) (4 1 2) 

(c) (4 1 2 3) (4 1 2 3) 

(d) (2 3 4 1) (4 1 2 3) 

(e) (2 3 4 1) (1 2) (3 4) 

(f) (1 3) (2 4) (1 3) (2 4) 

17) Put the following in one standard form of a 

permutation in S 4 . 

Example: 

(a) (1 2 3) (3 4) 

(b) (1 2) (1 3) (1 4) 

18) Let S be the set of four matrices S = {I, A, 

B, C} where 

B = , C = 

, A = , 

Construct a table and show that S closed under 


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q=fibonacci&client=ms-opera- 

isch&sa=X&ved=0ahUKEwjYmrHb3qnaAhVGWLwKHYJyAFc 

Q_AUIESgB#imgdii=ofZFO0GYzFBoBM:&imgrc=hmvSeIURcT 

-hPM: 

https://www.google.com/url? 

sa=t&source=web&rct=j&url=https:// 

www.math.stonybrook.edu/~irwin/ 

QegQIARAB&usg=AOvVaw1jvX-gf4CjrF_I7I8YNPP5 


mo- 

bile&channel=new&espv=1&prmd=ivn&source=lnms&tbm= 

https://www.did-you-knows.com/did-you-knowfacts/numbers.php 

algbk.pdf&ved=2ahUKEwjwzpDE0qraAhWIVLwKHTl5D34QFjA 

q=permutation+in+rubik%27s&client=ms-opera- 

&ved=0ahUKEwi5j__h66raAhWEv7wKHT4RBSIQ_AUIECgA& 

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AB&usg=AOvVaw0senwEU7TeB8dkW0iEBP4o 


sa=t&source=web&rct=j&url=https://www.goconqr.com/ 

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bile&channel=new&espv=1&prmd=vin&source=lnms&sa=X 

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AB&usg=AOvVaw3W-iSCDqJ8Bu-MdhgwVLd8&ampcf=1

A Dozen of Study Tips 

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e-Math M 

1. Set Study Goals. There is lots of credible research 

suggesting that goal setting can be used as part of a strategy 

to help people successfully effect positive changes in 

their lives, so never underestimate the power of identifying 

to yourself the things you want to achieve. Just make sure to 

ask yourself some key questions: Am I setting realistic goals? Will 

I need to work harder to achieve those goals? 

2. Make a Study Plan. Time is precious. Nobody is more 

aware of this than the poor student who hasn’t studied a thing until 

the night before an exam. By then, of course, it’s too late. The 

key to breaking the cycle of cramming for tests is to think ahead 

and create an effective study plan. Not only will this help you get 

organized and make the most of your time, it’ll also put your mind 

at ease and eliminate that nasty feeling you get when you walk 

into an exam knowing that you’re not at all prepared. As the old 

saying goes, fail to prepare and be prepared to fail. 

3. Take Regular Study Breaks. None of us are superhuman, 

so it’s important to realize that you can’t maintain an optimum level 

of concentration without giving yourself some time to recover from 

the work you’ve put in. This can take the form of a ten-minute walk, 

a trip to the gym, having a chat with a friend or simply fixing yourself 

a hot drink. If it feels like procrastination, then rest assured that 

it’s not: taking regular short breaks not only help improve your focus, 

they can boost your productivity too. 

4. Embrace New Technologies. Studying no longer means 

jotting things down with a pen on a scrap of paper. The old handwritten 

method still has its place of course, it’s just that now there 

are more options for personalizing study that ever before. Whether 

it’s through online tools, social media, blogs, videos or mobile 

apps, learning has become more fluid and user-centered. 

5. Test Yourself . It’s a strange thing, but sometimes simply entering 

an exam environment is enough to make you forget some of 

the things you’ve learned. The solution is to mentally prepare for 

the pressure of having to remember key dates, facts, names, 

formulas and so on. Testing yourself with regular quizzes is a 

great way of doing this. And don’t worry of you don’t perform 

brilliantly at first – the more you practice, the better you’ll become.

agazine 52 

6. Find a Healthy Balance. Take this 

opportunity to evaluate yourself both physically and mentally. Is your 

engine running on low? Instead of complaining “I never get enough 

sleep” or “I’m eating too much convenience food” take control and do 

something about it! Make the change and see how it positively affects 

your attitude and study routine. This should motivate you to maintain a 

healthy balance in the future. 

7. Be Positive Developing a Growth Mindset. Your attitude has 

a big impact on the level of study that you get done and the effectiveness of 

your learning process. If you keep saying that you can’t do it and won’t commit 

to the idea of learning, attempting to study is only likely to become more difficult. 

Instead, focus your mind on positive outcomes and on how you can use 

your own individual strengths to achieve them. When you think positively, the 

reward centers in your brain show greater activity, thereby making you feel 

less anxious and more open to new study tips. 

8. Collaborate with Study Partners. At this stage of the school year, 

you should know your classmates pretty well. This is a good point in time to select a couple 

of study partners who you know you work well with and are motivated to achieve good 

grades also. Don’t worry if you can’t meet up too often, you can use online tools such as 

GoConqr’s Groups tool to communicate and share study notes with one another. 

9. Turn lessons into stories. Everybody likes to read or listen to a good 

story, and with good reason – not only do stories entertain us, they help us to understand 

and memorize key details too. You can apply this to your studies by 

weaving important details or facts into a story – the more outlandish and ridiculous 

you can make it, the better (since you’ll be more likely to remember a particularly 

crazy story). 

10. Establish a Study Routine. Your study routine is comprised of 

more than planning what to learn and when. One of the main concerns is your 

study environment. 

11. Mark Small Challenges. When you have to face very long 

and dense subjects, you can set small challenges to keep your spirits high; a good way to 

focus on the day-to-day and find motivations while you study. According to scientific analysis, 

the more motivated and excited we are, the better our brain performs. 

12. Consult Teachers. Any questions you have about the exam, the best 

you can do is go to the teacher of the subject and expose your doubts. Not 

only is the person best suited to solve your questions, but your initiative 

will be well received and you’ll show good attitude by demonstrating that 

you’re interested in his subject.

53 

Rebus Puzzles e-Math M

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For Example: 

Answer: H2O or Water

Puzzle your mind… 

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There are seven types of teachers you’ll meet inside the classroom. Have 

you ever meet them? Which of them have you met?

agazine 58 

There are different 

types of student researchers 

you’ll encounter 

in your group! 

Whoever you are, as long as you 

contribute and know the topics very 

well, you are a great help to finish the 

tasks.

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Did you know that…

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Divisibility Rules 

I’m #2 and I’ll be your friend, 

A long as an even #’s on the end. 

#3 will work for me, you see, 

If the sum is divisible by 3. 

The #4 won’t be such a chore, 

If the last 2 are divisible by 4. 

The #5 is my biggest hero, 

He has to end in 5 and 0. 

The #6 will always go into me, 

As long as so does 2 and 3. 

#9 will go into me just fine, 

If the sum is divisible by 9.

THE 

END 

:gdb

emagazine

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