MATHEMATICAL SYSTEM

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Course Title:MATHEMATICS IN THE MODERN WORLD

MODULE 3
MATHEMATICAL SYSTEM
Module Introduction
Historically, a mathematical system consists of undefined terms, definitions, postulates or
axioms, propositions or theorems as contained on the classic Book I of the Elements written by a
Greek mathematician, Euclid of Alexandria. This treatise systematizes Geometry containing 23
definitions, 5 postulates, 5 common notions, and 48 propositions (Baltazar, Ragasa, &
Evangelista, 2018).
Notably, these geometric concepts were already
introduced and applied by the Egyptians and
Babylonians who came before Euclid. The Elements
of Euclid philosophizes these concepts putting them
into more coherent, abstract and consistent ideas. It
establishes “geometry as a primary inquiry” (Young
Ocean, 2019).
Nothing has changed about the system that Euclid
has in his geometry up to the present day. All of his
propositions can be reproduced by a contemporary
student with accuracy equivocal to how Euclid
demonstrated it. Geometry has become a concrete
mathematical system where the elements of a set are
concreted or illustrated into concrete figures, and on
how these objects of the set are related.
However, in this Module, the learners will learn that a mathematical system is also an abstract or
algebraic system with the concept of set as building block. The emphasis on this section is the
study of the important properties of a mathematical system of a set of integers, the remainder
function particularly modular arithmetic.

LESSON NO.2
LESSON TITLEMODULAR ARITHMETIC
DURATION/HOURS3 hours
Specific Learning
Outcomes:		During the learning engagement, the learners should be able to:
1. Evaluate the remainder function, a  b (mod m);
2. Perform the modular addition and multiplication operations,
and investigate their binary properties; and
3. Solve a Day-of-a-Week Problem applying the mod m concept,
and a write a reflective journal about it.
TEACHING LEARNING ACTIVITIES
PROCESSING
Activity 1. Defining the Function a  b (mod m)
The Modular Arithmetic, also known a the remainder function, is a mathematical system
defined on a set of integer Z by two binary operations: modular addition ()and modular
multiplication (◎).

Euclid of Alexandria and his Book Elements. Photo retrieved
from https://miro.medium.com/max/500/1*4ijBUB_OSisa7Y2
Fmrihw.jpeg
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You can view on this link: https://www.youtube.com/watch?v=Eg6CTCu8iio, the discussion
on the basic heuristic on modular arithmetic.
Can you now tell which of the following is a correct statement or not?
5  1 (mod 2) [answer: correct]
7  3 (mod 3) [answer: incorrect]
15  0 (mod 3) [answer: correct]
10  3 (mod 5) [answer: incorrect]
The statement 5  1 (mod 2) is correct because when 5 is divided by 2, the remainder is 1.
Also, when we subtract 1 from 5 it will give a result of 4 which is divisible by 2, that is,
5 – 1 = 4, where 2 divides 4.
The second statement is incorrect. Following the heuristics above, when 7 is divided by 3 the
remainder is 1 not 3; hence, the correct statement would be 7  1 (mod 3). Also, 3 does not
divide the difference of 7 and 3, that is:
7 – 3 = 4, and 3 does not divide 4. However,
7 – 1 = 6, and 3 divides 6, that is in the case of 7  1 (mod 3).
The discussion of the two remaining cases will be left as your exercise.
In the following video, you will be introduced with the classical definition of modular
arithmetic as follows: If a, b  Z and m is a positive integer, written as m  Z+, then a  b
(mod m) if and only if m divides a – b. Symbolically, a  b (mod m) iff m|a-b.
Activity 2. Modular Addition
Recall what you have learned in Lesson 1, and the definition of a  b (mod m). Perform the
arithmetic addition mod 7. Enter your answers on the matrix given. You can view from this
link how modular addition is done: https://www.youtube.com/watch?v=2POgCMpHRh0.
Verify whether  is closed, commutative,
and associative. You also determine
whether zero (0) is the additive identity
element of mod 7. If so what are the
inverses of each element?
7 0 1 2 3 4 5 60 1 2 3 4 5 6

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Set S = 0,1,2,3,4,5,6
Closure  Closure of 7exists. The operation 7 is
closed in S because the said operation is defined
in all elements of S.
Associativity 
Sol’n
27(4 75) = (274 ) 75
272 = 6 75
4=4
:. Associativity of operation 7 exists.
Commutativity 
4 75 = 574
2=2
:. Commutativity of operation 7 exists.
:. Commutativity of operation 7 exists as proven by
diagonal line of symmetry.
Additive Identity  0 is the additive identity of
operation 7
:. Additive Identity of operation 7 exists.
Additive Inverse  0 is the additive inverse of
operation 7
:. Additive Inverse of operation 7 exists.
0 7 0 = 0
7 0 1 2 3 4 5 60 0 1 2 3 4 5 61 1 2 3 4 5 6 02 2 3 4 5 6 0 13 3 4 5 6 0 1 24 4 5 6 0 1 2 35 5 6 0 1 2 3 46 6 0 1 2 3 4 5

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1 7 6 = 0
2 7 5 = 0
3 7 4 = 0
4 7 3 = 0
5 7 2 = 0
6 7 1 = 0
:. Therefore, the operation 7 exists.
DO ALL THESE NOW:
Activity 3. Modular Multiplication
The modular multiplication is similar to
how modular addition works as in Activity
2, only that the operation is multiplication.
Again, enter your answers on the matrix
given, then verify all 5 binary properties in
the set Z7.
The following video will be of great help for your to construct your table of multiplication mod
7: https://www.youtube.com/watch?v=LWyH25nzbt0.
(Intermission Drill. You can go to this link
https://www.sporcle.com/games/stephantop/modular-arithmetic. It contains a game on
modular arithmetic. This activity will provide you opportunity to master this mathematical
concept.)
◎7 0 1 2 3 4 5 60 1 2 3 4 5 6

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FORMATION ACTIVITY
Activity 4. Finding Day of Week
The following problem classically exposes the beauty of modular arithmetic. Seriously, go
over the solution to the case.
PROBLEM: In 2017, Albert’s birthday fell on a Saturday, 3rd of June. On what day
of the week does Albert’s birthday fall in 2020? Note that 2020 is a leap year.
SOLUTION: The number of days in a year is 365 except when it is a leap year where
there is one day added. A year is a leap year when it is divisible by 4. Let us
determine the number of days after June 3, 2017 to June 3, 2020.
Number of days: After June 3, 2017 to June 3, 2018: 365
After June 3, 2018 to June 3, 2019: 365
After June 3, 2019 to June 3, 2020: 366 (leap year)
Total number of days is equal to 1096 days.
Then, we take the relation 1096  x (mod 7), where x is Albert’s birthday in 2020. Can
you explain why we take mod 7? Solving for x, we arrive at 1096  4 (mod 7). Hence, x
= 4, and Albert’s birthday is the 4th day of the week that is a Wednesday.
Now, your turn. What day of the week would be your birthday in 2025? Show your solution
mathematically. How shall you celebrate that day? If you are to invite a very special person on
that day, who would that be? Why?

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SYNTHESIS ACTIVITY
ACTIVITY 5
1. Tell whether the following binary properties are satisfied in the given modular operation. If
not, explain why.
Addition
Multiplication
Binary Properties Modular
Addition
( or x)Modular
Multiplication
( or x)Remarks/
ExplanationClosureCommutativityAssociativityExistence of IdentityExistence of Inverse@3 0 1 2 3 40 1 2 3 4 @
3 0 1 2 3 40 1 2 3 4

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2. Respond to these reflective questions to show what you have learned and reflected, after
engaging in this entire module:
1. What have I LEARNED this day that has helped me do all aspects of this better?
2. What have I DONE this week that has made me better at doing all aspects of this?
3. How can I IMPROVE at doing all aspects of this?
ASSESSMENT
(encircle or highlight your
best choice)		1. Which of the following is FALSE?
A. 5  1 (mod 3) C. 15  1 (mod 7)
B. 7  2 (mod 5) D. 10  0 (mod 2)
2. In the set Z8, what is the product of 12 and 2?
A. 5 B. 3 C. 0 D. does not exist
3. Determine: 6  7 mod 3.
A. 3 B. 2 C. 1 D. 0
A. 4 B. 3 C. 1 D. 0
4. Determine the value of b: 35  b (mod 13).
A. 9 B. 22 C. 4 D. 6
5. Which of the following binary property of modular
arithmetic DOES NOT always exist to all m  Z+?
A. Commutativity C. Associativity
B. Existence of Inverse D. Existence of Identity
6. Which of the following is NOT a solution to 10  b (mod 3)?
A. 1 B. 7 C. 4 D. 3
RESOURCES:Burton (2005). Elementary Number Theory, 5th Ed. McGraw
Hill Education.
Connell (2013). Reading Response Forms and Graphic
Organizers. Retrieved on July 6, 2020 from
https://www.scholastic.com/teachers/blog-posts/genia
connell/reading-response-forms-and-graphic-organizers/
https://www.youtube.com/watch?v=LWyH25nzbt0
https://www.youtube.com/watch?v=2POgCMpHRh0

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https://www.youtube.com/watch?v=Eg6CTCu8iio
https://courses.lumenlearning.com/atd-hostos
introcollegemath/chapter/calculator-shortcut-for-modular
arithmetic/
Stephantop (n.d.). Science Quiz, Modular Arithmetic. Retrieved
on 5 July 2020 from
https://www.sporcle.com/games/stephantop/modular
arithmetic
Stein (2004). Elementary Number Theory. E-book retrieved on
12 May 2012 from
https://williamstein.org/edu/fall05/168/refs/stein
number_theory.pdf.
Stoll (2006). Introductory Number Theory. E-book retrieved on
12 May 2012 from
http://www.mathe2.unibayreuth.de/stoll/lecturenotes/Intro
ductoryNumberTheory.pdf.