MATM111 BY: PARAISO, JULIANNA ISSA B.

BS PHARM 1-Y1-1 INSTRUCTOR: Ms. Princess Joy Bogambal │ S.Y 2024-2025

PATTERNS AND NUMBERS IN NATURE AND

2. CHAOS
THE WORLD
 Shows apparent randomness whose
Mathematics
origins are extremely deterministic.
 A study of patterns and structures.
 e.g. Dried leaves and dry soil, ocean
Mathematics is fundamental to the physical
and sand dunes.
and biological sciences, engineering and
information technology, economics, and
social sciences.  Any form of these observations can be
 Mathematics is a useful way to think about modelled mathematically.
nature and our world.  The natural patterns, include symmetry,
 Mathematics is a tool to quantify, organize, spirals, tessellations, stripes and so on.
and control our world, predict, phenomena,
and make like easier for us. 1.0 FIBONACCI SEQUENCE
Patterns
Leonardo Fibonacci
 Patterns in nature are visible regularities of
form found in the natural world and can ● His real name was Leonardo Pisano Bogollo,
also be seen in the universe. and he lived between 1170 and 1250 in Italy.

 “Nature patterns which are not just to be ● The latin word for Fibonacci is “FILLIUS
admired, they are vital clues to the rules BONACCI” which roughly means “Son of Bonacci”

that govern natural processes.” ● He helped spread Hindu-Arabic Numerals

 Regular; repeated; recurring forms or (like our present numbers 0, 1, 2, 3, 4, 5, 6, 7, 8,
9) through Europe in place of Roman Numerals
designs; Identify relationships; Find logical (I, II, III, IV, V, etc.)
connections to form generalizations

 Fibonacci Sequence was discovered after an

TYPES OF PATTERNS
investigation on the reproduction of rabbits.

1. FRACTALS  The original problem Fibonacci investigated

 Fractals are geometric shape that was about how FAST rabbits can breed in

repeat their structure on an even ideal circumstances.

finer scale.
 e.g. Snowflakes, Radial symmetry
(starfish, plants)
 FIBONACCI SEQUENCE  First, the terms are numbered from 0
onwards like this:
 Simple rule: Add 2 successive numbers to
get next. n= 0 1 2 3 4 5 6 7 8
 Always starts with (0,1) 𝑿𝒏 = 0 1 1 2 3 5 8 13 21
 Indefinite  Simple Rule: Add the last two terms

Example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,... to get the next.

 𝐹𝑖𝑏(𝑛) = 𝑋𝑛−1 + 𝑋𝑛−2
 Just add 1 and 2 to get 3.
 Add 13 and 21 to get 34.

Example Problem: Pairs 10th method 2.0 THE GOLDEN RATIO

𝐹𝑖𝑏(10) = 55  The golden ratio can be expressed as the

ratio between two numbers.

FORMULA 𝑎 𝑎+𝑏
=
𝑏 𝑎
𝐹𝑖𝑏 (𝑛) = 𝑋𝑛−1 + 𝑋𝑛−2

EXAMPLE

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,…

𝑋𝑛−1 + 𝑋𝑛−2

𝐹𝑖𝑏(5) = 𝑋5−1 + 𝑋5−2 a = highest number b = lowest number

=𝑋4 + 𝑋3
 “φ” Phi (fi)
=3 + 2
 φ = 1.618034 (approximately)
𝐹𝑖𝑏(5) = 5
 Any two successive Fibonacci numbers and
its ratio is close to the GOLDEN RATIO
 Fibonacci Sequence and several biological (φ = 1.618034)
settings.
 The Golden ratio is also sometimes called
 Flower
the Golden section, Golden mean, Golden
 Plants
number, divine proportion, divine section
 Fibonacci Sequence in computer science.
and Golden proportion.
 Fibonacci Sequence
 The Fibonacci Sequence can be
written as a “Rule”
 TRANSLATING PHRASED INTO ALGEBRAIC
MATHEMATICAL LANGUAGE AND SYMBOLS EXPRESSIONS / EQUATIONS
Plus, More than, greater than,
ADDITION the sum of/ the total of, added
to
1.0 LANGUAGE Minus, the difference of, less
SUBTRACTION
 A system of conventional spoken, manual than, decreased by
Times, the product of,
(signed), or written symbols of by means of MULTIPLICATION
multiplied by, twice/trippled
which human beings, as members of a The quotient of, divided by, the
DIVISION
ratio of
social group and participants in its culture, EXPONENTS Squared (2), the cube of (3)
express themselves. Is equal to/equals, is, is the
EQUAL
same as, results, yields to

LANGUAGE ITSELF IS:

It can make very fine distinctions
PRECISE TRANSLATE
among sets of symbols
It can briefly express long
CONCISE
sentences
English to Math
It gives upon expressing complex
POWERFUL
thoughts 1. Six less than twice a number is forty five
 2x-6=45
SYMBOLS COMMONLY USED IN
MATHEMATICS 2. A number minus sever yields ten
 x=7=10
3. A total of 6 and some number
 6+x
4. Twelve added to a number
 x+12
5. The product of fourteen and a number
 14 · x = 14x
6. Twice a number minus eight
 2x-8
7. The quotient of a number and seven is two
 x÷7=2
𝑥
 =2
7

8. Three-fourths of a number
3
 x
4

9. The product of a number is eighty

 x · 10 = 80
10x = 80
10. Eight less than a number is five
 x-8=5
RELATION

Translate the following to mathematical

statements the solve for the unknown:

1. Twice a number diminished by two is equal

to 10.
2𝑥 − 2 = 10
2𝑥 = 10 + 2
2𝑥 12
=
2 3
𝑥=6

2. The square if a number is equal to thirty-six

𝑥 2 = 36
√𝑥 2 = √36  One is to one relation
𝑥=6  One is to many relation
 Many is to one relation
3. Twice the square of a number diminished  Many is to many relation
by ten yields to forty.
2𝑥 2 − 10 = 40
2𝑥 2 = 40 + 10 (x, y)
2
2𝑥 50 x = Domain y = Range
=
2 2
𝑥 2 = 25 Unique

√𝑥 2 = √25
𝑥=5
Sample 1: {(1,2)(3,4)(5,6)}

x y
FUNCTIONS & RELATIONS
1 2 One to one relation
 Function is defined relationship between
inputs where each input is related to 3 4 Function
exactly one output. 5 6
Sample 2: {(-1,5)(-1,2)(-1,0)}

A set is a well-defined collection of distinct

x y objects
5 One to many Relation It’s usually represented by CAPITAL
LETTERS
-1 2 Not function
The objects of a set are SEPARATED BY
0 COMMAS

The objects that belong in a set are the

Sample 3: {Children to Mother} ELEMENTS, or members of the set.

It can be represented by LISTING ITS

Children Mother ELEMENT BETWEEN BRACES
(x) (y) Squared (2), the cube of (3)
Many to one Relation
Panganay
Nanay Function
Gitna Sample 4: Set of even numbers less than 20.
Bunso
A = {0,2,4,6,8,10,12,14,16,18}

Inside the set are called ELEMENTS

2.0 SETS

 A set in mathematics is a collection of

Sample 5: Set of College Section
well-defined and distinct objects,
considered as an object in its own right. B = {Juan, Prince, John. Jane, Kenneth}
 Sets are one of the most fundamental
concepts in mathematics. CARDINALITY
 SET = a group/collection of data.  Tells how many things are in a set.
 When counting a set of objects, the last
word in the counting sequence names the
quantity for that set.
Sample 6:

Cardinality (No. of elements)

A={1}
1
B={A,B,C,D}
4
C={0,2,4,6,8,10}
6
D={5,5,5,5,5}
1
 SET
●UNIVERSAL The totality of ALL ELEMENTS SET ON VENN DIAGRAM
SET from two or more sets. (∪)
UNION (∪) INTERSECTION (∩)
●NULL SET/
A set with NO ELEMENTS (∅)
EMPTY SET

Sample 7:

A = {1,2,3,4,5} B = {0,2,4,6,8} C = {1,3,5,7,9}

∪= {0, 1,2,3,4,5,6,7,8.9}

OPERATIONS ON SET

The union of two sets X and Y Sample 9:

is defined as the set of
UNION (∪) elements that are INCLUDED A travel agent surveyed 55 people to find out how
either in the set X or set Y, or many of them had visited the cities of ANTIPOLO
both X and Y. and MARIKINA. Thirty-one people had visited
ANTIPOLO, 26 people had been to MARIKINA, and
●The intersection of two sets 12 people had visited both cities.
X and Y is defined as the set
INTERSECTION Draw a Venn diagram to find the number of people
of elements that belongs to
(∩) who had visited.
BOTH set X and Y.
●Common 1. Both cities
2. Antipolo but not Marikina
3. Antipolo or Marikina
Sample 8: 4. Marikina only
5. Marikina only Neither of the two cities
A = {1,2,3,4,5} B = {0,2,4,6,8} C = {1,3,5,7,9} M A

26
 A∪B ={0,1,2,3,4,5,6,8} 31
 A∩B = {2,4} 26-12= 14
12 31-12=14
 A∪C = {1,3,5}
14
 A∪B∪C = {0,1,2,3,4,5,6,7,8,9} (Universal) 19
 A∩B∩C = ∅

10

1. Both cities = 12
2. Antipolo but not Marikina = 19
3. Antipolo or Marikina = 14+12+19= 45
4. Marikina only= 14
5. Marikina only Neither of the two cities = 10
 ELEMENTARY LOGIC PROPOSITIONAL LOGIC

 Logic serves as a set of rules that govern Identify the statement using the given logical
proposition:
the structure and presentation of
p: It is raining
mathematical proofs. It allows us to
q: The ground is wet
determine the validity of arguments in and
r: The sun is shining
out of mathematics.
 Logic comes from the Greek word “Logos”
1. If it is raining then the sun is not shining.
which means speech and reasoning.
2. It is raining and the ground is wet.

LOGIC 3. The ground is wet if and only if it is raining

A statement that is, by itself, and the sun is shining.

either true or false. They can 4. ~pVr
PROPOSITION
be expressed in symbols P, Q,
R, or p, q, r. 5. r—›q
6. ~q‹—›r
PREPOSITIONAL A declarative statement that
LOGIC is either true or false. 7. q—›p
8. ~p
9. pVq
Types of Sentences

 Declarative: “.” (Proposition)

 Interrogative: “?” (Not proposition) Answers:
 Imperative: “command” (Not proposition) 1. p—›~r
 Exclamatory: “!” (Proposition) 2. pVq
3. q‹—›pVr

LOGICAL CONNECTIVES 4. It is not raining and the sun is shining.

5. If the sun is shining, then the ground is wet.
NEGATION “~” mean “not”
6. The ground is not wet if and only if the sun
CONJUNCTION “V” means “and” is shining.
DISJUNCTION “Λ” means “or” 7. If the ground is wet, then it is raining.
CONDITIONAL “—›” means “if, then” 8. It is not raining.
BICONDITIONAL “‹—›” means “if and only if” 9. It is raining and the ground is wet.
 Sample 11:
LOGICAL CONNECTIVES
(CONDITIONAL) P1 All humans are mortal (If True)
CONVERSE q—›p P2 Ariana is human. (If True) VALID
Therefore, Ariana is
INVERSE ~p—›~q C (TRUE)
mortal
CONTRAPOSITIVE ~q—›~p
BICONDITIONAL p‹—›q SOUND

 Sound contains TRUE premises and is

INFERENCE
VALID.
 Process of reasoning  Always TRUE, always VALID.

ARGUMENT UNSOUND
 A collection of proposition intended to  Opposite of sound argument.
support other propositions.  At least one is false; may be valid or invalid

SYLLOGISM, TAUTOLOGY AND FALLACY Sample 12:

A process of deductive
SYLLOGISM reasoning that’s drawn from P1 Dogs are blue (False) VALID
two or more premises P2 Fido is a dog. (True) but
A compound propositions Unsound
TAUTOLOGY where all statements are C Therefore, Fido is blue. (Valid)
TRUE.
The contradiction of Tautology
FALLACY where all propositions are ] 2.0 PROBLEM SOLVING BY INDUCTIVE
always FALSE. REASONING
Sample 10:

Momo is a cat (P1) POLYA’S STRATEGY

Cats are good companion. (P2)  Named after GEORGE POLYA (1887-1985)
Therefore, Momo is a good companion. (C)  It is a FOUR-STEP problem solving strategy

P = Premise C = Conclusion that are deceptive simple.

POLYA’S FOUR-STEP STRATEGY

1.0 VALIDITY & SOUNDNESS
1. UNDERSTAND the problem
VALIDITY  Can you restate the problem in your own
 An argument is VALID if and only if the words?

premises are TRUE then the conclusion is  Can you determine what is known about
TRUE. these types of problems.

 An argument is INVALID if it’s not valid.

  Is there a missing information that, if Sample 13: The sum of two numbers is 30. The first
known, would allow you to solve the number is twice as large as the second one. What
problem? are the numbers?
 Is there extraneous information that is not
1. UNDERSTAND
needed to solve problems?
 Two numbers
 What is the goal?
 Goal: what are the numbers?

2. Device a PLAN. 2. PLAN

 Two numbers = 30
 Make a list of the known information.
 1st no. = 2x
 Make the list of the information that is
 2nd no. = x
needed.
 2𝑥 + 𝑥 = 30
 Draw a diagram.
3. CARRY OUT THE PLAN
 Make an organize list that shows all the
 2𝑥 + 𝑥 = 30
possibilities.
3𝑥 30
 Make a table or chart. =
3 3
 Try to solve a similar but simple problem. x=10
 Write an equation. If necessary define each 4. REVIEW
variable represents.  2𝑥 + 𝑥 = 30
 Perform an experiment.  2(10) + 10 = 30
 Guess at a solution and then check your
results.
Sample 14: A sequence of four figures is shown
3. CARRY OUT the plan.
below. If the figures
 Work carefully. were continued how
 Keep an accurate and neat record of all many circles would
your attempts. be there in figure 1.
 Realize that some of your initial plans will
not work and that you may have to devise
another plan or modify your existing plan. 1. UNDERSTAND
 Fig. 1 – 1 circle
4. REVIEW the solution. Fig. 2 – 3 circles
 Work carefully. 3-5
4–7
 Keep an accurate and neat record of all
 Goal: Fig. 10
your attempts. 2. PLAN
 Realize that some of your initial plans will  All figures are equal to ODD
NUMBERS.
not work and that you may have to devise
 Adds 2.
another plan or modify your existing plan. 3. CARRY OUT THE PLAN
 5–9
6 – 11
7 – 13
8 – 15
9 – 17
10 -19