MATHEMATICS IN THE MODERN WORLD
The Nature of Mathematics
Mathematics is undisputedly the chief driving force that has propelled the evolution of a
highly sophisticated world and lifestyle. Consciously or unconsciously, man had to rely on the
value of mathematics in simple daily operations like doing grocery to the more sophisticated
undertakings like predicting the occurrence of thunderstorms or constructing state-of-the-art
infrastructure, be it physical or virtual in form. The practical appeal of mathematics can best be
understood if one gets to know its true nature and use. So, in this first part of the courseware,
we focus on the nature of mathematics, what it is, how it is expressed and represented, and how
it is used as a tool to better understand and improve the diverse universe. Specifically, Chapter 1
explores the indelible traces of mathematics in the multifarious universe, the patterns and
systems that govern them, Chapter 2 delves into the unique mathematical language and
symbolism, and Chapter 3 focuses on the logical structure of mathematics, its strength in terms
of how it can be used to predict occurrences in nature and in the world and consequently control
these occurrences.
Chapter 1. Mathematics in our World
Mathematics is exhibited not only in the technologies that has dominantly influenced
man’s daily pursuits. It is not only practiced by professionals like teachers, scientists, engineers
and economists. Mathematics is practically everywhere and for everyone. It is a thing that
perpetually exists in nature and propels development at varying degrees of usefulness. As an
ascendant expression for logical thinking, mathematics is vital in understanding natural
phenomena, human activities as well as social systems.
Chapter Objectives:
The universe, along with all life forms that exist in it, remains unfathomable by man’s
finite intellect. Yet, time has shown that one’s understanding of the universe continues to reach
greater bounds, and mathematics is rightfully credited for this. This chapter then serves to convey
that mathematics is a useful way to think about nature and the world. It takes the reader to a
journey of the trails which mathematics continues to create for humanity to comprehend,
appreciate and further enhance the universe where he exists. .
Section 1.1. Patterns and Numbers in Nature and the World
Quite so often, people immediately associate, and unfortunately, confine mathematics to
numbers and arithmetic. Such partial and superficial understanding of the nature of this body of
knowledge has restrained a liberal culture of learning in our educational system. In a way, the
amplified emphasis to numeric aptitude has brought about unfavorable outlook of the subject, a
culture that has been tolerated not only at the household setting but even in (early) formal
schooling of children. Students who exhibit masterful computational skills (e.g., doing arithmetic
operations with speed and precision) are immediately branded as “mathematicians”. While this
may boost the interest of students towards the subject, it may also pose deceptive expectations
especially when they get to encounter a different nature of mathematics. On the other hand,
students who exhibit numerical inadequacy find themselves desperately lurking in intimidation
along with a breeding distress over the subject. They are in a verge of losing interest of the subject
and ultimately depriving themselves of having to understand and appreciate the beauty of
creation, mainly because of a flawed belief about mathematics.
One thing must be clear at this point. Mathematics is not all about numbers. Rather, it is
more about reasoning, of making logical inferences and generalizations, and seeing relationships
in both the visible and invisible patterns in nature and in the world. One cannot simply base a
person’s potential in mathematics based on numeric skills in the same way that a good writer is
not judged from his or her penmanship.
One can draw inspiration from the famous mathematician David Hilbert, to whom the
influence to modern mathematics is given credit via the axiomatic treatment of geometry.
Believed to have a dyscalculia (a disability of learning and comprehending arithmetic), Hilbert
instead used abstraction rather than explicit computations to prove and establish essential
mathematical theories that led to breakthrough discoveries in both applied mathematics and
sciences. Other mathematicians and scientists who share the same background include N. N.
Luzin, George Gamov and Thomas Edison.
Mathematics goes beyond arithmetic, and this section is devoted to depicting
mathematics as a language by which the universe is elegantly designed, the value of which
transcends the intellectual, the practical and even the aesthetic values.
Intended Learning Outcomes
At the end of the section, students should be able to:
1. Explain the nature of mathematics;
2. Discuss how mathematics is exhibited in nature and in the world; and
3. Use the nature of mathematics to resolve issues that pertain to human activities,
natural occurrences and social systems.
Diagnostics:
 Learning Checkpoint
Instruction: Agree or Disagree. If you think the statement is correct, write AGREE, otherwise
write DISAGREE.
_________ 1. Mathematics is exhibited and demonstrated only through numbers.
_________ 2. Mathematics can progress even without numbers.
_________ 3. Every phenomenon, be it scientific or social, can be explained by mathematics.
_________ 4. Patterns that occur in nature are only for arts appreciation and not for
mathematical explorations.
_________ 5. Mathematics is not meant to be learned by everyone.
Patterns and Numbers in Nature and the World
Patterns are core topics in mathematics, in fact, mathematics is also known as the science
of patterns. Historically, mathematicians have primarily dealt with two types of patterns, the
numeric and geometric patterns (or more aptly, patterns of shapes). However, with the
increasing abstraction inspired by the drive to understand even the slightest detail in the
universe, these patterns have conceivably evolved to new perspectives bringing about a more
relevant approach of modelling the processes that are taking place in nature and in the world.
Thus, one may now also speak of patterns in structures, patterns in changes, random patterns in
both shapes (fractals) and occurrences (chaos).
While an appreciation of these patterns serves every individual well, an innate drive to at
least understand these occurrences has brought forth a variety of opportunities for man to better
understand and exploit the universe where he exists. It is this attempt to understand the universe
of patterns that has borne many significant theories in mathematics and which in turn had been
exploited by the other disciplines.
Abstract as they may be perceived, numbers and number patterns dominate most of
human endeavors. For example, the number sequence 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30,
31 may just seem a random collection of numbers until one realizes that these are the number
of days that make up each of the 12 months of the Gregorian Calendar. Everyone needs numbers
to operationalize gadgets like cellular phones (or telephones in general), home addresses are
conveniently assigned numeric codes for easier tracing, and a lot of similar instances. Of course
there are more systematic numerical patterns like 2, 4, 6, 8, 10, and so on, which may be used to
model the total savings a person makes when he starts saving 2 pesos on the first day and adding
up 2 pesos more for each succeeding day. In human anatomy, the heart beats rhythmically at a
certain pattern (which can be represented numerically) for the whole human system to sustain
its biological and chemical make-up.
Perhaps the most popular and unifying numeric pattern known to man is the Fibonacci
sequence (or simply, the Fibonacci numbers). Nature abounds with the testimonies about the
innateness of Fibonacci numbers. They appear in the numbers and arrangements of petals, leaves
and branches of plants.
The Fibonacci numbers started when Leonardo Pisano, working under the pseudo name
Fibonacci, proposed a problem about the breeding of rabbits. He started with a newly-born pair
(male and female) of rabbits which are able to mate after one month. Then another month of
gestation will produce exactly one pair (also male and female) of rabbit, that is, the female always
produces one new pair (one male, one female) every month from the second month on. The
problem then was to determine the number of pairs of rabbits after one year.
Going by the months, we have 1 pair (call it Pair A) in the initial period, still 1 pair (gestation)
a month after, then 2 pairs after two months (a new pair – pair B- is born at the end of second
month). On the third month, pair A produces new pair (pair C) while pair B is maturing, yielding
3 pairs. Going to the fourth month, pair A produces another new pair (pair D), pair C will now
produce an offspring (pair E), while pair D is maturing, accounting 5 pairs in all. From this, we
obtain the numbers 1, 1, 2, 3, 5. Figure 1 shows how the rabbits multiply and how the resulting
numeric pattern is formed. Note that the problem assumed that the female rabbits do not die
and are biologically reproductive in the entire process. So as one can notice, starting with the
third number, any other number in the set is just the sum of the two preceding numbers. In the
Credits: https://breakingfibonacci.weebly.com
Figure 1. Fibonacci Numbers from Rabbit Breeding
next section, we shall see how this Fibonacci sequence changed the way mathematicians and
scientists view nature and the world.
Aside from numeric patterns, there are also patterns of shapes which occur naturally like
the vibrant designs of flowers and leaves of certain plants and the fascinating designs in the skin
of some animals. There are also man-made patterns such as the symmetric designs of bridge
supports, wind mills, house roofs and many others.
In his book “Nature’s Numbers”, Ian Stewart explained that “by using mathematics to
organize and systematize our ideas about patterns, we have discovered a great secret: nature's
patterns are not just there to be admired, they are vital clues to the rules that govern natural
processes”.
One can look at the patterns exhibited by the
formations of clouds. Of course, clouds and cloud
formations are practically used to assess the possible
occurrence of an atmospheric phenomenon like rain or
even a storm depending on the type that can be observed
by the naked eye. In 2003, John Adams went a step ahead
when he was able to model the different cloud formations
by working on numeric data which were obtained by
observing these patterns. This model became in-demand
among physicists as it later became a significant factor in weather forecasting which can aid both
in air and marine navigations. He also used the same approach to model other naturally occurring
patterns like rainbows, river meanders, honeycombs and snowflakes.
In certain types of plants, exquisite spiral patterns may be found in their leaves and
flowers. These amazing formations have attracted not only mathematicians but also
practitioners in the applied sciences particularly
biology and botany. But beyond the genetic and
biological structure, mathematicians were able to
discover what is known as “Golden Angle” which
measures around 137.50.
The discovery of the Golden angle elicited
more interest from other fields (e.g., biochemists) as
the question of “Why would these plants choose (or
are they forced?) to grow in such fashion?” seem to be getting no affirmative answers.
Meanwhile, as mathematicians continued to explore the Golden angle among plants, they also
discovered the Golden ratio, which later became closely attached to the Fibonacci numbers.
What is noteworthy at this point is the fact that the original design pattern in plants has been
translated into numeric patterns via the Golden angle, the Golden ratio and the Fibonacci
numbers.
Animals also have their share of amazing patterns such as coat patterns in different
species of snakes, insects (like butterfly wings), peacock feathers, leopard spots, zebra or tiger
stripes, and a lot more. Over the years, biologists,
biochemists and of course mathematicians have been
working to find explanations to these spectacles. Simply
put, the mathematical justification arises from the
formulation of an equation (specifically, a unique system
of differential equations) which captures the interaction
between two chemical products that produce these skin
coats. Of course, chemists have yet to directly observe
the actual chemical reactions that are taking place in the
skin of these animals. The availability of data they have
gathered so far allowed mathematicians to simulate all possible factors and draw up the needed
mathematical models.
In the Philippines, one of the most prominent tourist attractions is the Mayon Volcano.
Renowned as the world’s most perfect natural conical formation due to (what used to be) an
almost flawless symmetrical features, this landmark has not only enticed local and foreign
nature-loving tourists but impelled scientific and mathematical explorations over the years.
The volcano has erupted almost 60 times over the past 100 years, making it one of the
most active volcanoes worldwide. So aside from the geometric aesthetics it offers, the volcano
has paved the way for probabilistic modelling to predict when and how it will erupt in the future.
At present, there is a multi-disciplinary team (with mathematicians taking crucial roles) that is
aggressively monitoring its activities.
In the southern part of the country, artistry and abstraction
is best exhibited in the intricate designs that are found in the textile
products and architectural designs and ornamentations. These
designs are themselves concrete testaments not only of the rich
cultural heritage but also to the mathematical ingenuity of
Filipinos, especially those coming from the southern regions. At
present, socio-cultural inquiries conducted on these designs have
established links with mathematics, giving birth to what is known
as ethnomathematics. It is believed that these elaborate designs
can be configured only with strong logical and mathematical proficiency. In-depth analyses of
these patterns have transcended core concepts of abstract algebra like groups, rings and fields.
In recent years, mathematics has allowed pattern exploration to get into deeper streams.
By further looking at the abstract details of the empirical occurrences, mathematicians have
succeeded in tracing patterns even in the most irregular shapes and events. Fractal, the type of
a random pattern with the property that each part has the same characteristic as the whole, is
gaining popularity in modelling natural structures and occurrences like mountain formation and
erosion, erratic wind directions, crystal structures and a lot more. Chaos, on the other hand,
examines superficial randomness originating from deterministic behavior. Weather conditions
are naturally chaotic, yet we have technologies that can already predict long-term patterns of
the weather conditions, be it in global or in localized settings.
Isaac Newton once said “What we know is a drop, what we do not know is an ocean”.
While mathematics remains in its full strength to uncover the mysteries of the universe, it is the
human intellect that is constantly challenged to exercise mathematics to discover the patterns,
learn from them and control the future by manipulating them. Every individual should then have
a focused mind to really embrace the abstract yet powerful nature of mathematics to use
patterns to nurture, advance and protect the universe that is entrusted to us by One supreme
being.
Exercise 1.1.1. Provide concise answers (maximum of 5 sentences) to the following questions.
1. To address the problem of traffic in a big city, several roads (assume straight) are being
constructed. It was noted that two roads will have at most one junction, three roads will
have at most three junctions, and so on.
a. Identify the pattern on the maximum number of junctions for each given number of
straight roads.
b. At most how many junctions are expected to be constructed if there will be seven
roads in the city?
2. Why are numbers important in our life? Explain your answer.
3. Other than those mentioned in the text, enumerate five (5) situations where you have
to use numbers.
4. Other than those mentioned in the text, describe three other patterns of shapes that
you see in nature. Discuss briefly the “mathematics” behind such patterns.
5. Identify and describe certain patterns (at least one) that you observe in your locality or
within your nearby environment.
Exercise 1.1.2. For this exercise, please follow the guidelines below:
 Form groups of 4-5 members, assign a leader
 Each member is to write an essay about the patterns (one for numeric patterns, another
for geometric patterns) that he/she has personal knowledge or experience.
 Each member will present his/her output before the members of the group
 Designated leader will synthesize the presentations of the members and present it
before the entire class
Assignment: For this assignment, it is assumed that you have already read and understood the
following readings:
- Nature’s Numbers by Ian Stewart, Chapters 1-4
Other than this, you need to watch the following video clips:
- Nature by Numbers (https://www.youtube.com/watch?v=kkGeOWYOFoA)
- One episode of the American crime drama television series Numbers
(https://www.youtube.com/channel/UCm0UJByEQHfKVnNFMXg9R1Q)
--------------------------------------------------------------------------------------------------------------------------1. (3) Things that I significantly learned from the readings/video clips
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2. (3) Things that are still unclear to me
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3. Complete the statement: I used to think that….
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4. (3) questions that I want to ask about the readings/video clips
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References
A) Readings
Adam, John A. Mathematics in Nature: Modeling Patterns in the Natural World,
Princeton University, 2003.
Manlapat, Ricardo. Mathematical Ideas in Early Philippine Society. Philippine Studies,
Ateneo de Manila Universirty, 2009.
Stewart, Ian A. Nature’s Numbers: The Unreal Reality of Mathematics, Basic
Mathematics Books, 1999.
B) Video Clips
Nature by Numbers (https://www.youtube.com/watch?v=kkGeOWYOFoA)
Painted with Numbers: Mathematical Patterns in Nature
(https://www.youtube.com/watch?v=07x3LBWn-Ao)
One episode of the American crime drama television series Numbers
(https://www.youtube.com/channel/UCm0UJByEQHfKVnNFMXg9R1Q)
Section 1.2. Modelling the Patterns to Address Issues in Nature and in the World
Patterns make up the entire universe, and everything in it (both static and dynamic
forms) should be the subject of inquiry of every mathematician. Over time, mathematics has
triumphantly organized these patterns allowing the human intellect to understand the order and
system by which the world operates and then made inferences out of these patterns to predict
the behavior of nature as well as other phenomena in the world. But more than understanding
these patterns, mathematics, this time through the arms of the applied and allied sciences, has
allowed man to manipulate and control such patterns either to derail what can be detrimental
occurrences in nature and in the world or hasten those that may appear as beneficial to the
human race. This is another nature of mathematics, it sets the systems and processes by which
man can understand and predict the behavior of and phenomena in nature and the world,
thereby controlling the predicted results for his or her favor.
This section journeys into how mathematicians have defined the course of scientific
inquiry through a comprehensive and intensive treatment of the patterns that occur in nature
and in the world.
Intended Learning Outcomes
At the end of the section, students should be able to:
1. Explain the significance of certain types of patterns;
2. Discuss how prediction is done by examining certain types of patterns; and
3. Demonstrate how certain patterns can be controlled to achieve desired results.
Diagnostics:
 Learning Checkpoint
Instruction: Agree or Disagree. If you think the statement is correct, write AGREE, otherwise
write DISAGREE.
_________ 1. Predictions as results of studying patterns do not necessarily yield precise results.
_________ 2. Mathematical models are simulations of patterns and are subject to error.
_________ 3. Mathematical models are not absolute truths.
_________ 4. Patterns allow mathematicians and scientists to control nature and the world.
_________ 5. Natural occurrences are random events and cannot be modelled by certain
designs.
Modelling the Patterns to Address Issues in Nature and in the World
As conveyed in the previous section, the universe is made up of a myriad of patterns, each
seemingly concealing a mystery about life and the entire creation. More than evoking
imagination for their discovery, patterns in nature and in the world continue to challenge the
human intellect for them to be understood and exploited for life’s preservation and
development. Arguably, mathematics has been the primary tool by which these patterns are
recognized, organized and operationalized.
The study of patterns dates back as early as the first known human civilization, at a time
when “number” was even a totally strange concept. For example, by taking records of the moon’s
shape and color, man could ascertain when is the best time to start a new cropping season.
Sophistication due to the concepts of numbers and shapes may have significantly changed the
game over time, but investigation of, and ultimately benefitting from patterns has always been
man’s retort towards natural as well as man-made phenomena.
Perhaps the Golden ratio is one of the biggest breakthroughs as far as numeric and
geometric patterns are concerned. Here, we take a line segment of length 𝐿. According to Euclid
(as written in his book “Elements”), it is always possible to divide the segment into two parts with
corresponding lengths 𝐿1 and 𝐿2 , where 𝐿1 > 𝐿2 (specifically, 𝐿1 and 𝐿2 are called the Golden
sections of 𝐿), such that the following equation holds:
𝐿
𝐿1
= .
𝐿1 𝐿2
The common value, which in this case is a constant (an irrational), is what we consider as the
Golden ratio, denoted as Φ ≈ 1.618 (actual value is non-terminating, non-repeating decimal
number). To show this, we re-write the equation as
𝐿1 + 𝐿2 𝐿1
=
𝐿1
𝐿2
If the common ratio
𝐿1
𝐿2
= Φ, then we have
Φ=
𝐿1 + 𝐿2
𝐿2
1
= 1+ = 1+
𝐿1
𝐿1
Φ
Consequently,
Φ2 − Φ − 1 = 0.
Using the quadratic formula, we obtain
Φ=
1 ± √5
2
In this case, Φ ≈ 1.618 or Φ ≈ −0.618. Based from the construction of 𝐿1 and 𝐿2 , we should
retain the positive value of Φ. Meanwhile,
1
Φ=1+
Φ
generates the “continued” fraction form
1
Φ=
1
1+
1+
1
1
1+1+⋯
In addition, from the equation Φ2 − Φ − 1 = 0, we also get
Φ2 = 1 + Φ
Φ = √1 + Φ
resulting in the “infinite” radical
Φ = √1 + √1 + √1 + √1 + ⋯
Perhaps, the most intriguing occurrence of the Golden ratio is with the Fibonacci
numbers. If we formalize the notation of the Fibonacci numbers as F1 = 1, F2 = 1, F3 = 2, F4 = 3, F5
= 5, F6 = 8, F7 = 13, F8 = 21, F9 = 34, F10 = 55, … , we can observe that
𝐹𝑛
𝐹
34
For example, 𝐹 9 = 21 ≈ 1.619 while
10
𝐹10
𝐹9
𝐹𝑛−1
55
→ Φ.
= 34 ≈ 1.618. This result allows for the determination
of any Fibonacci number without using the classical recursive formula 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 .
Φ𝑛
𝐹𝑛 =
√5
So, what has Φ now got to do with nature and the world?
The geometric construction that initiated the conception of the Golden ratio inspired
similar constructions which still led to the same value of Φ. The use of Golden triangles and
Golden rectangles paved the way for more practical value of the constant.
In human anatomy, Φ is exhibited in both the human physique and facial dimensions.
Examples are: (i) the width of the face forms a Golden section of the length of the face; (ii) the
width of the nose is a Golden section of the length of the mouth; (iii) the little finger is a Golden
section of the middle finger; (iv) the human lung is divided into sections based from the Golden
ratio; (v) the eardrum consists of chambers that are located at approximately Golden ratio to
optimize sound regulation and vibration; (vi) the ratio of systolic and diastolic pressure in the
blood pressure is ideally 1.6, a close approximation of the Golden ratio; (vii) the helix spirals of
the DNA molecule is configured at 34 angstroms and 21 angstroms, two consecutive Fibonacci
numbers whose ratio is close to Φ; and a lot more cases. All these are used to address and resolve
sensory impairments as well as other medical problems.
In architecture and design, the Golden ratio is not only used for aesthetic and visual
sensation but it has become a basis for structural stability in the construction of tall buildings and
similar large edifices. Examples of these are the pyramids of the ancient Egyptians, the Parthenon
temple of the Greeks, the Notre Dame in Paris, the CN Tower in Toronto, and many other
architectural structures.
Phi also paved the way for two- and three-dimensional spaces to be filled by two shapes
using five-fold symmetry (the so-called Penrose tiles). This rare configuration ushered more
advanced discoveries in crystal formations like those in sugar, salt and diamonds. In particular,
the idea of the five-fold symmetry facilitated the discovery of quasicrystals usually found in
aluminum-manganese alloy.
The Golden ratio is just one of the countless testaments that mathematics is all around
us, and it has shaped the way we understand nature and the world. In 2013, mathematician and
science author Ian Stewart published a book on "17 equations that changed the world" which
admittedly changed the way people look at mathematics.
First is the Pythagorean Theorem which established the way we study geometry. In
particular, this theorem, which specifies that in a right triangle the sum of the squares of the
lengths of the two shorter legs is equal to the square of the length of the longest side, drew the
line between Euclidean (flat-surface) geometry and non-Euclidean (curved-surface) geometries.
For example, right triangles drawn on the surface of a sphere does not necessarily follow the
Pythagorean theorem.
Second, there was the concept of logarithm, commonly defined as the reverse process of
exponentials. That is, we say that 𝑙𝑜𝑔𝑏 𝑁 = 𝑥 if and only if 𝑏 𝑥 = 𝑁. This concept facilitated an
easy way to shift multiplication into addition (as well as division into subtraction) with the
𝑀
property that 𝑙𝑜𝑔𝑏 𝑀𝑁 = 𝑙𝑜𝑔𝑏 𝑀 + 𝑙𝑜𝑔𝑏 𝑁 (similarly, 𝑙𝑜𝑔𝑏 𝑁 = 𝑙𝑜𝑔𝑏 𝑀 − 𝑙𝑜𝑔𝑏 𝑁). This concept of
logarithm was extensively and comprehensively used in the areas of physics, astronomy and
engineering.
Then there was Calculus – Differential and Integral, which focused on studying the
patterns of how quantities change in relation to other factors. The derivative, for example,
measures how one quantity changes as a related factor changes, like velocity being the derivative
of position with respect to time. In simple language, calculus became a powerful tool in
supplementing the tools of the basic geometric and algebraic processes. For example, the
concept of slope defined only for lines in (analytic) geometry has been extended to slope of any
curve using the derivative, while the computation of areas originally applied to regions bounded
by straight lines has been extended to areas bounded by curves using the concept of integrals.
In the allied field of Physics, the most celebrated
𝑚 𝑚
discovery is the equation 𝐹 = 𝐺 × 1𝑟 2 , known as the Law
of Gravity by Newton. Here, 𝐹 is the force of gravity
between the two objects, 𝐺 is a universal constant, 𝑚1 and
𝑚2 are the masses of the objects, and 𝑟 is the distance
between the two objects. This equation served as a model
Source:
for understanding the solar system, the orbit of the planets as well
http://www.alevelphysicstutor.com/fieldas other motion patterns as influenced by gravity. It took 200 gravit-1.php
years for another theory to come out, thanks to yet another
brilliant physicist Albert Einstein through his Theory of Relativity (the special case of which takes
the form 𝐸 = 𝑚𝑐 2 , E for energy, m for mass and c for the speed of light).
As the other fields progressed along with new theories in mathematics, mathematicians
continued to pursue and achieve new and greater results. Perhaps the most unprecedented
finding came with the discovery of the complex numbers. Inspired by the solution of the equation
𝑥 2 + 1 = 0, the imaginary number 𝑖 is defined as the square root of -1, that is, 𝑖 = √−1 or 𝑖 2 =
−1. In general, a complex number takes the form 𝑎 + 𝑏𝑖 where 𝑎 is called the real part while 𝑏
is the imaginary part. The complex numbers 2 + 𝑖 and 2 − 𝑖 (aptly called conjugates) comprise
the roots of the quadratic equation 𝑥 2 − 4𝑥 + 5 = 0. When used in calculus, the complex
numbers form an essential system in the areas of electronics and signal processing.
In statistics, the ideal model to use in describing the distribution of a large group of
independent processes is the normal distribution, also known bell-shape distribution or Gaussian
distribution. The normal curve is typically used to model experiments in both social and scientific
researches.
The readers are encouraged to study the other equations which were not mentioned in
this section.
Exercise 1.2.1. Provide concise answers (maximum of 5 sentences) to the following questions.
1. Give an example of a pattern that occurs in nature or in the world that does not involve
numbers.
2. Use the Golden ratio to divide the line segment L with length 12 units into two segments
𝐿1 and 𝐿2 . Specify the lengths rounded off to two decimal places.
3. Construct a logarithmic spiral using Golden triangles with isosceles side of length 3 units.
4. Construct a logarithmic spiral using Golden rectangles with sides measuring 4 units.
5. Assess the symmetry of your body measurements by computing the phi of your body.
Do the same with your facial measurements.
Exercise 1.2.2. For this exercise, please follow the guidelines below:
 Form groups of 4-5 members, assign a leader
 Each member is to write a mathematical vignette about “The 17 equations that changed
the World” Ian Stewart
 Each member will present his/her output before the members of the group
 Designated leader will synthesize the presentations of the members and present it
before the entire class
Assignment: For this assignment, it is assumed that you have already read and understood the
following readings:
- Nature’s Numbers by Ian Stewart, Chapters 1-4
Other than this, you need to watch the following video clips:
- Nature by Numbers (https://www.youtube.com/watch?v=kkGeOWYOFoA)
- One episode of the American crime drama television series Numbers
(https://www.youtube.com/channel/UCm0UJByEQHfKVnNFMXg9R1Q)
--------------------------------------------------------------------------------------------------------------------------a. (3) Things that I significantly learned from the readings/video clips
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
b. (3) Things that are still unclear to me
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
c. Complete the statement: I used to think that….
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
d. (3) questions that I want to ask about the readings/video clips
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References
A) Readings
Adam, John A. Mathematics in Nature: Modeling Patterns in the Natural World,
Princeton University, 2003.
Manlapat, Ricardo. Mathematical Ideas in Early Philippine Society. Philippine Studies,
Ateneo de Manila Universirty, 2009.
Stewart, Ian A. Nature’s Numbers: The Unreal Reality of Mathematics, Basic
Mathematics Books, 1999.
B) Video Clips
Nature by Numbers (https://www.youtube.com/watch?v=kkGeOWYOFoA)
One episode of the American crime drama television series Numbers
(https://www.youtube.com/channel/UCm0UJByEQHfKVnNFMXg9R1Q)
Section 1.3. The Indispensability of Mathematics
The German philosopher Immanuel Kant once said, “Mathematics is the indispensable
instrument of all physical researches.” A generation later, the post-Kantian philosopher and
educator Johann Friedrich Herbart also argued that “mathematics wards off the dangers of
philosophy” and that “it is not only possible, but necessary that mathematics be applied to
psychology”. With the unfolding of time, the value of mathematics gradually became more and
more appreciated even in the arts and social sciences. The versatility and potency of mathematics
as a powerful tool is another nature that is worth examining.
This section takes a survey of the link that mathematics has established with other fields,
its diverse applications in as far as patterns that exists in nature and in the world are concerned.
Intended Learning Outcomes
At the end of the section, students should be able to:
1. Discuss the usefulness of mathematics in one specific field;
2. Explain why mathematics is the proper tool for exploring nature and the world; and
3. Identify current issues or problems concerning human behavior, socio-political,
economic, technological or scientific in nature, and explain how mathematics can be
used to resolve them.
Diagnostics:
 Learning Checkpoint
Instruction: Agree or Disagree. If you think the statement is correct, write AGREE, otherwise
write DISAGREE.
_________ 1. Mathematical models are never accurate.
_________ 2. Theorems in mathematics are absolute truths which can stand usability over
time.
_________ 3. There is no aspect of human and social affair that does not involve mathematical
reasoning.
_________ 4. Mathematics is important in the arts and social sciences as it is in the physical
sciences.
_________ 5. The value of a mathematical concept or model is measured by its practical
benefits.
The Indispensability of Mathematics
Perhaps the most important nature of mathematics is its relevance to a broad spectrum
of human concerns. As usually said, knowledge is futile if it is not applied to daily life situations.
Inasmuch as mathematics is a rich body of knowledge, its practical value which transcends all
fields of human endeavor, even elevates itself from the rest of the other disciplines. With its
distinctive use of abstraction consequently formulating advanced natural concepts, the areas in
which they can be applied have been expanding at an unexpected pace.
The allied field of physics is one of the major recipients of these mathematical advances.
From Kepler’s discovery of planetary orbital periods, to Newton’s Law of Gravity, to Einstein’s
Theory of Relativity, to Charle’s Law of the expansion of gases, to Schrodinger’s equation of
quantum mechanics, and the list goes on, thereby naturally giving birth to new specialized areas
of study. What makes this work is the fact that physical models are formulated in the language
of mathematics, specifically through differential equations.
Take the case of the wireless technology that has widely influenced almost all aspects of
human concerns. It has paved the way for information to be accessible by everyone at the least
possible time even at the farthest possible place in the universe.
The theory behind the wireless technology can be traced from vibrating strings like those
that are normally seen in musical instruments like guitars and violins. It was the ancient Greeks
who are believed to have pioneered the inquiry about the relation between a string and the
sound it produces. They observed the sinusoidal behavior of the tones that the strings produce,
paving for the formulation of the concepts of frequency and amplitude of vibrations – pitch of a
tone is determined by the frequency while loudness is determined by the amplitude. So there
came Taylor’s book on vibrational mechanics, further enriched by d’Alembert’s concept of
dynamic sinusoidal waves, and eventually Euler’s formulation of “wave equation” for strings.
With Einstein’s relativity theory, the wave equations got liberated from the confines of
musical domain and established for itself the central focus of what later became the area of
mathematical physics. As it continued to evolve, the concepts of electricity and magnetism
blossomed, and this changed the historical course of man.
It was Michael Faraday who merged the theories of electricity and magnetism as a single
entity called electromagnetism. Later on, the theory of electromagnetism is described
mathematically by his successor James Clerk Maxwell by a system of partial differential equations
known widely as Maxwell’s equations.
With the help of Maxwell’s equations, humanity made a giant leap from vibrating strings
to wireless technology. Working with these equations, man discovered the possibility for
electromagnetic waves to travel anywhere at the speed of light. It was the German physicists
Heinrich Hertz (1886) who discovered electromagnetic waves –at the frequency we now know as
radio. Then Guglielmo Marconi successfully carried out the first wireless telegraphy in 1895 and
transmitted and received the first transatlantic radio signals in 1901. It did not take too long for
other technological innovations to follow that eventually opened doors for humanity into the
new world of digital universe. So, one must realize that whatever device or equipment that has
something to do with wireless communication can be traced to the concept of string vibrations.
Aside from physics, chemistry is another area that has established enduring and
expanding interdependence with mathematics. Algebra and calculus are typical tools in reaction
kinetics like balancing chemical reactions as well as in mixture problems, graph theory is used to
model molecular bonds, group theory is applied to study crystal structures, linear algebra and
matrix theory are used to characterize molecules when they change from one energy state to
another.
The Michaelis-Menten equation is perhaps one of the simplest and best-known models
in reaction kinetics (specifically, enzyme kinetics). For a product X with concentration C, the
reaction rate is described by the equation
𝑑𝑋
𝑉(𝐶)
=
𝑑𝑡 𝑀 + 𝐶
where 𝑉(𝐶) is the maximum rate achieved by the system and 𝑀 is the substrate constant (called
Michaelis constant). The model usually applies to biochemical reactions with a single substrate.
It has been employed in the treatment of some illnesses of the lungs (Yu et. al., 1997), in the
study of species richness (Keating and Quinn, 1998), in forensic investigations (Jones,2010) and
in bacteria control (Abedon, 2009).
Meanwhile, Arthur Cayley and J.J. Sylvester were credited to have pioneered the graphtheoretic approach of investigating some ideas in chemistry. In Chapter 6 of this book, we will
formalize the concept of graph, a kind of geometric structure that defy metric concepts like
distance and slope. One of the early discoveries of graph-theoretic treatment of chemistry is the
fact that molecules may exhibit different physical and chemical properties even if they have the
same number of carbon and hydrogen atoms. In the language of graph theory, the issue was
settled by the concept of “graph isomorphism”. The impact of the work of Cayley and Sylvester
saw the rise of other names in this area of mathematical chemistry: Harry Wiener, Alexandru
Balaban, Danail Bonchev, Ante Graovac, Ivan Gutman, Haruo Hosoya, Milan Randic to name a
few.
In biology, the role of mathematics is typically demonstrated in areas like (but not limited
to) biostatistics, bioscience, and medical research. Recently, mathematical biology is a new area
that has gained considerable attention from mathematicians. It focuses in developing new
models and carry out simulations of certain biological questions, usually at the cellular level
particularly those that has something to do with patterns in reaction-diffusion equations as well
as combinatorial problems concerning DNA-RNA configurations. The resulting practical values go
all the way to combating physical and mental illnesses like cancer and AIDS.
For example, in the two independent studies of Roy et. al. and Xue et. al., mathematical
models (using systems of partial differential equations) on the healing process of ischemic
wounds were formulated. Similarly, Friedman modeled the growth of cancer tumor that usually
occurs in fluid-like tissues like the mammary gland and the brain.
Appendix A provides a more extensive survey of how mathematics is used and applied in
various other fields. Readers are encouraged to read and re-read the material and conduct
further review of related materials available in the internet for more substantial details of the
concepts mentioned.
Exercise 1.3.1. Explain how mathematics is used in the following areas. Answer in not more
than five (5) sentences.
1. Physics
2. Chemistry
3. Biology
4. Engineering
5. Linguistics
Exercise 1.1.2. For this exercise, please follow the guidelines below:
 Form groups of 4-5 members, assign a leader
 Each member is to write a mathematical vignette about the application of mathematics
in any of the following areas: Music, Arts Designs and Architecture, Philosophy, Political
Science, and Psychology
 Each member will present his/her output before the members of the group
 Designated leader will synthesize the presentations of the members and present it
before the entire class
Assignment: For this assignment, you need to watch the following video clips:
- The Map of Mathematics
https://www.youtube.com/watch?v=OmJ-4B-mS-Y
-
Math is the hidden secret to understanding the world by Roger Antonsen
https://www.youtube.com/watch?v=ZQElzjCsl9o&t=921s
--------------------------------------------------------------------------------------------------------------------------1. (3) Things that I significantly learned from the video clips
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2. (3) Things that are still unclear to me
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3. Complete the statement: I used to think that….
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4. (3) questions that I want to ask about the video clips
___________________________________________________________________
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References
A) Readings
Adam, John A. Mathematics in Nature: Modeling Patterns in the Natural World,
Princeton University, 2003.
Manlapat, Ricardo. Mathematical Ideas in Early Philippine Society. Philippine Studies,
Ateneo de Manila Universirty, 2009.
Stewart, Ian A. Nature’s Numbers: The Unreal Reality of Mathematics, Basic
Mathematics Books, 1999.
B) Video Clips
-
The Map of Mathematics
https://www.youtube.com/watch?v=OmJ-4B-mS-Y
-
Math is the hidden secret to understanding the world by Roger Antonsen
https://www.youtube.com/watch?v=ZQElzjCsl9o&t=921s