Mathematics in the Modern World
GEC 14 Teachers
CHAPTER 0
The Nature of Mathematics
Mathematics may be undeniably commonplace in everyday life, yet it
receives little to no appreciation from most students. Although it is a
dreaded subject for many, it is required in university entrance examinations, in IQ tests during job applications and even in qualifying examinations for government service. This shows that aptitude in mathematics
is seen as a key skill affecting our capabilities as students, employees or
even in nation building. But is mathematics really as how many students
would think it is – “Mental Abuse To Humans”? In this preliminary chapter, we wish to re-introduce and re-imagine mathematics in a different
light. We will give emphasis on why mathematics is, instead, a useful
tool to understand our everyday existence better. More particularly, at
the end of this chapter, students are expected to:
(1) Identify patterns in nature and regularities in the world;
(2) Articulate the importance of mathematics in one’s life;
(3) Argue about the nature of mathematics, what it is, how it is expressed,
represented and used; and
(4) Express appreciation for mathematics as a human endeavor.
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1
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What is Mathematics?
Do you love Mathematics? One can bet this question would most likely
receive a big cringe and a loud “NO”. But it is said that “We can only
love those which we understand.” Thus, in order to appreciate mathematics
more, it is imperative that we understand first its very nature and learn
what mathematics really is.
If we Google the word mathematics, we can expect over half a billion of
results. That is quite daunting to start with. Thus, let us just examine
these three descriptions of mathematics:
(1) “Mathematics is the science of numbers, quantities and shapes and the
relations between them.” (Merriam-Webster Dictionary)
(2) “Mathematics, developed by human mind and culture, is a formal system of thought for recognizing, classifying, and exploiting patterns.” (Ian
Stewart,)
(3) “Mathematics is the alphabet with which God has written the Universe.”
(Galileo Galilei)
The first description reveals how many of us, especially students, view
mathematics – a purely technical and scientific field that only talks about
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abstract notions such as numbers, measurements, shapes, as well as operations and relationships of these concepts. While it is true that mathematics tackles these notions, there is more to it that we missed to learn or
our teacher may have missed to mention inside the classroom. We will
see and discuss this in the next two descriptions.
Ian Stewart, the author of the book “Nature’s Numbers”, implied that mathematics is invented. Just like the languages that we speak, mathematics is often viewed to be purely a product of human intellect, a means that
we developed and consistently use in order to communicate, make sense
and understand various phenomena happening around us. Being “manmade”, mathematics can therefore also be regarded as an art. Meanwhile,
the great scientist Galileo Galilei believes otherwise. For him, mathematics is already existent in the universe way long before people first
attempted to explore into it. That is, mathematics is EVERYWHERE as
an integral part of nature and we, mankind, just discovered it. In particular, the patterns that are observable in living and non-living objects are
already there even before we have detected and identified it as a regularity.
What do you think, is mathematics DISCOVERED or INVENTED?
We can sufficiently describe mathematics as follows:
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(1) Mathematics is the study of patterns. Needless to say, mathematics
includes the study of the existence and relationships of patterns in
the universe we live in. The discovery of the value of π came from
the observations on the ratios of the diameter and circumference
of circles. Mathematical formulas such as Einstein’s E = mc2 and
Pythagoras’s a2 + b2 = c2 also came about from recurrences in the
relationship of energy with mass and speed of light, and of the three
sides of right triangles, respectively. In fact, every mathematical
formula describes a pattern of relationships governing the elements
involved in that formula. We will discuss more of patterns in the
succeeding part of this module.
(2) Mathematics is a language. Using the layman’s terms and language to describe certain phenomena may be tedious and a bit of
a challenge. In watching a Korean TV series, understanding the
story will be difficult if there will be no subtitles. Imagine how it
would be much easier if we speak Korean! Mathematics also plays
the same role as the subtitles or the Korean language itself. We need
to understand how equations are written, how operations are executed and applied, and how the observed patterns are analyzed
and generalized. Indeed, mathematics is a universal language used
to express and communicate our observations about the universe
we live in. Later in this module, we will discuss certain rules of
expressing mathematical sentences and phrases into symbols and
the conventions in using definitions and variables in writing these
mathematical expressions.
(3) Mathematics is an art. Google defined art as “the expression or application of human creative skill and imagination ... producing works to be
appreciated primarily for their beauty or emotional power”. Majority of
published works in pure mathematics emanate from this characteristic of mathematics as an art. These outputs are conceptual ideas
from the mind of the mathematician and may or may not find application in the physical world. A mathematician has an endless trail
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of what-ifs, one idea branching out after another. This creative and
imaginative feature of mathematics is what makes it all the more
beautiful and worthy of appreciation. Also, we are unknowingly
making use of mathematics in our aesthetics. The rule of thirds in
photography, minimalism in fashion and symmetry in visual arts –
all of these are certainly in the scope of mathematics.
(4) Mathematics is a set of problem solving tools. We typically reach
into our toolboxes whenever we need to fix or build something.
Tools are certainly important in solving various problems that we
encounter. If we need to make generalizations from a data set obtained from a certain population, we make use of statistical tools. If
we need to build something with accuracy and precision, we need
our measuring tools. If we need to create a computer program to
make a process more convenient, we reach for our logical and computational tools. This role of mathematics as a tool primarily provides the greatest impacts and advantages to the general community. In solving certain problems, mathematics is helpful for organizing ideas, objects and data sets into a more meaningful form, for
predicting possible outcomes of certain events that may affect our
lives, and for controlling these outcomes so that the effects become
more favorable to us. This is why it is very important for all of us to
gain, enhance, and master our mathematical skills.
(5) Mathematics is a process of thinking. Logical reasoning and critical thinking are characteristics of mathematical thought. They are
most useful nowadays in the presence of the internet accompanied
by “information explosion”, where contents in social media are quite
overflowing, yet some remain unverified or are outright fake. Engaging in mathematics disciplines the mind to think rationally and
analytically so as not to readily accept information presented to us
without sound bases and irrefutable proofs.
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Mathematics is indeed everywhere and for everybody. We need not be
wary and get anxious in learning about and dealing with the mathematical variables x and y. All we need is to realize first that there is a significant reason why mathematics is inescapable and to see with wider eyes
and open minds its importance in our everyday existence.
2
Mathematics: A Study of Patterns
Patterns are regular, repeated and recurring forms or designs. When objects and phenomena occur in particular patterns, the observers are able
to anticipate what they might see or what happens next as the pattern
shows what may have come before. This is proven to be beneficial even
to ancient civilizations. The regularity of the flooding of the Nile River enabled the Egyptians to time the planting of their crops. Navigators used
the constellations and other astronomical patterns to arrive at their intended destinations. Development in medicine is backed by conducting
repeated tests and observing recurring positive results.
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Hence, patterns are important because they are used to analyze and solve
problems, and help individuals in understanding the world better by predicting what comes next, imagining what came before, and estimating
outcomes when some variables and conditions are changed.
In this section, we will look into different patterns that we will most likely
encounter everyday and in our mathematics sessions.
2.1
Geometric Patterns
When we talk about patterns, what usually comes to mind first are visual
arrangements of points, lines and shapes in space. These are called geometric patterns. A geometric pattern is a motif or design that depicts
abstract shapes such as lines, polygons, and circles, and typically repeats
like a wallpaper. The following pictures depict geometric patterns in our
surroundings:
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2.1.1
GEC 14 Teachers
Symmetry
An important element in geometric pattern is symmetry. The word symmetry came from the Greek word symmetria which means “agreement in
dimensions, due proportion, arrangement” and aptly so, it refers to “harmony, proportion and balance” when used in layman’s terms. However,
symmetry may have specific usage in different fields. In natural sciences,
symmetry may mean invariance and lack of change in a given physical
system (physics) or may mean regularity in the body shapes of organisms
(biology). In geometry, a shape is described as symmetric if the elements
in the object are transformed through movements or rearrangements of
these elements, but the overall shape remains unchanged. There are several
types of symmetries according to the transformation in the object:
(1) Reflectional Symmetry – also known as mirror symmetry, exists when
we can draw a line separating two portions of the object which are
mirror images of each other.
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(2) Rotational Symmetry – observed when an object can be rotated
about a fixed point and the shape of the object remain unchanged.
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(3) Translational Symmetry – exists if every point of the object can
be moved, or translated, on the space without changing its overall
shape.
(4) Scale Symmetry – also known as dilation, occurs when the object’s
shape remain unchanged when it is expanded or contracted.
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(5) Combined Types of Symmetry – when the object possesses two or
more types of symmetries.
Helical symmetry – occurs when the object undergoes translation
and rotation at the same time;
Glide reflection – occurs when a reflection is followed by a translation; and
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Rotoreflection – occurs when the object undergoes rotation and reflection.
Some geometric artworks made use a combination of these symmetric
transformations. When an arrangement of shapes are in such a way that
they are closely fitted together, especially of polygons in a repeated pattern without gaps or overlapping, what results is a tessellation. Common
examples of tessellations can be seen in floor tiles and fabric patterns.
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Fractals are infinitely complex patterns that are self-similar across different
scales. They are created by repeating a simple process over and over in
an ongoing feedback loop. It is a characteristic of a fractal that sections
in different levels of the object resembles its overall shape; hence, it is
sometimes classified under scale symmetry.
Read more about tessellations and fractals by exploring the following
web addresses:
(1) http://www.funmaths.com/fun_math_projects/what_is_a_tessellation.htm
(2) https://fractalfoundation.org/resources/what-are-fractals
2.2
Number Patterns
In many cases, we can observe some patterns from an array of quantities
or numbers. These patterns may be represented as number sequences or
through mathematical equations.
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2.2.1
GEC 14 Teachers
Sequences
The patterns in numbers are often written as sequences. A number sequence is an infinite progression of numbers following a particular pattern usually called an “nth term formula” as it tells us the “form” of the
specific number occupying the nth position (called the nth term) in the
sequence. A number sequence may be represented by writing its first
few terms separated by commas and followed by an ellipsis (...) when the
pattern is quite obvious and is often denoted as a1 , a2 , a3 , ... where a1 is
the first term, a2 is the second term and so on, or by writing its nth term
formula inside a pair of braces usually denoted as { an }.
Example 1. Consider the sequence 1, 4, 9, 16, 25, .... We can immediately
detect that the first five terms of the sequence are squares of the numbers
1, 2, 3, 4 and 5, respectively. That is, a1 = 1 = 12 , a2 = 4 = 22 , a3 = 9 = 32
and so on. Thus, we can predict that the sixth term is a6 = 62 = 36. This
pattern is equivalent to {n2 } in terms of its nth term formula.
Example 2. In the sequence { n
2 +n
2 }, the first term a1 is obtained by having
12 + 1
n = 1. That is, a1 = 2 = 1. In the same manner, the 34th term of this
2
sequence is a34 = 34 2+34 = 595.
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In some sequences, we need to determine first the preceding terms before
we can compute for the next term. If the terms of the sequence depend
on the values of the initial terms, this sequence is a recursively defined
sequence. Take a look at the following example:
Example 3. If { an } is the sequence where a1 = 1 and an = 2an−1 + 3, we
will see that:
a2 = 2a1 + 3 = 2(1) + 3 = 5,
a3 = 2a2 + 3 = 2(5) + 3 = 13,
a4 = 2a3 + 3 = 2(13) + 3 = 29,
and so on. Here, we cannot determine a5 unless we compute for the value
of a4 ;hence, this sequence is a recursively defined sequence.
2.2.2
Mathematical Formulas
A mathematical formula is a generalization of relationships existing between different quantities. It is simply a fact or a sort of a rule that is
typically represented using mathematical variables like x and y, and reR
lational or operational symbols like =, ×, and . The previously given
equations E = mc2 and a2 + b2 = c2 are some known examples introduced to us while learning physics and mathematics. But there are more
examples of mathematical formulas that are used everyday even by common people (even those who say they hate maths!). Take a look at some
of them in Example 4 and see if you can add more.
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Example 4. Here are some mathematical formulas we use regularly (sometimes, unknowingly) in real-life situations:
(1) I = Prt, where I is the simple interest for a loaned or invested
amount, P is the principal amount, r is the interest rate and t is the
duration of the loan or investment;
(2) W = ( R p − R0 )C, where W is the amount of the monthly water bill,
R p is the present month’s water meter reading, R0 is the previous
month’s water meter reading and C is the cost of water per cubic
meter; and

m,
d ≤ 4 km
F (d) =
, where F (d) is the amount
m + add − 4 kme d > 4 km}
in pesos of jeepney fare for a distance travelled d in kilometers, m
is the minimum fare for the first four kilometers and a is the added
amount per kilometer in excess of the 4 kilometers travelled. The
function d x e is called the ceiling function, which returns the value
of x if x is a whole number, and returns the next higher integer of x
whenever x contains a decimal. In this case d4e = 4 and d4.2e = 5.
Hence, if the minimum fare is Php 7.00 for the first four kilometers with additional Php 2.00 for every succeeding kilometer, and
you travelled 6.3 km, then your fare will be computed as follows:
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F (6.5 km) = Php7.00 + Php2.00(d6.3 − 4e)
= Php7.00 + Php2.00(d2.3e)
= Php7.00 + Php2.00 (3)
= Php7.00 + Php6.00
= Php13.00.
We must realize that even though numbers and mathematical formulas
are abstract concepts, their applications extend to our day-to-day activities. In the next section, we will dig deeper into a more subtle but extremely delightful interconnection of mathematics and the patterns in nature.
3
Fibonacci Sequence and the Golden Ratio
One of the most famous number sequences in mathematics is the Fibonacci sequence, named after the famous Italian mathematician Fibonacci,
who was also known as Leonardo of Pisa. In this module, we adopt the
special notation { Fn } for the Fibonacci sequence, where F1 = 1, F2 = 1
=
Fn−1 + Fn−2
F3 = F2 + F1 = 1 + 1 = 2
F4 = F3 + F2 = 2 + 1 = 3
F5 = F4 + F3 = 3 + 2 = 5
..
.
and
Fn
for
n
≥
3.
That
is,
In this form, the Fibonacci sequence is recursively defined where first ten
terms of the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34,and 55. Where can we
observe this sequence?
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3.1
GEC 14 Teachers
The Golden Spiral
We will now observe the Fibonacci sequence in terms of squares. Take a
square of side 1 unit (F1 ) and from this, extend a second square of side
1 unit (F2 ). Then, from the combined sides of the two squares, extend a
third square of side 2 units (F3 ). It would look like this:
From the combined sides of the second and third squares, extend another
square of side 3 units (F4 ). Continuing in this manner, we can obtain a sequence of squares, called the Fibonacci squares, whose sides correspond
to the terms of the Fibonacci sequence as can be seen in the following
image:
Now, in each square, draw an arc from one vertex to another using the
side as the radius as follows:
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This beautiful curve is called the golden spiral. This spiral can be observed aplenty in nature – in the minuteness of shells and fingerprints, in
arrangement of petals and whorls of vines, and even up to the formation
of typhoons and of galaxies.
This apparent pattern speak of the handiwork shown in the masterpiece
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of one perfect Divine Creator.
3.2
The Golden Ratio
One important number derived from the Fibonacci sequence is the golden
ratio. Recall the first few terms of the sequence { Fn }which are 1, 1, 2, 3, 5, 8, 13, 21, 34
and 55. Observe that if we divide compute for the ratio Fn /Fn−1 of two
succeeding terms, we will get the following values:
F2 /F1 = 1/1 = 1
F7 /F6 = 13/8 = 1.625
F3 /F2 = 2/1 = 2
F8 /F7 = 21/13 = 1.6153
F4 /F3 = 3/2 = 1.5
F9 /F8 = 34/21 = 1.6190
F5 /F4 = 5/3 = 1.6667
F10 /F9 = 55/34 = 1.6176
F6 /F5 = 8/5 = 1.6
F11 /F10 = 89/55 =1.6182
Continuing this process as n gets larger and larger, we will see that this
ratio gets closer and closer to the value ϕ = 1.6180339887.... This value
is known as the golden ratio. This constant exists when a line is divided
into two parts A and B and then the longer part A when divided by the
smaller part B is equal to the quotient when A + B is divided by A. This
only happens when both of the ratios A/B and ( A + B)/A is equal to
ϕ = 1.6180339887.... By using intermediate
mathematical calculations,
√
the value of ϕ is exactly given by ϕ =
1+ 5
2 .
How then is this inconspic-
uous number so significant and even more wondrous?
The golden ratio is much prevalent in design and aesthetics. Artists and
designers use this proportion to place objects into their design, although
some may be unaware while they are doing so. Many famous ancient
works of art contain this design element such as Leonardo da Vinci’s
Mona Lisa and the Parthenon. Product logos are also deliberately crafted
using the golden ratio. Women groom their eyebrows with this divine
proportion in mind.
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We need not look far to spot the existence of the golden ratio. The lengths
of the bones in our arms seem to satisfy this proportion as well – from the
phalanges, the carpals, up to the radius and ulna.
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To learn more of the Fibonacci sequence, watch the videos found in the
following web addresses:
(1) Nature by Numbers by Cristobal Vila, https://www.youtube.com/watch?v=kkG
(2) Gods Fingerprint – The Fibonacci Sequence - Golden Ratio and The Fractal
Nature of Reality, https://www.youtube.com/watch?v=4VrcO6JaMrM
(3) The Golden Ratio and Fibonacci Sequence in Music (feat. It’s Okay to be
Smart), https://www.youtube.com/watch?v=9mozmHgg9Sk
With this apparent link of mathematics and nature in the form of patterns,
one cannot deny that mathematics is part of us, whether we like it or not.
Opening our eyes to this exquisite interrelationship may be a good start to
fully embrace and appreciate the significance of mathematics in our daily
lives. It is therefore the goal – and the hope – of this course, GEC 14 Mathematics in the Modern World, to build up our understanding about
the nature of mathematics and, at the same time, to make more vivid the
presence of mathematics in nature in order to harness the full advantageous
potential of mathematics into our existence.
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