CHAPTER 1: The Nature of Mathematics
Daniel Bezalel A. Garcia
Instructor I, Pangasinan State University - Urdaneta City Campus
September, 2020
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
1
Introduction
Introduction
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
2
Introduction
Introduction
The coming of the ”Digital Age” contributes to our consumption
and production of data.
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
3
Introduction
Introduction
The coming of the ”Digital Age” contributes to our consumption
and production of data.
In this fast-paced society, how often have you stopped and thought
to appreciate the beauty of the things around you?
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
4
Introduction
Introduction
The coming of the ”Digital Age” contributes to our consumption
and production of data.
In this fast-paced society, how often have you stopped and thought
to appreciate the beauty of the things around you?
Have you every thought and pondered about the underlying
principles that govern the universe?
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
5
Introduction
Introduction
The coming of the ”Digital Age” contributes to our consumption
and production of data.
In this fast-paced society, how often have you stopped and thought
to appreciate the beauty of the things around you?
Have you every thought and pondered about the underlying
principles that govern the universe?
Have you ever paused and pondered about the processes and
mechanisms that make our lives easier and comfortable?
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
6
Introduction
Introduction
The coming of the ”Digital Age” contributes to our consumption
and production of data.
In this fast-paced society, how often have you stopped and thought
to appreciate the beauty of the things around you?
Have you every thought and pondered about the underlying
principles that govern the universe?
Have you ever paused and pondered about the processes and
mechanisms that make our lives easier and comfortable?
As rational creatures, we tend to identify and follow the patterns,
whether consciously or subconsciously. Through these processes
humans are able to survive up to this era.
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
7
Introduction
Introduction
The coming of the ”Digital Age” contributes to our consumption
and production of data.
In this fast-paced society, how often have you stopped and thought
to appreciate the beauty of the things around you?
Have you every thought and pondered about the underlying
principles that govern the universe?
Have you ever paused and pondered about the processes and
mechanisms that make our lives easier and comfortable?
As rational creatures, we tend to identify and follow the patterns,
whether consciously or subconsciously. Through these processes
humans are able to survive up to this era.
In this chapter, we will look at the patterns and regularities in the
world, and how mathematics played a huge part , both in nature
and human endeavors.
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
8
Patterns and Numbers in Nature and the World
What is Pattern?
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
What is Pattern?
Pattern
A pattern or patterns, are regular, repeated, or recurring forms or
designs.
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
What is Pattern?
Pattern
A pattern or patterns, are regular, repeated, or recurring forms or
designs. Studying these patterns enables us to identify relationships
and to find logical connections. Through these we are able to form
generalizations and make predictions.
Example 1.1
What do you think will be the next emoji in the sequence above?
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
11
Patterns and Numbers in Nature and the World
Other Examples: Logical Reasoning Test
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Other Examples: Logical Reasoning Test
Example 1.2
Source:
http://www.graduatewings.co.uk/how-to-improve-at-logical-reasoning
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Other Examples: Logical Reasoning Test
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
14
Patterns and Numbers in Nature and the World
Other Examples: Logical Reasoning Test
Example 1.3
Source:
http://afppracticeexams.com.au/abstract-reasoning-exam-tips/
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
15
Patterns and Numbers in Nature and the World
Other Examples: What Number Comes Next?
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
16
Patterns and Numbers in Nature and the World
Other Examples: What Number Comes Next?
Example 1.3
What number comes next in
1, 3, 5, 7, 9,
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
?
Mathematics
in
the
Modern
World
17
Patterns and Numbers in Nature and the World
Other Examples: What Number Comes Next?
Example 1.3
What number comes next in
1, 3, 5, 7, 9,
?
Answer: 11
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Other Examples: What Number Comes Next?
Example 1.3
What number comes next in
1, 3, 5, 7, 9,
?
Answer: 11
Example 1.4
1, 8, 27, 64, 125,
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
?
Mathematics
in
the
Modern
World
19
Patterns and Numbers in Nature and the World
Other Examples: What Number Comes Next?
Example 1.3
What number comes next in
1, 3, 5, 7, 9,
?
Answer: 11
Example 1.4
1, 8, 27, 64, 125,
?
Answer: 216
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Line Symmetry
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
21
Patterns and Numbers in Nature and the World
Patterns in Nature: Line Symmetry
Line/Bilateral Symmetry
This type of symmetry indicates that you can draw an imaginary line
across an object and the resulting parts are mirror images of each
other.
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
22
Patterns and Numbers in Nature and the World
Patterns in Nature: Line Symmetry
Line/Bilateral Symmetry
This type of symmetry indicates that you can draw an imaginary line
across an object and the resulting parts are mirror images of each
other.
Example 1.5: Bilateral Symmetry
Source:https://biologydictionary.net/bilateral-symmetry/
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
23
Patterns and Numbers in Nature and the World
Patterns in Nature: Rotational Symmetry
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
24
Patterns and Numbers in Nature and the World
Patterns in Nature: Rotational Symmetry
Rotational Symmetry
Rotational symmetry (or radial symmetry) is when an object is rotated
in a certain direction around a point while achieving the same
appearance.
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
25
Patterns and Numbers in Nature and the World
Patterns in Nature: Rotational Symmetry
Rotational Symmetry
Rotational symmetry (or radial symmetry) is when an object is rotated
in a certain direction around a point while achieving the same
appearance. The smallest angle that a figure can be rotated while still
preserving original formation is called angle of rotation. Let ρ be
the angle of rotation,
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
26
Patterns and Numbers in Nature and the World
Patterns in Nature: Rotational Symmetry
Rotational Symmetry
Rotational symmetry (or radial symmetry) is when an object is rotated
in a certain direction around a point while achieving the same
appearance. The smallest angle that a figure can be rotated while still
preserving original formation is called angle of rotation. Let ρ be
the angle of rotation,
ρ=
360◦
n
where in n is the order (n-fold rotational symmetry).
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Spiderwort
Example 1.6: Spiderwort
Source:
https://mathstat.slu.edu/escher/index.php/Rotational Symmetry
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Sea Star
Example 1.7: Sea Star
Source: https://www.pinterest.ph/pin/417145984205155939/
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
29
Patterns and Numbers in Nature and the World
Patterns in Nature: Snowflakes
Example 1.8: Snowflakes
Source: https://www.noaa.gov/stories/how-do-snowflakes-formscience-behind-snow
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
30
Patterns and Numbers in Nature and the World
Patterns in Nature: Translational Symmetry
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
31
Patterns and Numbers in Nature and the World
Patterns in Nature: Translational Symmetry
Translational Symmetry
Translational symmetry is when an object is relocated to another
position while maintaining its general or exact orientation.
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
32
Patterns and Numbers in Nature and the World
Patterns in Nature: Translational Symmetry
Translational Symmetry
Translational symmetry is when an object is relocated to another
position while maintaining its general or exact orientation.
Example 1.9: Honeycomb
Source: https://www.pinterest.ph/pin/294282156874810690/
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Honeycomb and Packing Problems
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
34
Patterns and Numbers in Nature and the World
Patterns in Nature: Honeycomb and Packing Problems
Packing Problem
Packing problems involve finding the optimum method of filling up a
given space such as circle or spherical container.
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Honeycomb and Packing Problems
Packing Problem
Packing problems involve finding the optimum method of filling up a
given space such as circle or spherical container.
Example 1.10
Suppose you have a circles of radius 1 cm, each of which will have an
area of πcm2 . We are then going to fill a plane with these circles using:
Square packing
Hexagonal packing
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Honeycomb and Packing Problems
Example 1.10 cont.
Square Packing
Source:
https://web.nmsu.edu/ snsm/classes/chem116/notes/crystals.html
For square packing, each square will have an area of
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
37
Patterns and Numbers in Nature and the World
Patterns in Nature: Honeycomb and Packing Problems
Example 1.10 cont.
Square Packing
Source:
https://web.nmsu.edu/ snsm/classes/chem116/notes/crystals.html
For square packing, each square will have an area of 4cm2 .
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
38
Patterns and Numbers in Nature and the World
Patterns in Nature: Honeycomb and Packing Problems
Example 1.10 cont.
Square Packing
Source:
https://web.nmsu.edu/ snsm/classes/chem116/notes/crystals.html
For square packing, each square will have an area of 4cm2 . The
percentage of the square’s area covered by circles will be
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
39
Patterns and Numbers in Nature and the World
Patterns in Nature: Honeycomb and Packing Problems
Example 1.10 cont.
Square Packing
Source:
https://web.nmsu.edu/ snsm/classes/chem116/notes/crystals.html
For square packing, each square will have an area of 4cm2 . The
percentage of the square’s area covered by circles will be
area of circles
area of square
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
× 100% =
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
40
Patterns and Numbers in Nature and the World
Patterns in Nature: Honeycomb and Packing Problems
Example 1.10 cont.
Square Packing
Source:
https://web.nmsu.edu/ snsm/classes/chem116/notes/crystals.html
For square packing, each square will have an area of 4cm2 . The
percentage of the square’s area covered by circles will be
area of circles
area of square
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
× 100% =
Clegg,
&
Epp
πcm2
4cm2
(2018).
× 100%
Mathematics
in
the
Modern
World
41
Patterns and Numbers in Nature and the World
Patterns in Nature: Honeycomb and Packing Problems
Example 1.10 cont.
Square Packing
Source:
https://web.nmsu.edu/ snsm/classes/chem116/notes/crystals.html
For square packing, each square will have an area of 4cm2 . The
percentage of the square’s area covered by circles will be
area of circles
area of square
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
× 100% =
Clegg,
&
Epp
πcm2
4cm2
(2018).
× 100% ≈ 78.54%
Mathematics
in
the
Modern
World
42
Patterns and Numbers in Nature and the World
Patterns in Nature: Honeycomb and Packing Problems
Example 1.10 cont.
Hexagonal Packing
Source:
https://www.researchgate.net/figure/Circle-packing-patterns-Square...
For hexagonal packing, we can think of each hexagon as composed of
six equilateral triangles with side equal to 2 cm.
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Honeycomb and Packing Problems
Example 1.10 cont.
Hexagonal Packing
Source:
https://www.researchgate.net/figure/Circle-packing-patterns-Square...
For hexagonal packing, we can think of each hexagon as composed of
six equilateral triangles with side equal to 2 cm. The percentage of the
hexagon’s area covered by circles will be
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Honeycomb and Packing Problems
Example 1.10 cont.
Hexagonal Packing
Source:
https://www.researchgate.net/figure/Circle-packing-patterns-Square...
For hexagonal packing, we can think of each hexagon as composed of
six equilateral triangles with side equal to 2 cm. The percentage of the
hexagon’s area covered by circles will be
area of circles
area of hexagon
Aufman,
Lockwood,
Nation,
× 100% =
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Honeycomb and Packing Problems
Example 1.10 cont.
Hexagonal Packing
Source:
https://www.researchgate.net/figure/Circle-packing-patterns-Square...
For hexagonal packing, we can think of each hexagon as composed of
six equilateral triangles with side equal to 2 cm. The percentage of the
hexagon’s area covered by circles will be
area of circles
area of hexagon
Aufman,
Lockwood,
Nation,
× 100% =
Clegg,
&
Epp
2
3πcm
√
6 3cm2
(2018).
× 100%
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Honeycomb and Packing Problems
Example 1.10 cont.
Hexagonal Packing
Source:
https://www.researchgate.net/figure/Circle-packing-patterns-Square...
For hexagonal packing, we can think of each hexagon as composed of
six equilateral triangles with side equal to 2 cm. The percentage of the
hexagon’s area covered by circles will be
area of circles
area of hexagon
Aufman,
Lockwood,
Nation,
× 100% =
Clegg,
&
Epp
2
3πcm
√
6 3cm2
(2018).
× 100% ≈ 90.69%
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Stripes and Spots
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Stripes and Spots
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Stripes and Spots
According to the theory of Alan Turing, chemical reactions and
diffusion processes in cells determine the growth patterns.
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
50
Patterns and Numbers in Nature and the World
Patterns in Nature: Stripes and Spots
According to the theory of Alan Turing, chemical reactions and
diffusion processes in cells determine the growth patterns.
A new model by Harvard University researchers predicts that
there are three variables that could affect the orientation of these
stripes:
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Stripes and Spots
According to the theory of Alan Turing, chemical reactions and
diffusion processes in cells determine the growth patterns.
A new model by Harvard University researchers predicts that
there are three variables that could affect the orientation of these
stripes: (1) substance that amplifies the density of stripe patterns;
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Stripes and Spots
According to the theory of Alan Turing, chemical reactions and
diffusion processes in cells determine the growth patterns.
A new model by Harvard University researchers predicts that
there are three variables that could affect the orientation of these
stripes: (1) substance that amplifies the density of stripe patterns;
(2) the substance that changes one of the parameters involved in
stripe formation; and
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Stripes and Spots
According to the theory of Alan Turing, chemical reactions and
diffusion processes in cells determine the growth patterns.
A new model by Harvard University researchers predicts that
there are three variables that could affect the orientation of these
stripes: (1) substance that amplifies the density of stripe patterns;
(2) the substance that changes one of the parameters involved in
stripe formation; and (3) physical change in the direction of the
origin of the stripe.
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
54
Patterns and Numbers in Nature and the World
Patterns in Nature: The Sunflower
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
55
Patterns and Numbers in Nature and the World
Patterns in Nature: The Sunflower
Source:
https://asknature.org/strategy/fibonacci-sequence-optimizes-packing/
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Nautilus
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Nautilus
Source: https://www.pinterest.ph/pin/292311832039678146/
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
58
Patterns and Numbers in Nature and the World
Patterns in Nature: Spiral Galaxy
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Spiral Galaxy
Source:
https://blogs.unimelb.edu.au/sciencecommunication/2018/09/23/theuniverse-in-a-spiral/
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
61
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
The role of mathematics in the world population is through
modeling the population growth or decay.
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
62
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
The role of mathematics in the world population is through
modeling the population growth or decay.
Exponential Growth Formula for Population
A = P ert
where in
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
63
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
The role of mathematics in the world population is through
modeling the population growth or decay.
Exponential Growth Formula for Population
A = P ert
where in
A is the size of the population after it grows
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
64
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
The role of mathematics in the world population is through
modeling the population growth or decay.
Exponential Growth Formula for Population
A = P ert
where in
A is the size of the population after it grows
P is the initial number of people
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
65
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
The role of mathematics in the world population is through
modeling the population growth or decay.
Exponential Growth Formula for Population
A = P ert
where in
A is the size of the population after it grows
P is the initial number of people
r is the rate of growth
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
66
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
The role of mathematics in the world population is through
modeling the population growth or decay.
Exponential Growth Formula for Population
A = P ert
where in
A is the size of the population after it grows
P is the initial number of people
r is the rate of growth
t is the time
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
67
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
The role of mathematics in the world population is through
modeling the population growth or decay.
Exponential Growth Formula for Population
A = P ert
where in
A is the size of the population after it grows
P is the initial number of people
r is the rate of growth
t is the time
e is the Euler’s constant
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
68
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
Example 1.11
Five years ago the population of Pangasinan is 2,787,326. Now, the
population of Pangasinan is 2,956,726.
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
Example 1.11
Five years ago the population of Pangasinan is 2,787,326. Now, the
population of Pangasinan is 2,956,726.
1
What is the rate of growth?
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
71
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
Example 1.11
Five years ago the population of Pangasinan is 2,787,326. Now, the
population of Pangasinan is 2,956,726.
1
What is the rate of growth?
2
What will be the population 10 years from the initial P ?
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
72
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
Example 1.11
Five years ago the population of Pangasinan is 2,787,326. Now, the
population of Pangasinan is 2,956,726.
1
What is the rate of growth?
2
What will be the population 10 years from the initial P ?
3
What will be the population 15 years from the initial P ?
Solution:
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
73
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
Example 1.11
Five years ago the population of Pangasinan is 2,787,326. Now, the
population of Pangasinan is 2,956,726.
1
What is the rate of growth?
2
What will be the population 10 years from the initial P ?
3
What will be the population 15 years from the initial P ?
Solution: For sub-item (1): Rearrange the equation,
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
74
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
Example 1.11
Five years ago the population of Pangasinan is 2,787,326. Now, the
population of Pangasinan is 2,956,726.
1
What is the rate of growth?
2
What will be the population 10 years from the initial P ?
3
What will be the population 15 years from the initial P ?
Solution: For sub-item (1): Rearrange the equation,
A = P ert ⇒
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
A
P
= ert
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
75
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
Example 1.11
Five years ago the population of Pangasinan is 2,787,326. Now, the
population of Pangasinan is 2,956,726.
1
What is the rate of growth?
2
What will be the population 10 years from the initial P ?
3
What will be the population 15 years from the initial P ?
Solution: For sub-item (1): Rearrange the equation,
A = P ert ⇒
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
A
P
= ert ⇒ ln ( PA ) = ln (ert )
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
76
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
Example 1.11
Five years ago the population of Pangasinan is 2,787,326. Now, the
population of Pangasinan is 2,956,726.
1
What is the rate of growth?
2
What will be the population 10 years from the initial P ?
3
What will be the population 15 years from the initial P ?
Solution: For sub-item (1): Rearrange the equation,
A = P ert ⇒
A
P
= ert ⇒ ln ( PA ) = ln (ert ) ⇒
A
)
ln ( P
t
=r
Thus,
A
ln ( P
)
t
=r⇒
)
ln ( 2,956,726
2,787,326
5
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
= r ⇒ r ≈ 0.01179997236 or r ≈ 1.18%
&
Epp
(2018).
Mathematics
in
the
Modern
World
77
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
78
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
Solution (cont.): For sub-item (2): Use the equation and plug in the
values,
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
79
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
Solution (cont.): For sub-item (2): Use the equation and plug in the
values,
A = P ert ⇒ A ≈ (2, 787, 326)e(0.0118)(10) ⇒ A ≈ 3, 136, 422
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
80
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
Solution (cont.): For sub-item (2): Use the equation and plug in the
values,
A = P ert ⇒ A ≈ (2, 787, 326)e(0.0118)(10) ⇒ A ≈ 3, 136, 422
For sub-item (3):
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
81
Patterns and Numbers in Nature and the World
Patterns in Nature: Population Growth and Decay
Solution (cont.): For sub-item (2): Use the equation and plug in the
values,
A = P ert ⇒ A ≈ (2, 787, 326)e(0.0118)(10) ⇒ A ≈ 3, 136, 422
For sub-item (3):
A = P ert ⇒ A ≈ (2, 787, 326)e(0.0118)(15) ⇒ A ≈ 3, 327, 038
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
82
The Fibonacci Sequence
Sequence
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
The Fibonacci Sequence
Sequence
Sequence
A sequence is an ordered list of numbers, called terms, that may have
repeated values. The arrangement of terms is set by a definite rule.
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
The Fibonacci Sequence
Sequence
Sequence
A sequence is an ordered list of numbers, called terms, that may have
repeated values. The arrangement of terms is set by a definite rule.
Example 1.12
Analyze the given sequence for its rule and identify the next term.
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
The Fibonacci Sequence
Sequence
Sequence
A sequence is an ordered list of numbers, called terms, that may have
repeated values. The arrangement of terms is set by a definite rule.
Example 1.12
Analyze the given sequence for its rule and identify the next term.
1
1, 10, 100, 1000, 10000, 100000
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
The Fibonacci Sequence
Sequence
Sequence
A sequence is an ordered list of numbers, called terms, that may have
repeated values. The arrangement of terms is set by a definite rule.
Example 1.12
Analyze the given sequence for its rule and identify the next term.
1
1, 10, 100, 1000, 10000, 100000
2
2, 5, 9, 14, 20, 27, 35, 44
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
The Fibonacci Sequence
Sequence
Sequence
A sequence is an ordered list of numbers, called terms, that may have
repeated values. The arrangement of terms is set by a definite rule.
Example 1.12
Analyze the given sequence for its rule and identify the next term.
1
1, 10, 100, 1000, 10000, 100000
2
2, 5, 9, 14, 20, 27, 35, 44
3
1, 1, 2, 3, 5, 8, 13, 21, 34, 55
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
88
The Fibonacci Sequence
The Fibonacci Sequence
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
The Fibonacci Sequence
The Fibonacci Sequence
Named after Leonardo of Pisa or widely known as Fibonacci.
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
The Fibonacci Sequence
The Fibonacci Sequence
Named after Leonardo of Pisa or widely known as Fibonacci.
While the sequence is widely known as Fibonacci sequence, the
pattern is said to have been discovered much earlier in India.
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
The Fibonacci Sequence
The Fibonacci Sequence
Named after Leonardo of Pisa or widely known as Fibonacci.
While the sequence is widely known as Fibonacci sequence, the
pattern is said to have been discovered much earlier in India.
The ratios of the successive Fibonacci numbers approach the
number φ, also known as the Golden ratio
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
The Fibonacci Sequence
The Fibonacci Sequence
Named after Leonardo of Pisa or widely known as Fibonacci.
While the sequence is widely known as Fibonacci sequence, the
pattern is said to have been discovered much earlier in India.
The ratios of the successive Fibonacci numbers approach the
number φ, also known as the Golden ratio
Aufman,
1
1
2
1
3
2
5
3
8
5
13
8
21
13
34
21
55
34
89
55
1
2
1.5
1.667
1.600
1.625
1.615
1.619
1.618
1.618
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
The Fibonacci Sequence
The Fibonacci Sequence
Named after Leonardo of Pisa or widely known as Fibonacci.
While the sequence is widely known as Fibonacci sequence, the
pattern is said to have been discovered much earlier in India.
The ratios of the successive Fibonacci numbers approach the
number φ, also known as the Golden ratio
1
1
2
1
3
2
5
3
8
5
13
8
21
13
34
21
55
34
89
55
1
2
1.5
1.667
1.600
1.625
1.615
1.619
1.618
1.618
Thus, the Golden ratio is approximately equal to 1.618. Next,
n
try calculating Fφn
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
The Fibonacci Sequence
The Fibonacci Sequence
Named after Leonardo of Pisa or widely known as Fibonacci.
While the sequence is widely known as Fibonacci sequence, the
pattern is said to have been discovered much earlier in India.
The ratios of the successive Fibonacci numbers approach the
number φ, also known as the Golden ratio
1
1
2
1
3
2
5
3
8
5
13
8
21
13
34
21
55
34
89
55
1
2
1.5
1.667
1.600
1.625
1.615
1.619
1.618
1.618
Thus, the Golden ratio is approximately equal to 1.618. Next,
n
try calculating Fφn
φn
Fn
Aufman,
1.6181
1
1.6182
1
1.6183
2
1.6184
3
1.6185
5
1.61810
55
1.61812
144
1.618
2.168
2.118
2.285
2.218
2.236
2.236
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
The Fibonacci Sequence
The Fibonacci Sequence
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
96
The Fibonacci Sequence
The Fibonacci Sequence
√
√
n
n
Hence, Fφn will approach the value 5. Rearranging Fφn = 5 to
φn
Fn ≈ √
gives us the idea that the nth Fibonacci number, Fn , is
5
the nearest whole number to
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
φn
√
.
5
(2018).
Mathematics
in
the
Modern
World
97
The Fibonacci Sequence
The Fibonacci Sequence
√
√
n
n
Hence, Fφn will approach the value 5. Rearranging Fφn = 5 to
φn
Fn ≈ √
gives us the idea that the nth Fibonacci number, Fn , is
5
the nearest whole number to
φn
√
.
5
The exact equation for the nth Fibonacci number is
Formula for the nth Fibonacci number
Fn =
φn
√
5
±
1√
φn 5
where in if n is even use − and else (odd), use +.
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
98
The Fibonacci Sequence
The Fibonacci Sequence
√
√
n
n
Hence, Fφn will approach the value 5. Rearranging Fφn = 5 to
φn
Fn ≈ √
gives us the idea that the nth Fibonacci number, Fn , is
5
the nearest whole number to
φn
√
.
5
The exact equation for the nth Fibonacci number is
Formula for the nth Fibonacci number
Fn =
φn
√
5
±
1√
φn 5
where in if n is even use − and else (odd), use +.
Try!
Find the
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
99
The Fibonacci Sequence
The Fibonacci Sequence
√
√
n
n
Hence, Fφn will approach the value 5. Rearranging Fφn = 5 to
φn
Fn ≈ √
gives us the idea that the nth Fibonacci number, Fn , is
5
the nearest whole number to
φn
√
.
5
The exact equation for the nth Fibonacci number is
Formula for the nth Fibonacci number
Fn =
φn
√
5
±
1√
φn 5
where in if n is even use − and else (odd), use +.
Try!
Find the
1
30th Fibonacci number
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
100
The Fibonacci Sequence
The Fibonacci Sequence
√
√
n
n
Hence, Fφn will approach the value 5. Rearranging Fφn = 5 to
φn
Fn ≈ √
gives us the idea that the nth Fibonacci number, Fn , is
5
the nearest whole number to
φn
√
.
5
The exact equation for the nth Fibonacci number is
Formula for the nth Fibonacci number
Fn =
φn
√
5
±
1√
φn 5
where in if n is even use − and else (odd), use +.
Try!
Find the
1
30th Fibonacci number
2
21st Fibonacci number
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
101
The Fibonacci Sequence
The Fibonacci Sequence
√
√
n
n
Hence, Fφn will approach the value 5. Rearranging Fφn = 5 to
φn
Fn ≈ √
gives us the idea that the nth Fibonacci number, Fn , is
5
the nearest whole number to
φn
√
.
5
The exact equation for the nth Fibonacci number is
Formula for the nth Fibonacci number
Fn =
φn
√
5
±
1√
φn 5
where in if n is even use − and else (odd), use +.
Try!
Find the
1
30th Fibonacci number
2
21st Fibonacci number
th Fibonacci number
100Lockwood,
Nation, Clegg, &
3
Aufman,
Epp
(2018).
Mathematics
in
the
Modern
World
The Fibonacci Sequence
The Fibonacci Sequence
Try to calculate the following (round off it to three decimal places):
1
1.618
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
103
The Fibonacci Sequence
The Fibonacci Sequence
Try to calculate the following (round off it to three decimal places):
1
1.618
= 0.618
1.6182
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
104
The Fibonacci Sequence
The Fibonacci Sequence
Try to calculate the following (round off it to three decimal places):
1
1.618
= 0.618
1.6182
= 2.618
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
105
The Fibonacci Sequence
The Fibonacci Sequence
Try to calculate the following (round off it to three decimal places):
1
1.618
= 0.618
1.6182
= 2.618 = φ + 1 (*)
1.6183
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
106
The Fibonacci Sequence
The Fibonacci Sequence
Try to calculate the following (round off it to three decimal places):
1
1.618
= 0.618
1.6182
= 2.618 = φ + 1 (*)
1.6183
= 4.236
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
107
The Fibonacci Sequence
The Fibonacci Sequence
Try to calculate the following (round off it to three decimal places):
1
1.618
= 0.618
1.6182
= 2.618 = φ + 1 (*)
1.6183
= 4.236 = 2φ + 1
1.6184
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
108
The Fibonacci Sequence
The Fibonacci Sequence
Try to calculate the following (round off it to three decimal places):
1
1.618
= 0.618
1.6182
= 2.618 = φ + 1 (*)
1.6183
= 4.236 = 2φ + 1
1.6184
= 6.854
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
109
The Fibonacci Sequence
The Fibonacci Sequence
Try to calculate the following (round off it to three decimal places):
1
1.618
= 0.618
1.6182
= 2.618 = φ + 1 (*)
1.6183
= 4.236 = 2φ + 1
1.6184
= 6.854 = 3φ + 2
1.6185
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
110
The Fibonacci Sequence
The Fibonacci Sequence
Try to calculate the following (round off it to three decimal places):
1
1.618
= 0.618
1.6182
= 2.618 = φ + 1 (*)
1.6183
= 4.236 = 2φ + 1
1.6184
= 6.854 = 3φ + 2
1.6185 = 11.089
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
111
The Fibonacci Sequence
The Fibonacci Sequence
Try to calculate the following (round off it to three decimal places):
1
1.618
= 0.618
1.6182
= 2.618 = φ + 1 (*)
1.6183
= 4.236 = 2φ + 1
1.6184
= 6.854 = 3φ + 2
1.6185 = 11.089 = 5φ + 3
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
112
The Fibonacci Sequence
The Fibonacci Sequence
Try to calculate the following (round off it to three decimal places):
1
1.618
= 0.618
1.6182
= 2.618 = φ + 1 (*)
1.6183
= 4.236 = 2φ + 1
1.6184
= 6.854 = 3φ + 2
1.6185 = 11.089 = 5φ + 3
Thus, this gives us the idea that φn = Fn φ + Fn−1 . Using (*) or known
as the Golden relation, we can derive the exact value of the Golden
ratio.
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
113
The Fibonacci Sequence
The Fibonacci Sequence
Golden Ratio
The Golden ratio φ is unique and φ ∈ R+ satisfying the Golden
relation.
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
114
The Fibonacci Sequence
The Fibonacci Sequence
Golden Ratio
The Golden ratio φ is unique and φ ∈ R+ satisfying the Golden
relation.
Rearranging the Golden relation as φ2 − φ − 1 = 0 then use Quadratic
formula,
√
√
√
−(−1)+ (−1)2 −4(1)(−1)
1+ 5
−b+ b2 −4ac
φ=
=
=
= 1.618033989... ∈ Q∗
2a
2
2(1)
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
115
The Fibonacci Sequence
The Fibonacci Sequence
Golden Ratio
The Golden ratio φ is unique and φ ∈ R+ satisfying the Golden
relation.
Rearranging the Golden relation as φ2 − φ − 1 = 0 then use Quadratic
formula,
√
√
√
−(−1)+ (−1)2 −4(1)(−1)
1+ 5
−b+ b2 −4ac
φ=
=
=
= 1.618033989... ∈ Q∗
2a
2
2(1)
Geometrically, it can also be visualized as a rectangle perfectly
formed by a square and another rectangle which can be repeated
infinitely inside each section.
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
116
The Fibonacci Sequence
The Golden Rectangle and Golden Spiral
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
The Fibonacci Sequence
The Golden Rectangle and Golden Spiral
Source:
https://www.sciencedirect.com/science/article/pii/S1110016815000265
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
118
The Fibonacci Sequence
Applications
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
The Fibonacci Sequence
Applications
Logo Creation
Source:
https://www.invisionapp.com/inside-design/golden-ratio-designers/
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
120
The Fibonacci Sequence
Applications
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
121
The Fibonacci Sequence
Applications
Architecture
Source: https://sites.google.com/site/funwithfibonacci/architecture/theparthenon
Aufman, Lockwood, Nation,
Daniel Bezalel A. Garcia
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
122
Mathematics for Our World
Mathematics for Our World
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Mathematics for Our World
Mathematics for Our World
Mathematics for Organization (specifically, Statistics and Big data
analysis)
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Mathematics for Our World
Mathematics for Our World
Mathematics for Organization (specifically, Statistics and Big data
analysis)
Mathematics for Prediction
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World
Mathematics for Our World
Mathematics for Our World
Mathematics for Organization (specifically, Statistics and Big data
analysis)
Mathematics for Prediction
Mathematics for Control
Therefore, mathematics is indispensable
Aufman,
Lockwood,
Nation,
Clegg,
&
Epp
(2018).
Mathematics
in
the
Modern
World