Mathematics for Our World
Lecture 03
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Introduction
Introduction
Mathematics has been humanity's companion for all of history. Perhaps, it started with farmers counting their cattle or kings sizing their armies. And as centuries passed by, mathematics never became less useful, instead, it grew and became more and more important for humans. Today, we use mathematics for almost everything! Pilots and seafarers calculate distances and directions in navigating the Earth; chefs maintain the correct proportion of ingredients in their recipes; epidemiologists model viral outbreaks using differential equations; agronomists calibrate farming methods to optimize agricultural profits, and many more.
Plimpton 322: one of the oldest mathematical tablets from Ancient Babylon | from Wikipedia
In the 2017 movie "The Wall", Sgt. Allen Isaac (Aaron Taylor-Johnson) uses mathematical calculations accurately locate his enemy. He did this by carefully measuring the difference between the time the enemy's bullet hit near him and the time he heard the sound of the rifle and then calculating the distance using the average speed of sound and of a sniper rifle bullet. | (c) FTZ - Youtube
In this lecture, we will see how mathematics is used for different purposes. We will focus on the various roles it plays namely organization, prediction, and control. Our aim is to appreciate mathematics for our world especially in the context of the modern times.
Mathematics for Organization
Mathematics for Organization
The world is extremely complex!
Every time, the world changes. Five thousand babies are born, 500 million tweets are posted online, 100 thousand airplanes fly from one airport to another, stock market prices move, and 463 exabytes (463 billion gigabytes) of new data are produced all in just a single day. To manage this complexity, we somehow need something that would help us organize them. Mathematics helps us on that.
In this section, we will be looking at a few mathematical tools that help us organize complex information. There should be several examples but we will feature only three: networks, the coordinate plane, and matrices.
Networks
Networks
A network is a set of objects (represented as nodes or vertices) connected by lines (called links or edges) [1]. Networks are widely used by computer scientists, mathematical modelers, statisticians, and even experts in other fields in order to effectively illustrate a system and the relationships between the components or elements of this system.
In the illustration in the right, the blue dots are nodes while the green lines are the links. Notice that links are connecting two nodes to represent some form of relationship between the two.
Network (simple undirected)| (c) MathInsight
Non-simple network | (c) MathInsight
The network above is an example of a simple network. A simple network is one in which all the links are connecting two different nodes and for every pair of nodes, there can only be a maximum of one link connecting them. If a network
has a link that connect one node to itself; or,
has a pair of nodes that are being connected by two or more links,
then the network is said to be non-simple. The network given on the left is a non-simple network.
Networks could also be either directed or undirected. Directed networks are those of which links are arrows instead of just lines. The arrows represent one-way relationships. For example, in Twitter, an account can follow another but not necessarily the other way around.
The example on the right is a directed network.
Directed network | (c) MathInsight
Networks are very useful in different fields of science and not just in mathematics. Take the following as examples.
Game of Thrones Social Network. Links represent relationship between characters as measured by the number of conversations they have in the series: the thicker the link, the more conversations they had. | Taken from a blog by (c) Lia Petronio - News@NorthEastern. For more info and further reading, click here.
The Human Disease Network. This shows the relationship between different human diseases. In the network on the left, two diseases are connected if the two are related via the same gene defect. In other words, a person with one disease would likely get the other diseases connected to it after some complications. The thicker the link, the more likely such complication to happen. This was taken from a research by Dr. Kamyar Hedayat of the Full Spectrum Health Center and Dr. Jean-Claude Lapraz. | (c) K. Hedayat - ResearchGate. For more info and further reading, click here.
British Royal Family Tree. A family tree is another example of a network. Here, the nodes are the members of the family. Those with blue linings in their pictures are the actual members of the family by blood while those with gray are members of the family through marriages. Black links represent marital bonds, gray links represent sons and daughters and red dashed links represent divorced marriages. | Image from Shayanne Gal - Business Insider. For more readings and info, click here.
Coordinate Plane
Coordinate Plane
In a story, the famous French mathematician René Descartes (pronounced re-NEY deh-CART) was known to be sickly. Many of the days of his life, he stays on his bed resting. One day he was looking at the door of his bedroom and noticed a spider crawling. (French doors are usually decorated with square tiles.) At that time, he thought of a way of determining the location of the spider on the door by counting how many squares from the left and how many squares from the bottom of the door the spider is located.
Descartes formalized this system in his book La Géométrie and is now called the Cartesian coordinate system (Renatus Cartesius was Descartes' Latin name) [2]. Today, the Cartesian coordinate system is one of the most useful mathematical inventions.
It starts with two perpendicular lines - one vertical (called the y-axis) and one horizontal (called the x-axis). These two lines intersect at a point called the origin. Every point on the plane is determined by an ordered pair (x,y) of coordinates x and y. The horizontal coordinate x or abscissa is the distance between the point and the y-axis while the vertical coordinate y or ordinate is the distance between the point and the x-axis. For example, the point shown below has the coordinates (5, 4).
Their are many coordinate systems today that were derived from the Cartesian coordinate system. Some examples are given below.
Matrices
Matrices
A matrix is a rectangular array of numbers arranged in rows and columns [3]. It is regarded as a mathematical tool with various applications in engineering, commerce, and other fields. A matrix with m rows and n columns is called an m Ă— n matrix. Take a look at the following matrices as examples.
2 Ă— 2 matrix
2 Ă— 3 matrix
3 Ă— 1 matrix
Matrices are useful because they are an effective tool to organize quantitative relationships between two variables. For example, suppose we have three cities: A, B, and C. The matrix below shows the amount of traffic that passes between cities (in 100 vehicles):
The figures in the matrix show the amount of traffic (in 100 vehicles) that pass through the roads connecting the corresponding cities in the rows and column per day. For example, there are 340 vehicles that travel between City A to City B everyday, and 240 vehicles between City A and City C. The figures 7.8, 10.2, and 6.1 means that there are 780, 1020, and 610 vehicles that travel only within A, B, and C respectively. This matrix could help governments decide, for example, which road projects that have to be prioritized due to the traffic volume which can be seen from the matrix.
The next matrix may be important for bakeries. Suppose they manufacture three kinds of cake, each having their own ingredient requirements. This matrix shows the amount (in PhP) of ingredients needed per cake. In industrial engineering, there is a course called Operations Research (Linear Programming) which uses matrices in order to optimize and decide how many cakes of each type should be produced in order to maximize profits (or minimize expenses).
There are many, many more uses of matrices: computer graphics, optics, cryptography, economics, chemistry, geology, robotics and animation, wireless communication and signal processing, finance, and many more. Check this page out for further readings.
Mathematics for Prediction
Mathematics for Prediction
The future is always uncertain -- but if we have a reliable way of seeing it, we could possibly save lives, ease up work, or make effective decisions now. Fortunately, mathematics gives us a little hope on telling what the future could be.
Usually, making prediction in mathematics starts with building a mathematical model. A mathematical model is a mathematical structure (such as equations, networks, matrices, etc.) that replicates or represents a real-world system. From these models, mathematicians could generate predictions of what may happen to the system.
Some examples of real-world systems in which models are developed for prediction are stock markets, astronomy and astrophysics, and weather forecasting.
Stock Market
Stock Market
It is possible for ordinary citizens to own part of large companies through buying shares of stocks. We can buy stocks from the stock market. Prices of stocks change every time (every second). People who buy stocks for investment could earn money if they bought it at a low price and sell at a higher price.
This is how stock market prices move per second | (c) Tanuya Bahirat - Great Learning
Due to demand for profits, several mathematical methods of predicting market prices were proposed. Some of these include technical analysis, the probabilistic concept of random walks, and even machine learning. For more about this, see this Wikipedia article.
Astronomy and Astrophysics
Astronomy and Astrophysics
Chelyabinsk meteor | (c) Ogg Theora - Wikimedia commons
Asteroids are dangerous. Even if large asteroid impacts are rare, they are still feared to cause catastrophe and destruction such as that of the extinction of dinosaurs. Many governments around the world invest in funding agencies that monitor, track, and predict asteroid movements around the Earth to possibly protect the planet from dangerous hits. However, despite the massive efforts by space agencies such as NASA, some meteors (with some large enough to be a threat) still escape their watch.
One day in February 2013, a meteor the size of a large house (20 meters in length) hit Chelyabinsk in Russia which brought bright light and caused a loud explosion. Despite its size, space agencies around the world were not able to detect the meteor. Rare incidents like these could potentially destroy cities and even cause another mass extinction!
To prevent this from happening again, mathematicians propose methods to try predicting asteroid movements especially those potentially hazardous heavenly bodies near Earth (called Near Earth Objects or NEOs). Their results could give the likelihood that a NEO would hit Earth at a given time.
Today, the largest threat to Earth is the 370-meter 99942 Apophis Asteroid with a 2.7% chance of hitting the Earth on April 13, 2029 [4]. NASA is hopeful this would likely not happen, at least in the next 100 years. See this video for their prediction.
Weather Forecasting
Weather Forecasting
If you Google "weather tomorrow", a forecast of tomorrow's weather in your locality will be returned. This weather prediction is powered by some mathematical models that predict weather using data from basic parameters such as air pressure, temperature, humidity, wind speed, and others.
Predicted path of Super Typhoon Haiyan (Yolanda) in 2013 | (c) Dumaguete.com
Predicting weather could save lives and millions worth of crops and property. For example, predicting the path of a typhoon would greatly help the people and governments in their decisions.
There are several mathematical methods of predicting weather. PAG-ASA, the Philippine weather agency, uses these models to advise Filipinos of thunderstorms, typhoons, and heat waves.
Mathematics for Control
Mathematics for Control
Sometimes, unpleasant elements get inside real-world systems. For example in agriculture, pests infest farms causing destruction to crops. In environmental science, human waste from cities would pollute lakes or rivers. In social science, a false information (now known as "fake news") would spread in the community.
To mitigate these "unwelcome elements", experts would apply control methods. For example, in the farm, farmers would apply pesticides or other pest control strategies to reduce pest populations. Environmental protectors would organize clean-up drives or other programs and projects to control pollution. Then, media outlets, governments, or other individuals or groups would launch campaigns to fight the spreading fake news.
Control Theory is a branch of mathematics that deals with the quantitative modeling of a control that is applied in a system to achieve desired results [5]. The goal of control theory is to understand the effects of applying a control method so that the field expert could make decisions on how to manage the application of the control method in order to effectively, quickly, or optimally reach the desired outcome.
Sugarcane borer | (c) pestnet.org
The research paper written by Rafikov & Limeira in 2012 models the control of the pest sugarcane borers (Diatraea saccharalis). They were able to recommend a proposed strategy of how to apply a certain parasitoid (Trichogramma galloi) which fights sugarcane borers in order to effectively control the pest population. The abstract of their paper can be accessed here.
Another paper by Thai mathematician N. Pochai introduced a model of controlling the transport of pollutants into a lake. The paper achieved a result of how to strategize this control while spending the least expenses. The paper can be accessed here.
Lake pollution | (c) NDRC - ndrc.org
(c) Peter Dazeley - Getty Images
In an article from Scientific Daily, M. Mukerjee discusses how mathematicians analyze the spread of fake news especially in the context of the internet and social media [6]. To read more, click here.
References
References
[1] Nykamp, D.Q., Network definition. Math Insight. from https://mathinsight.org/definition/network
[2] Cartesian coordinates. Math is Fun. from https://www.mathsisfun.com/data/cartesian-coordinates.html
[3] Application of Matrices in Science, Commerce, and Social Science FIelds. Vendatu. from https://www.vedantu.com/maths/application-of-matrices
[4] 99942 Apophis. Wikipedia - the Free Encyclopedia. from https://en.wikipedia.org/wiki/99942_Apophis
[5] Control Theory. Britannica. from https://www.britannica.com/science/control-theory-mathematics
[6] M. Mukerjee (2017). How fake news goes viral - here's the math. Scientific American. from https://www.scientificamerican.com/article/how-fake-news-goes-viral-mdash-heres-the-math/
Review Questions
Review Questions
In what fields can mathematics be used? Give at least three.
Give one example each on how mathematics is used for:
a. Organization
b. Prediction
c. Control
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