Applied Mathematics
Applied Mathematics is a language that allows us to model our world from atoms to galaxies, airplanes in flight and spacecraft in space, biological and geological activity, weather forecasting, social science and natural science. Mathematics allows engineers to design buildings, automobiles, computers, machines used in production and assembly, farm equipment, roads, tunnels and bridges, and electrical equipment.
“Why do we need mathematical models? A simple answer to this question is that we need math models to understand the world and how it works. It is useful in real-world applications. It simply signifies the world in simple models and shapes. These models should work with engineers and scientists to solve real-world problems using simple math. It will also involve equations.” 1
Virtually any educated individual will need the ability to2:
- Examine a set of data and recognize a behavioral pattern in it.
- Assess how well a given model matches the data.
- Recognize the limitations in the model.
- Use the model to draw appropriate conclusions.
- Answer appropriate questions about the phenomenon being studied.
Math is about more than finding the correct answer. It’s about using concepts, processes and reasoning to apply the best approach for solving a problem. The concepts, processes and reasoning are accumulated over time and start with arithmetic (i.e., numbers and counting, and basic operations), which is the basis for all subsequent mathematics. Once these basic concepts, processes and reasoning are mastered, additional concepts (i.e., geometry, algebra, trigonometry, complex numbers) can be added, and they culminate and are used in calculus.
One of the most powerful means to solve a problem is with a picture. Drawing a picture (shape), diagram (graph), or model is the most common problem solving strategy to understand a problem, and represents the problem in a way students can see, understand, and think about the problem while looking for the next step.
Let’s look at some of the math skills shown in the image above. Notice how each math skill builds upon the math skills below it.
Arithmetic
If I were again beginning my studies, I would follow the advice of Plato and start with mathematics.
Galileo Galilei
Arithmetic is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers — addition, subtraction, multiplication, division, exponentiation, and extraction of roots. 3
Contemporary mathematics is an extremely abstract and complex science. But originally, its essential task was to manipulate numbers that represented concrete quantities. There would not have been all subsequent developments without that initial impulse born of the need to harness the power of numbers to solve practical problems. But what are numbers?
A number is essentially an abstraction. It is the concept of a given quantity. It presupposes the cognitive ability to analyze our experience of the world, recognizing similarities and differences of all kinds, from the most evident to the most subtle. 4
Arithmetic Properties
The most important arithmetic properties (where a and b are real numbers) are the commutative laws of addition and multiplication, a + b = b + a and ab = ba; the associative laws of addition and multiplication, a + (b + c) = (a + b) + c and a(bc) = (ab)c; and the distributive law, which connects addition and multiplication, a(b + c) = ab + ac. These properties include subtraction (addition of a negative number) and division (multiplication by a fraction).
These arithmetic properties form the basis for all mathematics that follow.
Geometry
Geometry is the archetype of the beauty of the world.
Johannes Kepler
Geometry is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. 5
The most common types of geometry are plane geometry (dealing with objects like the point, line, circle, triangle, and polygon), solid geometry (dealing with objects like the line, sphere, and polyhedron), and spherical geometry (dealing with objects like the spherical triangle and spherical polygon). 6
A mathematical pun notes that without geometry, life is pointless. An old children’s joke asks, “What does an acorn say when it grows up?” and answers, “Geometry” (“gee, I’m a tree”).
Geometric shapes are any structure, open or closed, having a definite shape and properties made up of lines, curves and points. Some of the known geometric shapes are square, rectangle, circle, cone, cylinder, sphere, etc. All these shapes have some properties that make them unique and different from the other shapes. 7
Algebra
Algebra is a field of mathematics which is not rigorously delimited but can roughly be described as a focus, inside set theory, on a range of remarkable concepts and tools providing preliminary tools for model theory (the study of theories and their models).
“Foundations Of Algebra”. 2022. settheory.net. http://settheory.net/algebra/.
Algebra is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.8
Algebra helps solve the mathematical equations and allows to derive unknown quantities, like the bank interest, proportions, percentages. We can use the variables in the algebra to represent the unknown quantities that are coupled in such a way as to rewrite the equations.
The algebraic formulas are used in our daily lives to find the distance and volume of containers and figure out the sales prices as and when needed. Algebra is constructive in stating a mathematical equation and relationship by using letters or other symbols representing the entities. The unknown quantities in the equation can be solved through algebra. 9
The basic graphs are just that — basic. They’re centered at the origin and aren’t expanded or shrunken or jostled about. You can alter the basic graphs by performing translations to the left or right or up or down. These graphs include the quadratic polynomial graph, cubic polynomial graph, graph of the line y = x, absolute value function, reciprocal of x, reciprocal of x2, graph of the square root, graph of the cube root, graph of the exponential function, graph of the logarithmic function. 10
Let’s consider the arithmetic properties for matrices.
- Addition, subtraction and multiplication are the basic operations on the matrix, but not division.
- Matrix multiplication is associative but not commutative.
- The subtraction of matrices is NOT commutative or associative.
- Just like the addition of numbers, matrix addition also has similar properties like commutative law, associative law, additive inverse, additive identity, etc. 11
- In linear algebra, the trace of a matrix is defined as the sum of the elements along the leading diagonal. The trace of two matrices AB are equivalent to the trace of two matrices BA multiplied together. In group theory terms, the trace of two matrices is commutative.
Also, check out Functions II.
Trigonometry
Life and trigonometry both are the same. They both have formulas to solve problems but where which formula is applied; that’s difficult to understand.
Anonymous
Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. 12
To sketch the trigonometry graphs of the functions – Sine, Cosine and Tangent, we need to know the period, phase, amplitude, maximum and minimum turning points. These graphs are used in many areas of engineering and science. Few of the examples are the growth of animals and plants, engines and waves, etc. Also, we have graphs for all the trigonometric functions. 13
Calculus
Every one who understands the subject will agree that even the basis on which the scientific explanation of nature rests is intelligible only to those who have learned at least the elements of the differential and integral calculus, as well as analytical geometry.
Felix Klein
Calculus, originally called infinitesimal calculus or “the calculus of infinitesimals“, is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus and integral calculus; differential calculus concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. 14
The graph of a function f is the set of all points in the plane of the form (x, f(x)). We could also define the graph of f to be the graph of the equation y = f(x). So, the graph of a function is a special case of the graph of an equation. 15
Graphs help to present data or information in an organized manner, and there are eight main types: linear, power, quadratic, polynomial, rational, exponential, logarithmic, and sinusoidal. Learn more about the functions of different graphs and their visual differences. 16
Probability
If scientific reasoning were limited to the logical processes of arithmetic, we should not get very far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability.
Vannevar Bush
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes (“heads” and “tails”) are both equally probable; the probability of “heads” equals the probability of “tails”; and since no other outcomes are possible, the probability of either “heads” or “tails” is 1/2 (which could also be written as 0.5 or 50%). 17
In probability, a prediction is made based on a general model, which satisfies all aspects of the problem. This enables to quantify the uncertainty and the likelihood of occurrence of events in the scenario. Probability distribution functions are used to describe the probability of all possible events in the considered problem.
Another investigation in probability is the causality of events. Bayesian probability describes the likelihood of prior events based on the probability of the events caused by the events. This form is useful in artificial intelligence, especially in machine learning techniques. 18
Probability distributions can convey a fantastic amount of useful information, but there may be so much information to view that the important points get lost in the data. Because of this, it is very common to create a graphical representation of the data to highlight important or interesting values.
Tables, histograms and bar charts in particular are excellent means of visualizing the data from discrete probability distributions. If you use a histogram or bar chart, by enumerating the various outcomes along the x-axis and the expected probability of occurrence on the y-axis, you create a very concise and easily read summary of the distribution of outcome probabilities. 19
Statistics
Since all models are wrong the scientist cannot obtain a “correct” one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist so overelaboration and overparameterization is often the mark of mediocrity.
George Box
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as “all people living in a country” or “every atom composing a crystal”. Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. 20
Statistics supports theories for collection, analysis, and interpretation of data. The descriptive statistics and inferential statistics can be considered as a major division in statistics. Descriptive statistics is the branch of statistics which describe the main properties of a data set quantitatively. Inferential statistics is the branch of statistics, which derive conclusions about the concerned population from the data set obtained from a sample, subjected to random, observational, and sampling variations.
Descriptive statistics summarizes the data while inferential statistics is used to make forecasts and prediction, in general, about the population, from which the random sample was selected.
A statistical graph or chart is defined as the pictorial representation of statistical data in graphical form. The statistical graphs are used to represent a set of data to make it easier to understand and interpret statistical information. The four basic graphs used in statistics include bar, line, histogram and pie charts. 21
Pure Mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles. 22
Conclusion
“I’m so con-fused!”
Vinnie Barbarino – Welcome Back, Kotter
So what is mathematics? Mathematics is fun, hard, boring, challenging, frustrating, tedious, interesting, stressful and more. Mathematics is whatever you make, and sometimes what the teacher makes, it to be.
References
1 “Mathematical Modeling: Definition, Classifications – Turito”. 2022. US Learn. https://www.turito.com/learn/math/mathematical-modeling.
2 Watson, Saleem. “How Math Models the Real World (and how it does not)”. 2022. texmatyc.org. http://texmatyc.org/Highlights/Saleem%20Watson%20TCCTA-Talk.pdf.
3 “Arithmetic – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Arithmetic.
4 Diodati, Michele. “Number”. 2021. Medium. https://medium.com/not-zero/number-ccabb87ee52a.
5 “Geometry – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Geometry.
6 “Geometry — From Wolfram MathWorld”. 2022. mathworld.wolfram.com. https://mathworld.wolfram.com/Geometry.html.
7 “Geometric Shapes – Definition, List, Types, Properties”. 2022. CUEMATH. https://www.cuemath.com/geometry/geometric-shapes/.
8 “Algebra – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Algebra.
9 “Algebra (Definition, Basics, Branches, Facts, Examples) | What Is Algebra?” 2022. BYJUS. https://byjus.com/maths/algebra/.
10 “10 Basic Algebraic Graphs – Dummies”. 2022. dummies.com. https://www.dummies.com/article/academics-the-arts/math/algebra/10-basic-algebraic-graphs-203562/.
11 “Addition Of Matrices – Properties | What Is Matrix Addition?”. 2023. CUEMATH. https://www.cuemath.com/algebra/addition-of-matrices/.
12 “Trigonometry – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Trigonometry.
13 “Trigonometry Graphs For Sine, Cosine And Tangent Functions”. 2022. BYJUS. https://byjus.com/maths/trigonometry-graphs/.
14 “Calculus – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Calculus.
15 “Graphs Of Functions”. 2022. dl.uncw.edu. http://dl.uncw.edu/digilib/Mathematics/Algebra/mat111hb/functions/graphs/graphs.html.
16 “Probability – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Probability.
17 “Graphs: Types, Examples & Functions”. 2022. study.com. https://study.com/academy/lesson/graphs-types-examples-functions.html.
18 “Difference Between Probability And Statistics | Compare The Difference Between Similar Terms”. 2012. Compare The Difference Between Similar Terms. https://www.differencebetween.com/difference-between-probability-and-vs-statistics/.
19 Foundation, CK-12. 2022. “Visualizing Probability Distribution ( Read ) | Probability”. CK-12 Foundation. https://www.ck12.org/c/probability/probability-distribution/lesson/Visualizing-Probability-Distribution-PST/.
20 “Statistics – Wikipedia”. 2020. en.wikipedia.org. https://en.wikipedia.org/wiki/Statistics.
21 “Types Of Graphs In Mathematics And Statistics With Examples”. 2022. BYJUS. https://byjus.com/maths/types-of-graphs/.
22 “Pure Mathematics – Wikipedia”. 2023. en.wikipedia.org. https://en.wikipedia.org/wiki/Pure_mathematics.
Additional Reading
Ananthaswamy, Anil. “Babies Understand Numbers As Abstract Concepts”. 2022. New Scientist. https://www.newscientist.com/article/dn17264-babies-understand-numbers-as-abstract-concepts/.
Our ability to think of numbers as abstract concepts is probably innate and even babies barely a few hours old seem to have the ability, researchers say.
Abstract numerical thought is the ability to perceive numbers as entities, independently of specific things. It can be demonstrated by the humans capacity to link a certain number of objects to the same number of sounds, irrespective of what the specific sounds or objects are. But whether this ability is innate or learned through culture or language wasn’t known.
⭐ Chinaza, Favour. “MATHEMATICS”. 2022. Medium. https://medium.com/@favournaza.emma/mathematics-f86449834961.
Mathematics is the science of numbers and their operations, interrelations, combinations, generalizations, abstractions, space configurations, structure, measurement, transformations, and inferences.
It can also be defined as an area of knowledge that includes such topics as numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes.
Mathematics is an area of knowledge that includes such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis)
Put in simpler terms, Mathematics is the science and study of quality, structure, space, and change.
Kidwell, Eugene. 2021. “Real World Examples Of Mathematical Modelling”. Maths Careers. https://www.mathscareers.org.uk/real-world-examples-of-mathematical-modelling/.
“Matrix Multiplication – 2X2, 3X3 | How To Multiply Matrices?”. 2023. CUEMATH. https://www.cuemath.com/algebra/multiplication-of-matrices/.
“Subtraction Of Matrices – Properties | What Is Matrix Subtraction?”. 2023. CUEMATH. https://www.cuemath.com/algebra/subtraction-of-matrices/.
“What Is A Graph In Math? Definition, Solved Examples, Facts”. 2022. SplashLearn – Math Vocabulary. https://www.splashlearn.com/math-vocabulary/geometry/graph.
⭐ “What Is Math Modeling? | Society For Industrial And Applied Mathematics”. 2022. m3challenge.siam.org. https://m3challenge.siam.org/resources/whatismathmodeling.
Videos
Mathematical modeling provides answers to real world questions like “Which recycling program is best for my city?” “How will a flu outbreak affect the US,” or “Which roller coaster is the most thrilling?” In math modeling, you’ll use math to represent, analyze, make predictions or otherwise provide insight into real world phenomena. This is the first episode in this new math modeling video series and introduces the modeling process, setting the stage for the next six videos which dive into the specific steps to modeling.
The featured image on this page is from the Calculus Wallpapers collection on the WallpaperCave website.
⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.
Mathematical Mysteries 