Re-Introduction to Algebra with Intuition: Algebraic Foundation for Calculus, Computer Science, Physics, and Engineering

256 Pages·10.488 MB·English

by  Ruchira Sasanka

R -I A
E NTRODUCTION TO LGEBRA
I
WITH NTUITION

ALGEBRAIC FOUNDATION FOR
CALCULUS, COMPUTER SCIENCE, PHYSICS, AND ENGINEERING

Ruchira Sasanka, Ph.D.

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Copyright © 2017 by Ruchira Sasanka
Illustractions copyright © 2017 by Ruchira Sasanka

All rights reserved

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CONTENTS
PREFACE ........................................................................................................................................... VIII
SECTION I: FUNDAMENTALS.............................................................................................................. 1
1 A RELATIONSHIP GUIDE: BROUGHT TO YOU BY FUNCTIONS .................................................. 2
1.1 Modeling with Math ......................................................................................................................... 3
1.2 Relationship Guide: How to Build a Relationship (Tip: Use a Function) ..................... 4
1.3 Visual Model (Mental Model) For a Function ......................................................................... 4
1.4 Meet the Everyday Functions You Hardly Think About ..................................................... 5
1.5 Function Notation ............................................................................................................................. 7
1.6 Lazy Mathematicians .................................................................................................................... 11
1.7 Function Definition vs. Function Evaluation (Function Call) ........................................ 12
1.8 Where Do Equations Come From? ........................................................................................... 13
1.9 Taking Selfies: Visualizing Functions ..................................................................................... 15
1.10 The More the Merrier: Functions with More Than One Input ..................................... 16
1.11 Operators Are People, err…, Functions, Too! ...................................................................... 18
1.12 Feeding One Function into Another: Function Composition ........................................ 19
1.12.1 Adding and Multiplying Functions As Composition ............................................................ 24
1.13 Oops! Hit Undo! Hit Undo! Undo with the Inverse of a Function ................................. 25
1.14 Bad Root! You Produced Nothing! ........................................................................................... 31
1.15 There’s More Than One Way to Skin a Cat (or a Relationship) .................................... 34
1.16 The Story So Far .............................................................................................................................. 38
2 NUMBERS: WHAT FUNCTIONS CONSUME AND PRODUCE .................................................... 40
2.1 Invasive Numbers: How Numbers Breed More Numbers .............................................. 40
2.1.1 Addition and Multiplication ........................................................................................................... 41
2.1.2 First Signs of Trouble: Subtraction ............................................................................................. 41
2.1.3 More Trouble: Division .................................................................................................................... 43
2.2 Unwholesome Whole Numbers ................................................................................................ 43
2.2.1 Poor Abstraction of Reality ............................................................................................................ 44
2.3 Operations Revisited (with Negative Numbers) ................................................................ 45
2.3.1 Addition and Subtraction ................................................................................................................ 45
2.3.2 Multiplication as Two Operations ............................................................................................... 46
2.4 Go Forth and Multiply, err…, Exponentiate .......................................................................... 49
2.5 The Story So Far .............................................................................................................................. 52
3 FUNCTION TOOLBOX ................................................................................................................. 54
3.1 Power Function Family: How to Grow Your Power .......................................................... 54
3.1.1 Linear Term .......................................................................................................................................... 56
3.1.1.1 Reducing Non-Linear Models to Linear Models ..................................................................... 59
3.1.2 Quadratic Term ................................................................................................................................... 62
3.1.3 Cubic Term ............................................................................................................................................ 64
3.2 Polynomial Family: How to Accumulate Different Powers ............................................ 66
3.2.1 Linear model (Linear Function) ................................................................................................... 67
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3.2.1.1 Significance of the Absence of a Constant Term .................................................................... 68
3.2.1.2 Linear Models of Multiple Inputs (Linear Combinations) ................................................. 69
3.2.2 Quadratic Model ................................................................................................................................. 71
3.2.3 Cubic Model .......................................................................................................................................... 73
3.2.4 Generalization: Polynomial Functions ...................................................................................... 73
3.2.4.1 Polynomials as Linear Combinations .......................................................................................... 74
3.2.4.2 The Fundamental Theorem of Algebra ...................................................................................... 75
3.3 Exponential Family ........................................................................................................................ 77
3.3.1 THE (Natural) Exponential Function ......................................................................................... 82
3.4 The Inverse Functions of The Families We Met ................................................................. 83
3.4.1 Inverse of Power Functions ........................................................................................................... 84
3.4.2 Inverse of Polynomials .................................................................................................................... 86
3.4.2.1 Inverse of the Linear Model ............................................................................................................ 87
3.4.2.2 Inverse of Polynomials ...................................................................................................................... 88
3.4.3 Inverse of Exponential Functions: Logarithmic Model ...................................................... 89
3.5 Reciprocal Functions of the Families We Met ..................................................................... 93
3.5.1 Reciprocal of Power Functions ..................................................................................................... 94
3.5.1.1 Reciprocal of The Linear Term ...................................................................................................... 94
3.5.1.2 Inverse Square Model ........................................................................................................................ 96
3.5.2 Reciprocals of Polynomials ............................................................................................................ 98
3.5.3 Reciprocal of Exponential Models (Exponential Decay) .................................................... 99
3.6 Trigonometric Family ................................................................................................................. 101
3.6.1 Periodic Functions .......................................................................................................................... 101
3.6.2 Communicating with Waves ....................................................................................................... 107
3.6.3 Inverse and Reciprocal Models of Trigonometric Functions ........................................ 112
3.7 Growing Faster than Exponential: The Factorial Function .......................................... 113
3.8 Building Larger Models from Simpler Models .................................................................. 116
3.9 Summary of Function Families ............................................................................................... 119
3.10 The Story So Far ............................................................................................................................ 121
4 SERIES: FUNCTIONS UNLIMITED ........................................................................................... 122
4.1 A Polynomial as a Series ............................................................................................................ 122
4.2 The Sky is the Limit: Power Series ........................................................................................ 125
4.3 Maclaurin and Taylor Series .................................................................................................... 128
4.4 Fourier Series ................................................................................................................................. 133
4.5 The Story So Far ............................................................................................................................ 133
SECTION II: BEYOND FUNDAMENTALS ....................................................................................... 135
5 FUNCTIONS THAT CAUSE (YOUR HEAD TO) SPIN ............................................................... 136
5.1 A “Rotated” Number .................................................................................................................... 139
5.2 Functions with Complex (“Rotated”) Input and Output ................................................ 141
5.2.1 Basic Arithmetic Operators (Functions) ............................................................................... 141
5.2.2 Complex Multiplication as Two Operations ......................................................................... 143
5.3 Complex Roots of Polynomials ................................................................................................ 144
5.3.1 Why do They Come in Pairs? ...................................................................................................... 146
5.4 Graph of a Complex Function .................................................................................................. 147
5.5 Complex Numbers as “Complete” Numbers....................................................................... 150
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5.6 Non-real Exponents (Advanced Topic) ................................................................................ 153
5.6.1 Exponentiation with an Imaginary Input .............................................................................. 154
5.6.2 Exponentiation with a Complex Input .................................................................................... 158
5.7 The Story So Far ............................................................................................................................ 160
6 FUNCTIONS IN 3D SPACE ....................................................................................................... 162
6.1 Vectors: Representing Objects in Space .............................................................................. 162
6.1.1 Where do Babies, err…, Vectors, Come From? .................................................................... 163
6.2 Vector Difference (Subtraction) ............................................................................................. 164
6.3 Vector Addition ............................................................................................................................. 165
6.4 Multiplication by a Scalar (Scaling) ....................................................................................... 166
6.5 Examples of Vector Addition and Subtraction .................................................................. 166
6.6 Vector Functions You Meet Every Day ................................................................................. 168
6.7 Product Between Two Vectors ................................................................................................ 169
6.7.1 Dot Product (Scalar Product) ..................................................................................................... 169
6.7.2 Cross Product (Vector Product) ................................................................................................ 173
6.7.3 Why is Vector Product Not Commutative? ........................................................................... 178
6.7.4 2D Vectors vs. Complex Numbers ............................................................................................ 178
6.8 Linear Combinations of Vectors ............................................................................................. 179
6.9 Component Representation and Algebraic Vectors ........................................................ 180
6.9.1 Dot Product in Component Representation ......................................................................... 181
6.9.1.1 Dot Product as a Linear Combination of Components ..................................................... 183
6.9.2 Cross Product in Component Representation ..................................................................... 186
6.10 What Unites Algebraic and Geometric Vectors? ............................................................... 187
6.11 Modeling with Vectors (Scalar Fields and Vector Fields) ............................................. 189
6.11.1 Scalar Fields ...................................................................................................................................... 189
6.11.2 Vector Fields ..................................................................................................................................... 192
6.11.3 Vector Valued Functions of a Parameter ............................................................................... 196
6.12 Function Space .............................................................................................................................. 197
6.13 The Story So Far ............................................................................................................................ 198
7 MATRICES: EXTENDING LINEAR FUNCTIONS....................................................................... 201
7.1 Linear Combination Revisited ................................................................................................. 201
7.2 Multiple Linear Combinations ................................................................................................. 202
7.3 What Does a Matrix Represent? ............................................................................................. 207
7.4 Modeling with a Matrix .............................................................................................................. 210
7.4.1 Extending Linear Models ............................................................................................................. 212
7.5 Prelude to Matrix Multiplication ............................................................................................ 213
7.6 The Meaning Behind Matrix Multiplication ....................................................................... 214
7.6.1 Capturing All Linear Combinations ......................................................................................... 216
7.7 Matrix Multiplication as Function Composition ............................................................... 219
7.8 Row View vs. Column View of a Matrix ............................................................................... 222
7.9 Linear Transformations ............................................................................................................. 225
7.9.1 Performing Multiple Transformations at Once .................................................................. 227
7.9.2 Multiplying by a Matrix as Two Operations in One........................................................... 228
7.10 Systems of Linear Equations .................................................................................................... 229
7.11 The Story So Far ............................................................................................................................ 231
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8 SUMMARY: FUNCTIONS IN PERSPECTIVE.............................................................................. 233
8.1 The Meaning of Life, err…, Math ............................................................................................. 233
8.2 Objects .............................................................................................................................................. 233
8.3 Functions ......................................................................................................................................... 234
8.3.1 Linear Models ................................................................................................................................... 235
8.3.2 Non-Linear Models ......................................................................................................................... 238
8.4 The Story .......................................................................................................................................... 240
EPILOGUE ........................................................................................................................................ 243
ACKNOWLEDGEMENTS .................................................................................................................. 245
INDEX .............................................................................................................................................. 246
ABOUT THE AUTHOR ..................................................................................................................... 248

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P
REFACE
This is the math book I wish I had when I was in high school and college. If you were like
me, by the time you got to high school or college, you would have learned a considerable
amount of math, mostly algebra. However, you may feel that what you have learned is a
bunch of jumbled up ideas and a myriad of ways to manipulate numbers and symbols. To
make matters worse, you may feel that you could hardly explain why you ever learned
those things, or how one thing you learned is related to another. That’s how I felt about
math for a very long time. If you feel the same way, this book will give you an opportunity
to take a fresh look at what you have learned, and re-learn previously learned concepts
(and more) in a new light — one that will develop your intuition and help you see how
everything is connected to serve a common purpose in the real world. That intuition will
build a solid algebraic foundation to help you pursue other areas like Calculus, Computer
Science, Physics, and Engineering.

This book attempts to present a narrative of the function dynasty: how they came to be,
their purpose in life, what they have in common, what they consume and produce, their
kith and kin, how they appear in various forms in disguise, and how everything we learn in
algebra is related to them in one way or the other. If you are a student who is trying to
make sense of what you are learning, or a parent or an educator looking for ways to explain
math to kids in a cohesive and intuitive way, this book will give you many fresh ideas. It
offers a lot of analogies, figures, visual models, graphs, real-world examples, and gratuitous
explanations to help anyone connect with the fundamentals of math. Instead of formalism,
this book places emphasis on developing intuition.

There is another unique feature of this book. It tries to explain the connection between
math and computer programming with numerous examples of very short computer
routines that correspond to their mathematical counterparts. These routines are written in
JavaScript-like syntax to help anyone digest them easily. Computer science students and
anyone who is even remotely interested in computer programming will find the connection
between math and computer programming quite enlightening. You will soon realize that
understanding one helps you better understand the other. Similarly, this book should help
you better understand models used in many other scientific disciplines.

Finally, I should also mention what this book is not. This is not a textbook. This is not a test
preparation guide. This is not a cheat sheet. This book was written to serve one purpose —
to develop your intuition and help you see the underlying connections between different,
seemingly unrelated math concepts. That understanding will allow you to appreciate math
as a real-world tool, the same way engineers and scientists do. There is no guarantee that
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this book will improve your test scores. On rare occasions, it might lower your scores.
Other side effects may include nausea, dizziness, and allergic reactions to dry humor. If
symptoms persist, you should immediately stop reading this book and read a standard text
book.

Ruchira Sasanka
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S I: F
ECTION UNDAMENTALS
1 A RELATIONSHIP GUIDE: BROUGHT TO YOU BY FUNCTIONS
Why do we study math? What’s the meaning of all this number crunching and formula
juggling we learn in school? Although it may sound like a deep philosophical question that
needs lots of soul searching, the answer is simpler than you think.
With math, we try to build various models about the world we live in.
“But”, you may object, “that looks like a really difficult thing that requires a lot of expertise”.
Not so! It is much easier than you think.
Let’s start small. Let’s start with your dog.

Modeling your poodle with functions

Let’s say you realize that if you feed your dog some amount of dogfood a day, she produces
half that amount of poop (in weight) the next morning. For instance, if you give her 6
ounces of dogfood, she poops 3 ounces (sometimes, don’t you wonder whether she poops
more than what she eats!)
This is an example of a real-world relationship. The relationship is between the weight of
dogfood your dog eats and the weight of poop she produces. You can simply model this as:
Weight of poop = Weight of dogfood / 2
Congratulations! You have built a “mathematical model”, also known as a “mathematical
relation”. That is, a relationship between input and output.
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