Doing Math with Python

D o i n g M a t h with Python

U s e P r o g r a m m i n g t o Explore Al g e b r a , S t a t i s t i c s , Calculus, and More!

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Bri e f Cont e n t s

Acknowledgments	xii
Introduction	xv
Chapter 1: Working with Numbers	1
Chapter 2: Visualizing Data with Graphs	27
Chapter 3: Describing Data with Statistics	61
Chapter 4: Algebra and Symbolic Math with SymPy	93
Chapter 5: Playing with Sets and Probability	121
Chapter 6: Drawing Geometric Shapes and Fractals	149
Chapter 7: Solving Calculus Problems	177
Afterword	209
Appendix A: Software Installation	213
Appendix B: Overview of Python Topics	221
Index	237

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Cont e n t s in De tail

ACKNOWLEDGMENTS	xiii
INTRODUCTION	xv
Who Should Read This Book	xv
What’s in This Book?	xv
Scripts, Solutions, and Hints	xvi
1
WORKING WITH NUMBERS	1
Basic Mathematical Operations	1
Labels: Attaching Names to Numbers	4
Different Kinds of Numbers	4
Working with Fractions	5
Complex Numbers	6
Getting User Input	8
Handling Exceptions and Invalid Input	9
Fractions and Complex Numbers as Input	11
Writing Programs That Do the Math for You	12
Calculating the Factors of an Integer	12
Generating Multiplication Tables	15
Converting Units of Measurement	17
Finding the Roots of a Quadratic Equation	20
What You Learned	22
Programming Challenges	22
#1: Even-Odd Vending Machine	22
#2: Enhanced Multiplication Table Generator	23
#3: Enhanced Unit Converter	23
#4: Fraction Calculator	23
#5: Give Exit Power to the User	24
2
VISUALIZING DATA WITH GRAPHS	27
Understanding the Cartesian Coordinate Plane	28
Working with Lists and Tuples	29
Iterating over a List or Tuple	31
Creating Graphs with Matplotlib	32
Marking Points on Your Graph	33
Graphing the Average Annual Temperature in New York City	35
Comparing the Monthly Temperature Trends of New York City	38
Customizing Graphs	41
Saving the Plots	45
Plotting with Formulas	46
Newton’s Law of Universal Gravitation	46
Projectile Motion	48
What You Learned	54
Programming Challenges	55
#1: How Does the Temperature Vary During the Day?	55
#2: Exploring a Quadratic Function Visually	55
#3: Enhanced Projectile Trajectory Comparison Program	56
#4: Visualizing Your Expenses	56
#5: Exploring the Relationship Between the Fibonacci Sequence and the Golden Ratio	59
3
DESCRIBING DATA WITH STATISTICS	61
Finding the Mean	62
Finding the Median	63
Finding the Mode and Creating a Frequency Table	65
Finding the Most Common Elements	66
Finding the Mode	67
Creating a Frequency Table	69
Measuring the Dispersion	71
Finding the Range of a Set of Numbers	71
Finding the Variance and Standard Deviation	72
Calculating the Correlation Between Two Data Sets	75
Calculating the Correlation Coefficient	76
High School Grades and Performance on College Admission Tests	78
Scatter Plots	81
Reading Data from Files	83
Reading Data from a Text File	84
Reading Data from a CSV File	86
What You Learned	89
Programming Challenges	89
#1: Better Correlation Coefficient–Finding Program	89
#2: Statistics Calculator	89
#3: Experiment with Other CSV Data	89
#4: Finding the Percentile	89
#5: Creating a Grouped Frequency Table	90
4
ALGEBRA AND SYMBOLIC MATH WITH SYMPY	93
Defining Symbols and Symbolic Operations	94
Working with Expressions	96
Factorizing and Expanding Expressions	96
Pretty Printing	97
Substituting in Values	100
Converting Strings to Mathematical Expressions	103
Solving Equations	105
Solving Quadratic Equations	106
Solving for One Variable in Terms of Others	106
Solving a System of Linear Equations	108
Plotting Using SymPy	108
Plotting Expressions Input by the User	111
Plotting Multiple Functions	113
What You Learned	115
Programming Challenges	115
#1: Factor Finder	115
#2: Graphical Equation Solver	115
#3: Summing a Series	116
5 PLAYING WITH SETS AND PROBABILITY	121
What’s a Set?	121
Set Construction	122
Subsets, Supersets, and Power Sets	124
Set Operations	126
Probability	131
Probability of Event A or Event B	133
Probability of Event A and Event B	134
Generating Random Numbers	134
Nonuniform Random Numbers	137
What You Learned	140
Programming Challenges	140
#1: Using Venn Diagrams to Visualize Relationships Between Sets	140
#2: Law of Large Numbers	143
#3: How Many Tosses Before You Run Out of Money?	144
#4: Shuffling a Deck of Cards	144
#5: Estimating the Area of a Circle	145
6 DRAWING GEOMETRIC SHAPES AND FRACTALS	149
Drawing Geometric Shapes with Matplotlib’s Patches	150
Drawing a Circle	151
Creating Animated Figures	153
Animating a Projectile’s Trajectory	156
Drawing Fractals	158
Transformations of Points in a Plane	158
Drawing the Barnsley Fern	163
What You Learned	168
Programming Challenges	168
#1: Packing Circles into a Square	168
#2: Drawing the Sierpiński Triangle	170
#3: Exploring Hénon’s Function	171
#4: Drawing the Mandelbrot Set	172
7 SOLVING CALCULUS PROBLEMS	177
What Is a Function?	178
Domain and Range of a Function	178
An Overview of Common Mathematical Functions	178
Assumptions in SymPy	180
Finding the Limit of Functions	181
Continuous Compound Interest	183
Instantaneous Rate of Change	184
Finding the Derivative of Functions	185
A Derivative Calculator	186
Calculating Partial Derivatives	187
Higher-Order Derivatives and Finding the Maxima and Minima	188
Finding the Global Maximum Using Gradient Ascent	191
A Generic Program for Gradient Ascent	195
A Word of Warning About the Initial Value	196
The Role of the Step Size and Epsilon	197
Finding the Integrals of Functions	200
Probability Density Functions	201
What You Learned	205
Programming Challenges	205
#1: Verify the Continuity of a Function at a Point.	205
#2: Implement the Gradient Descent	205
#3: Area Between Two Curves	206
#4: Finding the Length of a Curve	207
AFTERWORD	209
Things to Explore Next	209
Project Euler	210
Python Documentation	210
Books	210
Getting Help	211
Conclusion	211
A
SOFTWARE INSTALLATION	213
Microsoft Windows	214
Updating SymPy	215
Installing matplotlib-venn	215
Starting the Python Shell.	215
Linux	216
Updating SymPy	217
Installing matplotlib-venn	217
Starting the Python Shell.	217
Mac OS X	217
Updating SymPy	220
Installing matplotlib-venn	220
Starting the Python Shell.	220
B
OVERVIEW OF PYTHON TOPICS	221
if name == ‘main‘	221
List Comprehensions	223
Dictionary Data Structure	224
Multiple Return Values	226
Exception Handling	228
Specifying Multiple Exception Types	228
The else Block.	230
Reading Files in Python.	230
Reading All the Lines at Once.	232
Specifying the Filename as Input.	232
Handling Errors When Reading Files.	232
Reusing Code	235
INDEX	237

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Acknowl edg me n t s

SymPy forms a core part of many chapters in this book and I would like to thank everyone on the SymPy mailing list for answering my queries patiently and reviewing my patches with promptness. I would also like to thank the matplotlib community for answering and clearing up my doubts.

I would like to thank David Ash for lending me his Macbook, which helped me when writing the software installation instructions.

I also must thank every writer and thinker who inspired me to write, from humble web pages to my favorite books.

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