Preface In the past 10 years or so, there has been a growing interest in applyingmath and statistics to oureverydaywork and lives. Why is that? Does it have to do with theaccelerated interest in “datascience”and “big data.”Or is it the promise ofmachinelearning and “artificialintelligence”changing our lives? Media headlinesareindundated with storiesaboutstudies, polls,and research findings. But thesesame headlinescause us to ask critical questions,and weare uncertainwhat thesecritical questionsactually are. I willmaketheargument that the disciplines ofmath and statistics havecaptured mainstreaminterest because ofthe growing availability of data,and we need math and statistics to makesense ofit. Yes, we do havescientifictools, machinelearning,and otherautomations thatcallto us likesirens. Weareto blindly trust these“black boxes,” devicesand softwares we dont understand but we usethemanyway. Whileit iseasy to believecomputersaresmarter than us (and this ideais frequentlymarketed), thereality cannot be morethe opposite. This disconnectcan be precarious on so many levels. Do you reallywantan “algorithm” or“AI” performing criminalsentencing or driving a vehicle, but nobody including the developercan explainwhy itcameto aspecific decision? Explainability is the next frontier ofstatisticalcomputing and AI. This can only beginwhenwe open up the“black box”and uncover the math. Youmay also ask howcan a developer not knowhowtheir own algorithmworks? We willtalk about that in thesecond half ofthe book whenwe discuss machinelearning techniques,and emphasize whywe need to understand the math behind the black boxes we build. To another point, thereason datais being collected on a massivescaleis largely dueto connected devicesand their presencein oureveryday lives. We no longer solely usetheinternet on a desktop or laptop computer. We nowtakeit with us in our smart phones,cars,and household devices. This has subtly enabled atransition over the past two decades. Data has nowevolved froman operationaltoolto something that iscollected and analyzed for less defined objectives. Asmartwatch isconstantly collecting data on our heart rate, breathing, walking distance,and other markers. Then it uploads that datato acloud to beanalyzed with other users. Our driving habitsare being collected by computerized cars,and being used by manufacurers to learn our driving style orenable“self-driving” vehicles. Even “smart toothbrushes”arefinding their way into drug stores, which track brushing habitsand storethat datain acloud. Whether smart toothbrush datais usefuland essentialisanother discussion! All ofthis datacollection is permeating every corner of our lives. Itcan be overwhelming,and a whole book can be written on privacy concernsand ethics. But thisavailability of dataalso creates opportunities to leverage math and statistics in newways,and create moreexposure outsideacademic environments. Wecan learnmoreabout the human experience, improve product design and application,and optimizecommercialstrategies. If you understand theideas presented in this book, youwill beableto unlock the value held in our data-hording infrastructure. This does not imply that data and statisticaltoolsareasilver bullet to solveallthe world’s problems, but it has given us newtools that wecan use. This growing availability of data has made way for“datascience”and “big data”to become demanded professions. We defineessentialmath asan exposureto probability, linearalgebra, statistics,and machinelearning. If you areseeking acareer in datascience, machinelearning, orengineering, thesetopicsare necessary. I willthrowin justenough college math and calculus necessary to better understand what goes in the“black box”libraries youwillencounter. With this book, Iaimto givereadersan exposureto differentmathematicaland statisticalareas that will beapplicableto real-world problems. Thefirst fourchapterscover foundationalmath concepts including practicalcalculus, probability, linearalgebra,and statistics. Thelast threechapters willsegue into machinelearning. The ultimate purpose ofteachingmachinelearning is to integrateeverythingwelearn,and demonstrate practicalinsights in using machinelearning and statisticallibraries beyond a“black box” understanding. The only toolthat is needed to followexamples isa Windows/Mac/Linux computerand a Python 3 environment of yourchoice. The primary Python libraries we will need are numpy, scipy, sympy,and sklearn. If you are unfamiliar with Python, it isafriendly and easy-to-use programming language withmassivelearning resources behind it. HerearesomeI recommend: Data Sciencefrom Scratch 2nd Edition (O’Reilly) by JoelGrus - Thesecond chapter ofthis book has the bestcrash coursein Python I haveencountered. Even if you have never written code before, Joel doesafantasticjob getting you up and runningwith Python effectively in theshortest time possible. It isalso a great book to have on your shelfand to apply your mathematical knowledge! Python for the BusyJava Developer (Apress) byDeepak Sarda- If you areasoftwareengineercoming fromastatically-typed, objectoriented programming background, this is the book to grab. As someone who started programmingwith Java, I havea deep appreciation howDeepak shares Python featuresand relates themto Java developers. If you have done.NET, C++, or other C-likelanguages youwill probably learn Python effectively fromthis book as well. This book will notmake you an expert or give you PhDknowledge. Iavoid mathematicalexpressions full ofGreek symbols,and instead striveto use plainEnglish in its place. But, what this book will do is make youmorecomfortabletalking aboutmath and statistics, giving you essential knowledge to navigatetheseareas successfully. I believethe widest path to success is not having deep, specialized knowledgein onetopic, but instead having exposureand practical knowledgeacross severaltopics. That is the goal ofthis book,and youwilllearn justenough to be dangerousand ask those onceelusivecritical questions. So let’s getstarted! Chapter 1. Basic Math and Calculus Review A N O T E F O R E A R L Y R E L E A S E R E A D E R S WithEarlyReleaseebooks, you get books in theirearliest form—theauthor’s rawand unedited contentas theywrite—so you can take advantage ofthesetechnologies long beforethe officialrelease ofthesetitles. This will bethe 1stchapter ofthefinal book. If you havecommentsabout howwe might improvethecontentand/orexamples in this book, or if you notice missingmaterialwithin thischapter, pleasereach out to theauthorat thomasnield@live.com. We will kick offthefirstchaptercoveringwhat numbersareand howvariablesand functions work on a Cartesian system. We willthen cover exponentsand logarithms. After that we willlearn thetwo basic operations ofcalculus: derivativesand integrals. Before we diveinto theapplied areas ofessentialmath such as probability, linearalgebra, statistics,and machinelearning, weshould probably review afewbasic math and calculusconcepts. Before you drop this book and run screaming, do not worry! I will present howto calculateslopesand areas forafunction in a way youwere probably not taught in college. We got Python on our side, nota penciland paper. I willmakethesetopicsas tightand practicalas possible, focusing only onwhat will help us in laterchaptersand fall under the“essentialmath” umbrella. T H I S I S N O T A F U L L M A T H C R A S H C O U R S E ! This is by no means a comprehensive review of high schooland college math. If you want that, a great book to check out is No Bullshit Guide to Math and Physics by Ivan Savov. The first few chapters contain the best crash course on high schooland college math I have ever seen. The book Mathematics 1001 by Dr. Richard Elwes has some great content as well, and in bite-sized explanations. Number Theory Whatare numbers?I promiseto not betoo philosophicalin this book, butare numbers notaconstruct we have defined? Why do we havethe digits 0 through 9,and not have more digits than that? Why do we havefractionsand decimalsand not just whole numbers? Thisarea ofmathwhere we muse about numbersand whywe designed themacertainway is known as number theory. Numbertheory goesallthe way back to ancient times, where mathematicians study different number systemsand whywe haveaccepted themthe waywe do today. Hereare different number systems that youmay recognize: Natural Numbers Thesearethe numbers 1, 2, 3, 4, 5… and so on. Only positive numbersareincluded here,and aretheearliest known system. Natural numbers areso ancientcavemen scratched tallymarks on bonesand cave walls to keep records. Whole Numbers Adding to natural numbers, theconcept of“0”was lateraccepted and wecallthese“whole numbers.”The Babyloniansalso developed the useful ideafor place-holding notation forempty “columns” on numbers greater than 9, such as“10”,“1000”, or“1090.”Thosezeros indicate no value occupying thatcolumn. Integers Integers include positiveand negative whole numbersas wellas 0. We may takethemfor granted, butancientmathematicians were deeply distrusting oftheidea of negative numbers. But when you subtract 5 from3, you get -2. This is usefulespeciallywhen itcomes to finances where we measure profitsand losses. In 628 AD,an Indianmathematician named Brahmaguptashowed why negative numbers were necessary for arithmeticto progress,and thereforeintegers becameaccepted. Rational Numbers Any number that you can expressasafraction, such as 2 3 , isarational number. This includesallfinite decimalsand integers sincethey can be expressed as fractions too, such as 687 100 = 6 . 87 and 2 1 = 2 respectively. They arecalled rational becausethey areratios. Rational numbers were quickly deemed necessary bececause becausetime, resources,and other quantitiescould notalways be measured in discrete units. Milk does notalwayscomein gallons. We may haveto measureitas parts ofa gallon. IfI run for 12 minutes, Icannot beforced to measurein whole miles when in actuality I ran 9 10 ofa mile. Irrational Numbers Irrational numberscannot beexpressed asafraction. This includes thefamous Pi π , squareroots ofcertain numbers like 2 ,and Euler’s numbere whichwe willlearn about later. These numbers havean infinite number of decimal digits, such as π = 3 . 141592653589793238462