book

Math for Programming

by Ronald T. Kneusel

April 2025

Intermediate to advanced

504 pages

13h 6m

English

No Starch Press

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Who Is This Book For?What I Expect You to Know AlreadyWhat Can You Expect to Learn?How to Use This Book
Numbers and Number BasesBinary, Octal, and Hexadecimal NumbersConversions Between Number BasesHow Computers Represent NumbersIntegersFloating-Point NumbersWhat You Need to Know About Floating-Point ArithmeticRound-Off ErrorUnrepresentable NumbersSummary

Concerning SetsSpecial SetsSet OperationsSet Operations in PythonVenn DiagramsSubsets and SupersetsPower SetsThe Laws of Set TheoryExperimenting with Number Sets in PythonAbstract Algebra and GroupsExamples of Groups, Magmas, and MonoidsSpecial GroupsCantor: To Infinity and BeyondThe Cardinality of Infinite SetsThe Continuum Hypothesis and Infinities Without EndComputable and Uncomputable NumbersUncomputable PythonSummary
Definition and LawsBoolean Algebra Leads to Digital LogicSet Theory as a Boolean AlgebraPropositional Logic as a Boolean AlgebraThe Set {0,1} as a Boolean AlgebraThe Truth Tables of Digital LogicBoolean FunctionsDuals and ComplementsBoolean Functions in CodeCanonical Normal FormsSum of ProductsProduct of SumsKarnaugh MapsAlgebraic Simplification of Boolean ExpressionsDigital CircuitsBoolean Functions as Digital CircuitsAll You Need Is NAND (or NOR)A Binary AdderLogic Gate ConstructionSummary
FunctionsVisualization of FunctionsOnto, One-to-One, and Inverse FunctionsComposition of FunctionsBinary RelationsCartesian ProductsDefinitionRelations as MatricesCounting RelationsRelations on Infinite SetsCompositionConverse and ComplementBinary Relations as Graphsn-ary RelationsBinary Relations in CodeEquivalence Relations and ClassesEquivalence ClassesNumber SetsPartial OrderingsSummary
Weak InductionStrong InductionThe Chicken Nuggets ProblemThe Fundamental Theorem of ArithmeticCautionary TalesForgetting the Base CaseReversing the ImplicationShowing the Inductive Hypothesis Holds for All mProving with Loop InvariantsInteger Square RootBubble SortBinary SearchGnome SortSummary
Recurrence RelationsLinear Recurrence RelationsThe Collatz SequenceNonlinear Recurrence RelationsLogistic MapMandelbrot SetSolving Recurrence RelationsFirst-Order RelationsSecond-Order RelationsRecursionPowersFactorialsFibonacci SequenceQuicksortTowers of HanoiSummary
PrimesProving Euclid’s TheoremLocating PrimesIdentifying Prime SubsetsExploring Quirky PrimesRunning the Primality TestInvestigating the Goldbach ConjectureDivisibilityDivisibility PropertiesThe Least Common Multiple and the Greatest Common DivisorModular ArithmeticCongruencesModular EquationsDiophantine EquationsLinearNonlinearInteger SequencesEuler’s Totient FunctionAliquot Sum and SequenceHarshad NumbersNarayana’s CowsDeranged SubfactorialsSmith and Ruth–Aaron NumbersSequence AudioSummary
The Principles of CountingAn Example: Hats in the ParkWomen Without HatsMen with (and Without) HatsLoose EndsSum and Product RulesInclusion-ExclusionThe Pigeonhole PrinciplePermutations and CombinationsPermutationsCombinationsThe Binomial TheoremSummary
Basic Graph ConceptsComplete GraphsGraph IsomorphismsRepresenting Graphs in CodeAdjacency ListsAdjacency MatricesBreadth-First and Depth-First Traversal and SearchingBreadth-First TraversalDepth-First TraversalTraversals as SearchesThe Shortest Path Between NodesUnweighted Shortest PathDijkstra’s Algorithm for Weighted GraphsDirected Acyclic Graphs and Topological SortWorking Through an ExampleCoding Topological SortSummary
Defining TreesSpanning TreesLocating Spanning TreesFinding Minimum Spanning TreesRooted and Ordered TreesBinary TreesBinary Tree Traversals and SearchesUsing Breadth-First Tree TraversalUsing Depth-First Tree TraversalSearching a TreeBinary Trees in CodeCreating Traversals in CodeUsing the CodeThe Animals GameSummary
Events and Random VariablesThe Rules of ProbabilityProbability of an EventSum RuleProduct RuleSum Rule UpdateConditional ProbabilityTotal ProbabilityJoint and Marginal ProbabilitiesJoint Probability TablesChain Rule for ProbabilityProbability DistributionsHistogramsDiscrete DistributionsContinuous DistributionsThe Central Limit Theorem and the Law of Large NumbersBayes’ TheoremSummary
Types of DataNominalOrdinalIntervalRatioSummary StatisticsMeans, Median, and ModeMeasures of VariationQuantiles and Box PlotsQuantiles and PercentilesBox PlotsRobustness and Sensitivity to OutliersCorrelationPearson CorrelationSpearman CorrelationA Cautionary Tale: Anscombe’s QuartetHypothesis TestingA GedankenexperimentIndependent SamplesPaired SamplesConfidence IntervalsConfidence Intervals and Hypothesis TestsConfidence Intervals and Sample SizeSummary
Vectors and Vector SpacesDefinitionsVectors as Geometric ObjectsVectors in ℝnRow and Column VectorsVector OperationsTreasure Island ReduxVector Space ConceptsSubspaces, Linear Combinations, and Spanning SetsLinear Independence and Basis SetsMatricesExploring Matrix ExamplesForming Vector SpacesSolving Systems of Linear EquationsGauss–Jordan EliminationLinear Vector EquationsSquare MatricesProperties of Square MatricesMatrix RankDeterminantsLinear TransformationsAffine TransformationsIterated Function SystemsEigenvalues and EigenvectorsFinding EigenvaluesFinding EigenvectorsSummary
DerivativesBasic RulesRules for Trigonometric FunctionsThe Chain RuleRules for Exponentials and LogarithmsMinima and Maxima of FunctionsUsing the Second Derivative TestLocating Stationary Points with Newton’s MethodPartial DerivativesMixed Partial DerivativesThe Chain Rule for Partial DerivativesDerivatives in CodeNumeric DifferentiationAutomatic DifferentiationSymbolic DifferentiationOptimization with Gradient DescentSummary
Curves and AreasIndefinite IntegralsA Few Indefinite IntegralsAntiderivativesA Fistful of Integration TricksDefinite IntegralsThe Throwing Darts MetaphorMonte Carlo IntegrationTrapezoidal RuleSimpson’s RuleDefinite Integrals with SymPyAdaptive Numerical IntegrationSummary
Ordinary Differential EquationsFirst-OrderSecond-OrderSolving ODEs NumericallyEuler’s MethodRunge–Kutta MethodWorked ExamplesSimple PendulumProjectile Motion with DragSIR Epidemic ModelLorenz AttractorSummaryFinal Words

Overview

Every great programming challenge has mathematical principles at its heart. Whether you’re optimizing search algorithms, building physics engines for games, or training neural networks, success depends on your grasp of core mathematical concepts.

In Math for Programming, you’ll master the essential mathematics that will take you from basic coding to serious software development. You’ll discover how vectors and matrices give you the power to handle complex data, how calculus drives optimization and machine learning, and how graph theory leads to advanced search algorithms.

Through clear explanations and practical examples, you’ll learn to:

Harness linear algebra to manipulate data with unprecedented efficiency
Apply calculus concepts to optimize algorithms and drive simulations
Use probability and statistics to model uncertainty and analyze data
Master the discrete mathematics that powers modern data structures
Solve dynamic problems through differential equations

Whether you’re seeking to fill gaps in your mathematical foundation or looking to refresh your understanding of core concepts, Math for Programming will turn complex math into a practical tool you’ll use every day.