Mathematics

For a reader beginning from arithmetic, Kiselev's Arithmetic is an English adaptation of Kiselev's classic text that teaches why each method works before the procedure, as a foundation for algebra and later problem solving,[26] and Mathematics for the IGCSE is a 680-page teacher's textbook for the IGCSE examination that covers the syllabus and goes beyond it where the author thought it worthwhile.[25] Simple Calculus puts the trigonometry and algebra a reader needs before calculus in the same hundred pages as limits, derivatives, integration and differential equations.[27] A reader who wants to see where mathematics is used before studying it has Applied Mathematics for the Modern World, which assumes no background and introduces vectors as shopping lists and recipes,[8] and Patterns We Don't See, a tour of symmetry, spirals, fractals and queues in daily life that promises no formulas.[30]

For the undergraduate or the self-taught reader at that level, three free textbooks carry most of the weight of this article. Linear Algebra is a standard first course, written for students with a semester of calculus, that proves everything but takes a developmental approach, with motivation and many examples, and whose free sample is the entire fourth edition.[1] APEX Calculus is a three-volume calculus sequence, also complete in its sample, that omits most proofs on purpose so that students are not confused by them.[2] Discrete Mathematics for Computer Science supplements a Coursera specialisation with 464 problems, 225 Python snippets and 219 automatically graded exercises, and asks the reader to try each problem before it explains anything.[3] Its larger sibling, Discrete Mathematics, is the text for a year-long university course and runs to 837 pages.[4] For statistics, An Introduction to Statistics is written for first-year business students with no prior knowledge,[24] and Probability for Lemurs explains probability one illustrated page at a time, with lemurs.[10]

For the programmer, Logic for Programmers is addressed to developers with little or no formal mathematics and chooses accessibility over rigour,[13] Certainty by Construction teaches the Agda proof assistant by constructing mathematics in it,[15] and The Hundred-Page Guide to the Core Mathematics of Machine Learning asks for some Python and school mathematics and ends with regression models written from scratch in NumPy.[9] The research-level books at the far end of the shelf, the group theory theses of Two Essays on Group Theory and Supersolubility and Finitary Groups, the topology of General Topology as Ordered Semigroup Actions: Book 1 and the dynamical systems of Notes on Dynamical Systems for Actor-Critic Learning, assume graduate mathematics and are described below for completeness rather than as places to begin.[40][41][42][43]

Mathematics is older than writing in the sense that counting and measuring are, and its written record begins with the arithmetic and geometry of Mesopotamia and Egypt. The Greeks made it a deductive subject: Euclid's Elements, about 300 BC, organised geometry and number theory as theorems proved from axioms, and the method of proof it established is still the one the library's textbooks teach. One of the library's essays introduces Euclid as the ancient Greek mathematician sometimes called the father of geometry, before using his algorithm for the greatest common divisor to solve a programming puzzle.[14] Algebra took its name and much of its early form from the Islamic world of the ninth century, and the Hindu-Arabic numerals that make modern arithmetic possible reached Europe by the same route.

The calculus is the seventeenth century's contribution. The free calculus text in the library opens with the two problems that eluded the mathematicians of the time, the area of an arbitrary shape and the rate of change of a quantity that does not change uniformly, and credits Isaac Newton and Gottfried Leibniz with independently formulating a system of computing that solved both and showed how they were connected.[2] The differential equations of mathematical physics followed in the next century. A short course on partial differential equations begins its history in 1746, when Jean le Rond d'Alembert set out to describe a vibrating string and found that its displacement is the superposition of two waveforms travelling in opposite directions, which raised the question of what data is needed to pick out one solution from the infinitely many the equation allows.[6] Its companion volume adds Joseph Fourier's 1822 treatise on heat and Pierre-Simon Laplace's equation for gravitational and electrostatic potential, three equations whose solutions behaved so differently that they seemed to belong to separate subjects, until Augustin-Louis Cauchy's formulation of the problem of initial data and Paul du Bois-Reymond's classification explained why.[7]

The nineteenth century made the subject rigorous and abstract: analysis was rebuilt on the limit, groups and other algebraic structures were defined in their own right, and Georg Cantor's set theory became the common foundation. The twentieth century tested that foundation, with Gödel's incompleteness theorem and the undecidability of the halting problem, both of which appear as applications in one of the library's discrete mathematics texts,[4] and then built alternatives to it: one author describes category theory as an alternative foundation for mathematics, as opposed to the traditional set-theoretic one, and repeats the comparison that doing mathematics in sets is like programming in assembly language while doing it in categories is like a modern language.[15] The same century put mathematics inside the computer. A general relativity primer recalls that Einstein's physical insights were later expressed precisely in the language of differential geometry,[19] and the newest books on the shelf describe the subject as the hidden blueprint of the technology around the reader, from the graph theory in a phone's directions to the prime numbers that protect a purchase.[8]

The library's shelf begins before the university. Kiselev's Arithmetic is an edited English translation and adaptation of Alexander P. Kiselev's Arithmetic, the text that, in its publisher's words, trained generations of mathematicians by teaching them to think rather than memorise; the edition keeps the original's logical structure, definitions and graduated exercises, puts concepts before procedures, and is addressed to students, parents and tutors, and adults rebuilding their fundamentals.[26] Mathematics for the IGCSE, subtitled "and a bit more", is the book its author, a teacher, says he would have wanted as a student taking the IGCSE: it covers the basics and the repetitive, algorithmic parts of the syllabus and expands above it where it can.[25] Simple Calculus bridges school and university by treating trigonometry and algebra, then limits, derivatives, function analysis, integration and differential equations, with worked examples and graphs for each derived result.[27]

Popular mathematics is the other way in. Patterns We Don't See was written by a cadet at an engineering school in Voronezh whose curiosity, its description says, grew beyond the classroom when a few plants on a windowsill led to questions about symmetry and Fibonacci sequences, snowflakes and honeycombs to geometric optimisation, and a piano to the structure of music; it goes on to fractals, probability, social networks, queues and cities, and is explicitly not a textbook.[30] 50 Math Puzzles - Solved using functional Julia offers fifty classic puzzles, the Twelve Days of Christmas, Wheat on a Chessboard, Fibonacci's Rabbits and the Census Puzzle among them, each solved in the Julia language, for amusement by the author's own account.[31] Ten Essays on Fizz Buzz belongs here too, though it is a Python book first: its ten essays each solve the Fizz Buzz interview problem a different way, and the second essay is a short course in prime numbers, trial division, the sieve and prime factorisation, with timings that show a sieve marking a boolean array finding every prime below ten million faster than a sieve that rebuilds its candidate list found those below a hundred thousand.[14] At the lighter end of statistics, How to Lie with Numbers, Stats & Graphs is a box set of two books, Truth, Lies and Statistics in eight chapters and Graphs Don't Lie in nine, in which a statistician walks through the ways politicians, corporations and the press misrepresent data with numbers, percentages and pie charts.[32]

Calculus is the mathematics of change and accumulation, built on the limit. APEX Calculus defines the word itself as a method of calculation or reasoning, so that computing a sales tax or finding the area of a polygon by cutting it into triangles is each a calculus, and the one Newton and Leibniz built became the calculus; its foundation, the book says, is the limit, and the first chapter approaches limits graphically and numerically before giving the formal definition and the properties that make finding them tractable.[2] The text is a three-volume series matching the usual three-semester sequence, limits, derivatives and the basics of integration, then integration, sequences, series and Taylor polynomials, then parametric and polar curves, vectors and functions of several variables. It is unusual in two ways its preface explains: most proofs are left out, because students are often confused by them or worry that they should have constructed them, and nearly every graph of an object in space in the PDF can be rotated and zoomed. It is published by a consortium of authors, APEX, under a Creative Commons licence, and the free sample is the whole book.[2] The shorter Simple Calculus covers the same ground for a first encounter in a hundred pages.[27]

Analysis is calculus made rigorous and carried into higher dimensions, and the library's one text on it is a companion to a classic. Analysis on Manifolds and Calculus on Manifolds: Solution Manuals of Selected Exercise Problems is a solution manual for selected exercises from James R. Munkres' Analysis on Manifolds, with a contents list that mirrors the textbook's chapters, the algebra and topology of n-dimensional space, differentiation, integration, and on to manifolds and differential forms; the sample's first solutions concern matrix norms, left and right inverses and compactness. Its description adds that the solutions are detailed from chapter five on, that the closing summary serves as a crash course on manifolds and differential forms, and that a manual for Spivak's Calculus on Manifolds was added in 2024.[5]

Linear algebra studies vectors, the spaces they form and the linear maps between them, and it has become the working language of data. Linear Algebra, Jim Hefferon's text for a first US undergraduate course, opens chapter one with two problems from high-school science, three masses on a balance and the proportions in which toluene and nitric acid combine, and shows that both come down to a system of linear equations. It then presents Gauss's method, the algorithm that always works: swap two equations, multiply one by a nonzero constant, or add a multiple of one to another, and the solution set does not change, a theorem the book proves before using the method throughout.[1] From there the course runs through echelon form, vector spaces, linear maps and their matrices, determinants, and eigenvalues and Jordan form, with short topics closing each chapter on lines of best fit, Markov chains, computer graphics, page ranking and coupled oscillators. Its preface explains the developmental approach: early courses focus on calculating and later ones demand mathematical maturity, and a linear algebra course is the ideal place to make that transition, early enough to pay off and late enough that the room holds only students serious about mathematics. Every exercise is answered in full in a separate answers book.[1]

Two other texts take the computational side. A Quick Steep Climb Up Linear Algebra is a free and open-source linear algebra book from the same author and companion site as the discrete mathematics walk described below, aimed, by its description, at the mathematics that computer science students will actually use.[29] Computing Matrix Algebra is a 73-page booklet its author calls a cheat sheet: fifteen short chapters from assigning a matrix through multiplication, determinants, inverses and linear systems to the LU, Cholesky, QR and eigen factorisations, each in five sections that state the mathematics, compute the operation in R, implement it in Rcpp and then call the Armadillo and Eigen C++ libraries, so that the reader sees both the algebra and the cost of doing it from scratch. It has, by its own account, no theorems or proofs and says nothing about numerical stability.[12]

The applied guides begin here because machine learning does. Applied Mathematics for the Modern World introduces the scalar as a single number, the vector as an ordered list and the matrix as a grid, with a four-by-four grayscale image as its first matrix, and shows a matrix-vector product stretching a vector as the transformation behind computer graphics; its application chapter builds a user-by-movie ratings matrix, measures the similarity of two users by the dot product of their rows, and names singular value decomposition as what a real recommendation engine uses.[8] The Hundred-Page Guide to the Core Mathematics of Machine Learning makes the same move with a house described by its size and number of bedrooms, a vector that is at once a point and an arrow in a two-dimensional feature space, and calls it the first great leap of abstraction in machine learning that every sample in a dataset is a point in a high-dimensional space; later chapters, by its contents, treat the dot product as the heart of the neuron, matrix multiplication as the language of neural networks and principal component analysis from scratch.[9]

Discrete mathematics is the mathematics of finite and countable structures, proofs, counting, graphs, logic and sets, and it is the branch the library holds most of, because it is what computer science asks of its students. Discrete Mathematics for Computer Science, by five authors including Alexander S. Kulikov, is built around active learning: interactive puzzles that let the reader invent a key idea, quizzes graded on the spot, code snippets to tweak and run, and programming challenges, on the principle, quoted from Donald Knuth, that one does not understand a thing until one has tried to program it. Its first chapter asks why proofs are necessary and answers with the library function that runs billions of times and the air traffic controller who receives one wrong value, then teaches by example: a chessboard missing one corner cannot be tiled by dominoes because sixty-three cells would need thirty-one and a half tiles, and a board missing two opposite corners cannot be tiled either, because every domino covers one dark and one light cell and the board has thirty of one and thirty-two of the other, the difference between a necessary condition and a sufficient one being the lesson.[3] The four parts cover mathematical thinking, combinatorics and probability, graph theory, and number theory and cryptography, with projects on the 15-puzzle, dice games, the Gale-Shapley algorithm and attacks on RSA.[3]

Discrete Mathematics, by Nikolai Chukhin and Kulikov, accompanies a year-long course in a computer science and artificial intelligence degree at Neapolis University Pafos. Its five chapters, mathematical thinking, logic and set theory, combinatorics, probability and graph theory, each carry applications, theory problems, optional sections and a project: proofs of existence, optimality and algorithm correctness lead to the 15 puzzle and A-star search; propositional logic, satisfiability and Boolean circuits lead to synthesising an optimal circuit with a SAT solver; set theory reaches Cantor's diagonal argument, the undecidability of halting and Gödel's first incompleteness theorem; and combinatorics runs from arrangements and Catalan numbers through backtracking and branch and bound to recurrences and generating functions, with portfolio selection as its project.[4] A Cool Brisk Walk Through Discrete Mathematics is a free and open-source alternative from a lecturer at the University of Mary Washington, with a companion site, selected for what computer science students use.[28] Combinatorial Thinking in AI VOL-1 treats combinatorics as the backbone of artificial intelligence, with counting principles, Stirling's formula, permutations, combinations and Pascal's triangle, partitions and lattices in its first part and the state-space explosion of search as the running application.[33] Discrete Mathematical Algorithm, and Data Structure pairs Boolean algebra, De Morgan's laws, set theory and propositional logic with their implementation as data structures, showing each example in seven languages.[34]

Algorithms are where discrete mathematics meets code, and Learning Algorithms Through Programming and Puzzle Solving, by Kulikov and Pavel Pevzner, is the companion to a Coursera specialisation that its sample says has enrolled over 700,000 students since 2016. It argues that learning algorithms without implementing them is like learning surgery from an anatomy book, and backs the argument with an automated grader built from the ten to fifteen most frequent errors students make on each problem, over 20,000 test cases in all; the book's own thirty challenges begin with Fibonacci numbers and their last digits, the greatest common divisor, the least common multiple and making change.[20]

Probability measures uncertainty and statistics draws conclusions from data, and the library teaches both at the introductory level with unusual variety. Probability for Lemurs is an illustrated handbook in which every idea is a page with a lemur story and a worked calculation. Its sampled chapter on Bayes' theorem supposes that one banana in thirty-six has a bug and that a green spot indicates a bug 99 times in a hundred in either direction, lays the four outcomes out as a table, matching false positives and false negatives to the two types of error from hypothesis testing, and works out that a banana without a spot has about three chances in ten thousand of a bug, so that acting on the base rate alone would be 96 times too fearful, while a spotted one has about a 73 per cent chance, not the 99 the clue alone suggests. The chapters that follow begin the Poisson process with nocturnal lemurs listening for falling coconuts and Markov chains with lemurs leaping from tree to tree; the contents mark several chapters as not yet written.[10]

The conventional courses are also here. An Introduction to Statistics is a first-year text for business degrees that moves from presenting data with histograms, box-and-whisker and stem-and-leaf plots through measures of location and spread, Chebyshev's inequality and z-scores, classical probability and its rules of addition and multiplication, to distributions and inference; its description singles out two derivations rarely found in introductory texts, the Poisson from the binomial and the least-squares regression formulae, and its appendices include guides to the statistics mode of the Casio calculators permitted in university examinations.[24] Introductory Statistics with Randomization and Simulation is OpenIntro's college-level introduction through randomisation and simulation, free to every student with paperbacks sold at cost, and its description says a successor edition under a new title is in development.[23] What Just Happened: Descriptive Statistics is a student's guide to descriptive statistics, with 17 practice problems, two quizzes and 36 graphs, that works each measure of position, centre and dispersion in turn and adds chapters on types of data, the normal distribution and visualisation.[11] Multivariate Analysis – The Simplest Guide in the Universe is a plain-English guide to choosing, running and interpreting multivariate tests once univariate analysis is done, on the premise that most relationships found in data are wrong and a holistic strategy makes the real ones more likely to survive.[39]

Statistics is also the first chapter of the machine learning books, and The Orange Book of Machine Learning - Green edition states the division cleanly: a statistical model, built for explanation, assumes a parametric form and seeks the best parameters, while a non-parametric machine learning model forgoes explanatory power for predictive power; both rest on the assumption that data is signal plus irreducible noise. Its opening chapter separates interpolation from curve fitting, error from residual with a chocolate bar that never weighs exactly its printed hundred grams, aleatoric from epistemic uncertainty, and confidence intervals on parameters from prediction intervals on predictions; its statistics chapter then gives the mean, median and mode, with a NumPy example in which one outlier drags a mean of four to 146, the variance and the Bessel correction that pandas applies and NumPy does not, the median absolute deviation, quartiles and the Gaussian.[17] Applied Mathematics for the Modern World covers the same basics, probability, descriptive statistics and the normal distribution, with A/B testing as its application.[8]

Abstract algebra studies structures defined by axioms, groups first among them, and the library's holdings run from a first course to two theses. Introduction to Group Theory is written for an introductory abstract algebra course and delegates much of its content to activities that students work through with an instructor.[35] Two Essays on Group Theory binds together a 1997 MSci thesis from Queen Mary, University of London, on characterisations of finite supersoluble groups, through supersoluble series, Sylow towers and a theorem of Philip Hall, with the author's notes for a combinatorics study group on finitary permutation groups, generalised wreath products, primitivity and the Jordan-Wielandt theorem, and locally soluble and locally nilpotent groups.[40] Supersolubility and Finitary Groups is the same author's PhD thesis of 2000, revised for print, on generalised solubility conditions in finitary permutation and linear groups: local systems, the Hirsch-Plotkin radical, Prüfer rank and generalisations of Platonov's theorems, a theory of height, and Engel elements.[41]

Topology, the study of the properties a space keeps under continuous deformation, is represented by a research programme rather than a textbook. General Topology as Ordered Semigroup Actions: Book 1, the first volume of Victor Porton's Algebraic General Topology, introduces funcoids and reloids, generalisations of proximity spaces and uniform spaces that use filters rather than sets in their domains and ranges, and claims a single algebraic formula for continuity that covers ordinary, proximity and uniform continuity at once; it runs to 457 pages.[42] The general relativity primer described below gives the standard definition instead, a topology on a set as a family of subsets closed under arbitrary unions and finite intersections and containing the set and the empty set, and insists that the word open means nothing more than membership in that family.[19]

The newest books on the shelf teach mathematics because of what it now runs. The Hundred-Page Guide to the Core Mathematics of Machine Learning sets out its philosophy as intuition through action: its goal is not to train research mathematicians but effective engineers, and it compares the two ways of learning to cook, the chemistry of the Maillard reaction or the knowledge that searing a steak makes it taste good, choosing the second. Each topic follows a loop from concept to intuition to application to code, and the four parts are linear algebra as the language of data, calculus as the engine of learning, probability as the science of uncertainty, and projects in NumPy.[9] Behind the Black Box: Building Neural Networks From Scratch does the calculus in full for a two-layer classifier in NumPy, for readers who have called a library's fit method and want to know what happened inside. Its sample derives the linear forward step with its array shapes, a sigmoid that avoids overflow, binary cross-entropy computed from logits so that no probability rounds to zero or one, and backpropagation as the chain rule applied in reverse, where the gradient at an output logit simplifies to the prediction minus the label; it then checks the analytical gradients against finite differences, and goes on to gradient descent, disjoint training, validation and test sets, gradient clipping, mini-batches and L2 regularisation.[18]

Two larger books assume more. Advanced Machine Learning Made Easy - Volume 1, the first of three volumes on supervised and unsupervised learning, prefers in its preface a semester each of linear algebra, multivariate calculus, probability and statistics, takes the bottom-up route of building the mathematical framework before each learner, and promises that every formula will be converted directly into a single line of Python array manipulations; this volume covers generalised linear models, from linear regression through optimisation to logistic regression.[16] The Orange Book of Machine Learning - Green edition grew out of a five-day course its author taught across Spain and treats the statistics of regression and classification on tabular data; its contents reach principal component analysis, the LASSO and a chapter asking why no neural networks, and its description says it was written without generative AI.[17] Applied Conformal Prediction:Practical Uncertainty Quantification for Real-World ML is a guide to conformal prediction, a method for prediction sets with guaranteed coverage from finite samples, written by a student of the method's creator Vladimir Vovk; its description starts from exchangeability, nonconformity scores and prediction regions and goes on to calibration, regression, time series and covariate shift, with chapters still being released.[50] Notes on Dynamical Systems for Actor-Critic Learning, described under differential equations above, is the mathematical end of the same shelf.[43]