Where to start
These books are a sequence, not a shelf. Answer two questions and you will land on the stage that fits — or read the whole ladder below.
How old is the student?
What have they already worked through?
Ages are the usual range, not a rule. Adults working alone start wherever the contents look unfamiliar — the tables of contents on each book page are there for exactly that.
The whole sequence
Arithmetic
Whole numbers, divisibility, fractions, decimals, ratio and proportion — every rule derived rather than asserted.
The text
Kiselev's Arithmetic
by A. P. Kiselev
Ages 9–12 · Grades 3–6
The foundational arithmetic textbook used in Russian schools for over a century. 7 parts, 214 sections. Whole numbers through proportional quantities, with mathematical proofs for every operation.
Kiselev's Systematic Arithmetic
by A. P. Kiselev
Ages 9–13 · Grades 3–7
Kiselev's definitive systematic arithmetic — the expanded companion to his classic arithmetic textbook. 207 pages, 264 numbered articles across seven divisions: whole numbers; concrete (denominate) numbers and the metric system; divisibility, primes, and factorisation; common fractions; decimal fractions with limits and repeating decimals; ratio and proportion; and a full course of word problems on proportional quantities. Includes elementary treatments most textbooks leave out — Euclid's proof of the infinitude of primes and the concept of a limit used to prove that 0.999… equals 1.
Malinin & Burenin
by A. Malinin & K. Burenin
Applied arithmetic · Flexible level
A complete English translation of Malinin and Burenin's Imperial Russian arithmetic textbook. 11 chapters, 301 sections. Covers numeration, operations on whole numbers, divisibility, fractions, ratios, proportions, percentage, interest, and Diophantine problems.
Problems and enrichment alongside it
Algebra, first course
Algebraic notation, signed numbers, equations and inequalities, polynomials, fractions, ratio and proportion, graphs.
The text
Kiselev's Algebra I
by A. P. Kiselev
Ages 12–15 · Grades 6–8
6 chapters, 126 sections, 239 exercises. From relative numbers and polynomials through first-degree equations, powers, roots, and quadratics.
Elements of Algebra & Analysis I
by A. P. Kiselev
Ages 14+ · Grades 9–12
Part One builds a connected foundation: algebraic notation and operations, signed numbers, identities, equations and inequalities, polynomials, fractions, ratio and proportion, functions, coordinates and graphs, powers, roots and irrational numbers, progressions, logarithms with four-figure tables, compound interest and the binomial theorem. A distinct work from the 1888 Elementary Algebra.
Problems and enrichment alongside it
Geometry of the plane
The straight line, the circle, similar figures, regular polygons, areas — proof-based from the first page.
The text
Problems and enrichment alongside it
Algebra, second course
Powers and roots, quadratics, progressions, logarithms, combinations and the binomial theorem, limits.
The text
Kiselev's Algebra II
by A. P. Kiselev
Ages 14–17 · Grades 8–10
12 chapters plus Supplements, 187 sections. Functions, quadratic systems, inequalities, logarithms, progressions, complex numbers, the binomial theorem, and continued fractions.
Elements of Algebra & Analysis II
by A. P. Kiselev
Ages 15+ · Grades 10–12
Part Two carries the numbering of Part One forward into analysis: the theory of limits and its applications to elementary geometry, derivative functions, slopes of straight lines and curves, increasing and decreasing functions, and maxima and minima.
Problems and enrichment alongside it
Trigonometry
Goniometry, the trigonometric functions and their identities, triangle solution, trigonometric equations.
The text
Problems and enrichment alongside it
Geometry of space
Lines and planes, dihedral and polyhedral angles, the regular polyhedra, prisms, pyramids, cylinders, cones, the sphere, volumes.
The text
Problems and enrichment alongside it
Introduction to analysis
Limits, the derivative, maxima and minima, the beginnings of the integral calculus.
The text
University mathematics
Analytic geometry, differential and integral calculus, differential equations, series, complex variables, probability.
The text
Gunter & Kuzmin, Part I
by N. M. Gunter & R. O. Kuzmin
University level
The first half of the standard Russian problem book for higher mathematics, as a volume on its own: analytic geometry in the plane and in space, differential calculus and its applications including differential geometry, higher algebra, and indefinite and definite integration. 448 pages.
Gunter & Kuzmin, Part II
by N. M. Gunter & R. O. Kuzmin
University level
The second half completes the collection: multiple, curvilinear and surface integrals, field and potential theory, ordinary and partial differential equations, improper integrals and special functions, series and Fourier series, approximate computation, functions of a complex variable, mathematical physics, the calculus of variations and probability. 379 pages.
Gunter & Kuzmin
by N. M. Gunter & R. O. Kuzmin
University level
The standard Russian problem collection for higher mathematics — the book used across Soviet technical universities. 814 pages spanning mathematical analysis, differential and integral calculus, series, differential equations, and probability theory, in the first complete English translation.
Problems and enrichment alongside it