Sixteen chapters of advanced algebra, from irrational expressions to limits. The second half of a problem collection used for more than seventy years, complete in English.
Part Two is where Shaposhnikov and Valtsov's collection turns demanding. The elementary operations are assumed; the work now is quadratic and higher-degree equations, logarithms, progressions, combinatorics, inequalities, and the first approach to limits. The method is unchanged - abundant variation, deliberate progression, and enough practice for a technique to become genuinely secure - but the material is that of a strong final-years course rather than an introduction.
Inside this volume:
- 1,531 numbered problems across sixteen chapters (IX-XXIV)
- Irrational expressions: radicals, rationalizing denominators, fractional exponents, imaginary numbers
- Functions and their graphs, and the graph of a quadratic function
- Quadratic equations, properties of the roots, and equations composed to order
- Equations of higher degrees, irrational equations, and systems above the first degree
- Arithmetic and geometric progressions
- Logarithms: general properties, decimal logarithms, exponential and logarithmic equations, compound interest
- Combinatorics and Newton's binomial
- Divisibility of polynomials, inequalities, and indeterminate equations of the first degree
- Continued fractions, investigation of equations, and limits
- An answer key with more than 1,380 entries
The collection began appearing in 1887-1890. It was used for more than seventy years, reached at least fifty-two editions, and was recognized with the Peter the Great Prize. Its longevity reflects the architecture of the exercises rather than any novelty of presentation: the same idea is met again and again, in forms that keep changing, until the method belongs to the reader.
Two authors, one mathematical lineage
Nikolai Shaposhnikov learned mathematics at the Fourth Moscow Gymnasium under Alexander F. Malinin. He completed his gymnasium education with a gold medal, studied at Imperial Moscow University, earned a master's degree in pure mathematics, and later taught at the Imperial Moscow Technical School. Nikolai Valtsov was also trained at Imperial Moscow University and brought that tradition into daily classroom work as a teacher of mathematics and physics, remembered for the time he gave to pupils who needed more of it.
Together they joined rigorous mathematical training to first-hand knowledge of how students learn. Part Two is where that combination is tested hardest: the problems get harder without getting arbitrary.
Built to be worked through
This volume suits students who have finished a first algebra course, homeschooling families building a rigorous mathematics library, teachers and tutors who need a large ordered exercise bank at this level, and adults rebuilding fluency before pre-calculus or analysis. It includes concise answers for checking work; it is a problem collection, not a book of fully worked solutions.
Part One of the collection, covering notation, signed numbers, polynomials, factoring, fractions, roots, first-degree equations, and quadratic equations with numerical coefficients, is available separately. The two volumes together cover the full course.
Part of Russian Math Classics - English Editions. Translated by Valery Manokhin, PhD (Royal Holloway, University of London). Published by Northern Star Academic Press. Full catalogue at russianmathbooks.com.
