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Essays on Numbers and Figures
V. V. Prasolov, Independent University of Moscow, Russia

Mathematical World
2000; 75 pp; softcover
Volume: 16
ISBN-10: 0-8218-1944-5
ISBN-13: 978-0-8218-1944-9
List Price: US$19
Member Price: US$15.20
Order Code: MAWRLD/16
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This is the English translation of the book originally published in Russian. It contains 20 essays, each dealing with a separate mathematical topic. The topics range from brilliant mathematical statements with interesting proofs, to simple and effective methods of problem-solving, to interesting properties of polynomials, to exceptional points of the triangle. Many of the topics have a long and interesting history. The author has lectured on them to students worldwide.

The essays are independent of one another for the most part, and each presents a vivid mathematical result that led to current research in number theory, geometry, polynomial algebra, or topology.


Advanced high school and undergraduate students, mathematics educators at secondary and university level; general mathematical audience.


"This volume is quite delightful ... Each essay ... engages the reader's interest in an immediate and lively manner ... can be highly recommended for well-prepared students aspiring to a top-level mathematics degree and will prove a marvelous source for teachers ... a good bedtime read for Putnam competitors ... a pleasure indeed for the general lover of mathematics."

-- Mathematical Reviews

"A worthwhile resource for mathematics departments to have on hand ... this collection serves as a nice resource for accessible topics beyond the standard curriculum around the level of calculus."

-- MAA Online

Table of Contents

  • Conjugate numbers
  • Rational parametrizations of the circle
  • Sums of squares of polynomials
  • Representing numbers as the sum of two squares
  • Can any knot be unraveled?
  • Construction of a regular 17-gon
  • The Markov equation
  • Integer-valued polynomials
  • Chebyshev polynomials
  • Vectors in geometry
  • The averaging method and geometric inequalities
  • Intersection points of the diagonals of regular polygons
  • The chromatic polynomial of a graph
  • Brocard points
  • Diophantine equations for polynomials
  • The Pascal lines
  • One butterfly and two butterflies theorems
  • The Van der Waerden theorem on arithmetical progressions
  • Isogonal conjugate points
  • Cubic curves related to the triangle
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