Mathematics Small Series

Author: Hua Luogeng
Publisher:
Publish Date: 2002-05-01
Features:
1. Book The triangle of coefficients representing the binomial expansion, known as "Origin of Squaring and Extraction of Square Roots," is now simply called the "Yanghui Triangle." This book begins by analyzing the basic properties of the Yanghui Triangle, discussing the binomial theorem, square roots, and various series, and finally presents an approximate calculation method for estimating the sum of an infinite series as an example.
2. Book Symmetry, literally speaking, means two things are opposite and balanced, so if you swap them, it seems as if nothing has changed. This book primarily introduces the mathematics of symmetry, starting with algebraic symmetry, then geometric symmetry, and finally introducing the concept of "groups." The concept of "groups" is one of the important concepts in modern mathematics, not only for algebra and geometry but also for mathematical analysis and theoretical physics. Through these discussions, the author also attempts to help readers understand that mathematical theory is abstracted from concrete reality and has practical applications in reality.
3. Book The ancient Chinese mathematician Zu Chongzhi proposed the approximate and exact rates for calculating π, which gave rise to the problem of using rational numbers to optimally approximate real numbers. The concept of "approximation" is extremely important in modern mathematics. This book begins by addressing why the Soviet Union's artificial satellite will approach Earth again in 2113 and some interesting astronomical phenomena, building on middle school math knowledge such as the greatest common divisor, the Euclidean algorithm, and continued fractions to derive the principles of using rational numbers to optimally approximate real numbers. This method is applicable to any periodic recurrence or repetition and often appears in the study of sound waves, light waves, electric waves, water waves, and air waves.
4. Book The application of mathematics in mechanics is obvious. For example, some calculations in mechanics use mathematics. However, mechanics' application to mathematics, such as its use in geometry, may not be widely known. In fact, as early as 2,000 years ago, Archimedes already used the law of equilibrium in mechanics to prove some set propositions. Middle school students who have studied physics are familiar with concepts such as the center of gravity and force equilibrium in mechanics. This book uses these mechanical concepts to illustrate how they are used to prove geometric propositions, with content limited to the physics and geometry knowledge covered in middle school curricula and not involving advanced theories.
5. Book This book explores some interesting mathematical problems around the concept of "average." It begins by discussing the relationship between arithmetic mean, geometric mean, and harmonic mean, then introduces its interesting applications: solving practical problems such as what shape a food can should have to be most economical or how high a lamp should be hung to cast the brightest light on a table, as well as proving some useful mathematical inequalities. It then further generalizes the concept of average, introducing "power mean" to unify arithmetic mean, geometric mean, and harmonic mean, and introduces some properties of power mean. Finally, it discusses "weighted average," which is another type of average frequently encountered in real life and can also be linked to the problem of the center of gravity in mechanics. The book includes many exercises, allowing readers to further appreciate the practical applications of the theories discussed.
6. Book A grid paper with two sets of parallel lines, with equal distances between each pair of lines, forms the intersection points known as lattice points. In a finite region on a plane, the number of lattice points is always an integer. How to calculate the area of a finite region on a plane using the number of lattice points, or conversely, how many lattice points are there in a finite region on a plane with a given area, is the problem this small book discusses. It particularly discusses a theorem called the "Fundamental Theorem of Number Geometry." To prove this theorem, the book also introduces a theorem called the "Principle of Overlap." Related to the overlap principle, it discusses how to use rational numbers to approximate irrational numbers. This small book discusses some useful mathematical problems around the theme of lattice points and area.
7. Book A "" is an interesting geometric problem. In world history, since the Renaissance, people have begun to notice some geometric phenomena and problems beyond Euclidean geometry, and a is one of them. It is the ancestor of the geometric discipline now called "graph theory." For centuries, it was regarded as a mathematical game, but in China in the 1950s, the problem of mail delivery routes found practical applications. This small book presents its arguments and reasoning in a clear and rigorous manner, aiming to help readers familiarize themselves with common mathematical approaches and learn methods of analysis and argumentation. It includes a small number of exercises for practice. At the end, it includes a historical document—Euler's report of 1735—which shows the gradual deepening of this mathematician's thoughts while studying the problem.
8. Book The ancient Chinese mathematician Liu Hui calculated the area of a regular hexagon inscribed in a circle, then doubled the number of sides repeatedly to calculate the areas of hexagons, dodecagons, and, gradually approaching π. This method is called Liu Hui's method of cutting the circle. The characteristic of Liu Hui's method is to use the finite to approximate the infinite, a thought that continues to play an extremely important role in modern mathematics and will continue to do so in the future. This small book applies this idea of Liu Hui's method to solve some area and volume problems and introduces the principle of area to calculate the sums and limits of certain series.
9. Book This small book is written for middle school students. It begins by using some real-life examples to illustrate the nature of extreme problems; then, based on middle school mathematics, it discusses several types of extreme value problems involving quadratic functions, without involving advanced mathematics. It also provides some interesting examples that are appropriately connected to reality; finally, it unifies all these types of problems under a general theorem. At the end, it includes some exercises, allowing readers to better understand and apply the theories discussed. The proofs of some theorems in the book, although not involving advanced mathematics, are somewhat similar in method to advanced mathematics, but they are within the range that readers at the middle school level can understand. This may help improve readers' logical thinking skills and serve as a guide for studying advanced mathematics.
10. Book The Sunzi Suanjing is an excellent ancient Chinese mathematical work that includes the problem "Unknown Numbers." Such problems had many interesting names in ancient times, including "Miraculous Calculation." Mathematicians both in China and abroad call this theorem the "Chinese Remainder Theorem." This work not only holds a place in the history of mathematics but also plays an important role in modern mathematics, such as in the design of electronic computers. The book introduces the problem and solution of "Miraculous Calculation" in an accessible and profound way, extracting the fundamental principles and methods, and leading to the theory of congruences and other important branches of mathematics in a popular and profound manner, making it easy for middle school students to understand. At the same time, this small book provides methods for thinking about problems, offering inspiration for learning mathematics and problem-solving.
11. Book The classic example of an isoperimetric problem is: "Among all closed plane curves with equal perimeter, which curve encloses the largest area?" This small book mainly introduces its elementary solution and a series of interesting applications. Readers who have studied plane geometry and trigonometry can fully understand it. This book begins with simple triangles, then discusses that among all quadrilaterals with given side lengths, the inscribed quadrilateral has the largest area; among all N-sided polygons with given perimeters, the regular N-sided polygon has the largest area. It then provides two proofs of the solutions to the above isoperimetric problems and an extension of Heron's formula. Finally, it proves that among all solids with the same volume, the sphere has the smallest surface area.
12. Book In the first chapter of this book, the proof of Euler's theorem on convex polyhedra only requires knowledge of middle school solid geometry. In the second chapter, through analysis and discussion of this theorem and its proof, as well as the intuitive description of topological transformations using rubber membranes to form shapes, it introduces the generalization of Euler's theorem to closed polyhedra. In the final chapter, theorems 3 and 4 perfectly solve some of the problems raised by theorem 1 and also provide the topological classification of curved surfaces.
13. Book This small book illustrates some convenient and interesting applications of complex numbers in plane geometry through many examples. Section 1 briefly reviews the basic knowledge of complex numbers. Section 2 lists some general examples of the application of complex numbers in geometry. Sections 3, 4, 5, and 6 respectively explain the applications of complex numbers in collinearity, concyclicity, concurrency, circles, fractional linear transformations of complex numbers, and uniform circular motion. While explaining these applications, it also introduces some commonly used mathematical thinking methods. The small book also includes exercises and answers or hints for exercises, providing readers with opportunities for practice.
14. Book A unit fraction is a fraction with a numerator of 1 and a denominator of a natural number. Expressing fractions as unit fractions has many interesting properties, leading to some interesting problems, some of which are still unsolved number theory problems and conjectures. This book begins with an ancient problem related to unit fractions, discusses some important properties and applications of unit fractions, and finally introduces an interesting infinite series and its method of summation.
15. Book This book first provides an in-depth yet accessible analysis of the principle of mathematical induction. Then, through discussions of some "variations" of mathematical induction and its applications in recursive functions, permutations and combinations, algebraic identities, differences, inequalities, and geometry, it gradually helps readers understand the thinking, methods, and techniques for discovering and solving problems. On this basis, the book naturally introduces the mathematical basis of mathematical induction—the Peano axioms—in the final section. The book is rich in content, lively and intuitive, progressing from simple to complex, and includes carefully selected examples that are not limited to problem-solving techniques but also provide sufficient space for reflection. For thoughtful readers, the gains will extend beyond the book.
16. Book This book introduces mathematical problems through the structure of honeycombs, then discusses various types of extreme value problems from different perspectives, methods, and tools in an accessible and profound way, naturally leading to some important and classic results. Through these discussions, it very naturally introduces the ideas and methods of modern mathematics, such as tiling and filling, combinatorics, group theory, inequality theory, and variational calculus. The book emphasizes how to refine mathematical models, solve problems, and engage in deeper reflection after solving problems to propose and address more general and broader problems. The book is rich in content and engaging. Although it was originally intended as extracurricular reading for middle school students, readers at different levels can gain a lot from it. In particular, many insightful observations scattered throughout the book are highly inspiring for mathematicians.
17. Book In the 400s AD, Zu Chongzhi published an astonishing inequality: 3.1415926 < π < 3.1415927. However, the method Zu Chongzhi used to derive this inequality was lost, turning it into a mystery. This small book, based on Liu Hui's method of cutting the circle, introduces a method for calculating π using current middle school mathematics, which may be related to the lost method of Zu Chongzhi. The latter part of the book describes a possible development of the lattice point cutting circle method. This development will inevitably lead to calculus, providing background material for readers who want to study calculus.
18. Book In 1637, the French mathematician Fermat proposed a conjecture, which was proven by Andrew Wiles in 1994 and is considered a major achievement in pure mathematics in the 20th century. The proof used many significant recent developments in algebra, number theory, and geometry. This book introduces in a relatively accessible way the historical process over 300 years in which Fermat's conjecture was solved, the innovative ideas and methods that emerged in the process, and their role in advancing mathematics.