Focusing on key topics, this book encourages critical thinking among students by offering in-depth coverage and proofs of significant mathematical results often overlooked in standard curricula. Its approach aims to enhance understanding and appreciation of the historical context of mathematics.
This book is about the rise and supposed fall of the mean value theorem. It discusses the evolution of the theorem and the concepts behind it, how the theorem relates to other fundamental results in calculus, and modern re-evaluations of its role in the standard calculus course. The mean value theorem is one of the central results of calculus. It was called “the fundamental theorem of the differential calculus” because of its power to provide simple and rigorous proofs of basic results encountered in a first-year course in calculus. In mathematical terms, the book is a thorough treatment of this theorem and some related results in the field; in historical terms, it is not a history of calculus or mathematics, but a case study in both. MVT: A Most Valuable Theorem is aimed at those who teach calculus, especially those setting out to do so for the first time. It is also accessible to anyone who has finished the first semester of the standard course in the subject and will be of interest to undergraduate mathematics majors as well as graduate students. Unlike other books, the present monograph treats the mathematical and historical aspects in equal measure, providing detailed and rigorous proofs of the mathematical results and even including original source material presenting the flavour of the history.
Exploring the historical development of elementary probability, the book emphasizes the significant problems that influenced the field rather than traditional methods. It features numerous excerpts from foundational texts and their commentators, including new English translations, providing context and depth. Designed for readers without a Calculus background, it offers an accessible approach to understanding probability concepts through a rich narrative of its evolution.
The binomial theorem is a fundamental concept in algebra, providing a formula for expanding expressions raised to a power. It explores the coefficients of the expanded form, known as binomial coefficients, and their applications in probability, statistics, and combinatorics. This book delves into the theorem's historical background, its mathematical significance, and offers practical examples to illustrate its use in solving complex problems. Ideal for students and enthusiasts, it enhances understanding of this essential mathematical tool.
Exploring the origins of mathematics, the book delves into the balance between intuition and formalism in mathematical practice. It highlights how intuition serves as a foundation, while formalism emerges to tackle complex problems. However, it warns of the inherent dangers associated with formalism, categorizing it into three distinct types. Through this examination, the text invites readers to reflect on the interplay between intuitive understanding and formal methods in the evolution of mathematical thought.