A Treatise on the Binomial Theorem (Texts in Mathematics) by Craig Smorynski (PDF)

Description

“The binomial theorem is usually quite rightly considered as one of the most important theorems in the whole of analysis.” Thus wrote Bernard Bolzano in 1816 in introducing the first correct proof of Newton’s generalisation of a century and a half earlier of a result familiar to us all from elementary algebra. Bolzano’s appraisal may surprise the modern reader familiar only with the finite algebraic version of the Binomial Theorem involving positive integral exponents, and may also appear incongruous to one familiar with Newton’s series for rational exponents. Yet his statement was a sound judgment back in the day. Here the story of the Binomial Theorem is presented in all its glory, from the early days in India, the Moslem world, and China as an essential tool for root extraction, through Newton’s generalisation and its central role in infinite series expansions in the 17th and 18th centuries, and to its rigorous foundation in the 19th. The exposition is well-organised and fairly complete with all the necessary details, yet still readable and understandable for those with a limited mathematical background, say at the Calculus level or just below that. The present book, with its many citations from the literature, will be of interest to anyone concerned with the history or foundations of mathematics.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a very nice short treatment of various applications of the binomial theorem. The author does an excellent job of balancing mathematical complexity and conversational tone. This is not a “history of mathematics” book, nor is it a text book. It is a walk through different ways the binomial theorem can be used in practice. It includes computer/calculator examples and programs that can be used to explore the different applications, such as calculating roots of numbers. It omits probability and statistics applications, as these topics are covered in the author’s other book on the subject.

⭐This is a great little book, although it isn’t little. It has 60 pages of pre-Newton and 200 more on Newton’s work and all its repercussions. Both a history and a tutorial, it is thorough and thoroughly readable. Recommended for anyone with a fixation on the subject.

⭐The author has no idea about the history of binomial theorem. He picks his information up here and there and in secondary literature. It is not a book of a real historian of mathematics.

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