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Continued Fractions (Dover Books on Mathematics)
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Original Language: Russian
Top Customer Reviews
The book is brilliant and necessary for understanding continued fractions, but can't stand alone without supplemental material unless one is a professional mathematician. Khinchin frequently employs contrapositive proof formats, and there are occasional translation errors from Russian. The errors range from minor (awkward usage) to major (in one place, the translation is "negative" when it should be "non-negative", which confused me for half a day).
In this short book the first two chapters contain a very clear development of the theory of simple continued fractions, culminating in a proof of Lagrange's theorem on the periodicity of the continued fraction representation of quadratic surds. Chapter three presents Khinchins beautiful and original work on the measure theory of continued fractions. The proofs of the theorems in this chapter are also entirely elementary.
All in all a very good start for understanding this profound mathematical tool.
Never heard of them? You're not alone. The first recorded instance of continued fractions was by Lord Brouncker in the 17th century which makes them a relatively new addition to mathematics. Nor are they taught in typical undergraduate scientific curricula. Notwithstanding, if they were discovered by the Pythagoreans, history may have been much different.
The Pythagoreans were a mystical sect that believed that all things geometric could be described by rational numbers (i.e., wholes and fractions). Something like the square root of two was clearly geometric (the diagonal of the unit square) yet, irrational. Legend has it that Hippasus (5th century B.C.) was expelled from (or killed by) the Pythagorean school for proving the irrationality of a number such as the square root of 2 or the golden ratio. This ultimately destroyed the Pythagorean religion.Read more ›
Measure theory is the theory of the fraction of the extent of a domain that is mapped from the range of inputs to a function. Measure theory is used primarily by Khinchin and his students, but similar work is posed in terms of probability theory and other contexts by other mathematicians. Once this is understood (and it is hinted at by a footnote or two from the translator), this material becomes as accessible as the rest of the material.
The book begins with a minor aside in a proof of convergence of continued fractions that have real partial numerators and denominators, whose partial numerators are all unity, and the sum of whose partial denominators diverges. Since the simple classical number-theoretic continued fractions are the subject of the book, this proof clearly includes all such continued fractions. This minor excursion from number theory and algebra is a significant advantage to this particular book as it provides a bedrock for later rate-of-convergence discussions.
The related field of analytic theory of continued fractions that was explored by Riemann, Stieltjes, Tchebychev, Padé, Hamburger, Cesàro, and others that are contemporary to Khinchin (memorable classic by H.S. Wall was published in 1948, long after this book was written), is not ignored entirely.
Most Recent Customer Reviews
This is a short book about continued fractions as they occur in analysis. The machinery of continued fractions is worked out in the first part of the book, mainly inequalities... Read morePublished 21 days ago by Jordan Bell
Short, readable book, wealth of applications, this little gem provides a doorway into deep number theory. Read morePublished 6 months ago by nicholas c. strauss
Covers some ideas using Continued Fractions. It doesn't really go into depth, but it is a pretty thin book. Read morePublished 9 months ago by William Schram
The first two chapters are an excellent introduction for the subject. Though the book lacks examples and exercises, the first chapter is very well explained and organised letting... Read morePublished on January 9, 2010 by Damian Scelato