Continued Fractions : Under which branch of mathematics do they come? I wanted to know in which branch of Mathematics do Continued Fraction come?
By branch I mean for example Geometry or Differential Equation are a branch of maths so is there any particular branch of continued fraction.
 A: Short version: Continued fractions fall under Number theory.
Long version: Continued fractions is one of the areas of mathematics that has applications in many other areas of mathematics, so it is difficult to place into just one branch. However, it has more easy applications to Number theory than to other branches of mathematics, so it is usually first encountered there by math students.
It is similar in this to Combinatorics, which is a "cross-cut" field of study that has uses in just about every branch of mathematics. However, Combinatorics is usually first seen by students in a Probability class, since that class needs the simple uses of the counting principle, permutations, combinations, and so on. It would be a great mistake, however, to think that Combinatorics is only useful for probability. Combinatorics is a large enough area that it does get its own classes and is not placed under any of the math branches in the Dewey Decimal classification system. (It is placed under 511 General principles of mathematics.)
Continued Fractions has simple and direct applications to many problems in Number theory, such as approximating real numbers with rational numbers of small denominator, solving Diophantine equations (with integral solutions) such as linear and Pellian equations, and finding multiplicative inverses in modular arithmetic. These are important topics in Number theory, so a complete Number theory class often includes the study of Continued fractions.
I first met Continued fractions in a Number theory class at the Ross Mathematics Program. The Dewey Decimal system (see www.oclc.org/content/dam/oclc/webdewey/help/500.pdf) recommends placing many books on Continued fractions under 512.7, Number theory (though some references put it at 512.81, still Number theory). The book Elements of Number Theory by I. M. Vinogradov includes the topic. Elementary Number Theory by J. V. Upensky and M. A. Heaslet does not include it but says "A chapter on continued-fractions would have been particularly desirable, but ... this would have extended the book beyond reasonable limits."
That said, remember the many applications to other branches of mathematics. For example, the book Continued Fractions by A. Ya. Khinchin includes a little number theory, but the last third of the book is on measure theory and the Dewey Decimal classification for this book is under Analysis.
