quantity idea is a department of arithmetic which attracts its energy from a wealthy ancient history. it's also routinely nourished via interactions with different components of study, reminiscent of algebra, algebraic geometry, topology, complicated research and harmonic research. extra lately, it has made a miraculous visual appeal within the box of theoretical desktop technology and in questions of conversation, cryptography and error-correcting codes. supplying an user-friendly advent to the primary themes in quantity idea, this booklet spans a number of parts of study. the 1st half corresponds to a sophisticated undergraduate direction. all the statements given during this half are in fact followed by means of their proofs, with probably the exception of a few effects showing on the finish of the chapters. A copious record of routines, of various trouble, also are integrated the following. the second one half is of a better point and is appropriate for the 1st 12 months of graduate institution. It includes an creation to elliptic curves and a bankruptcy entitled “Developments and Open Problems”, which introduces and brings jointly a variety of issues orientated towards ongoing mathematical examine. Given the multifaceted nature of quantity thought, the first goals of this booklet are to: - offer an summary of many of the different types of arithmetic precious for learning numbers - show the need of deep and classical issues equivalent to Gauss sums - spotlight the position that mathematics performs in sleek utilized arithmetic - contain fresh proofs comparable to the polynomial primality set of rules - strategy topics of up to date examine equivalent to elliptic curves - illustrate the wonderful thing about mathematics the must haves for this article are undergraduate point algebra and a bit topology of Rn. it is going to be of use to undergraduates, graduates and phd scholars, and should additionally entice specialist mathematicians as a reference text.