Abstract mathematics, sometimes called pure mathematics, refers to mathematics qua mathematics, as contrasted to applied mathematics. The three primary branches of abstract mathematics are the study of shape, called geometry at the undergraduate level, extended to topology at the graduate level (if not sooner); the study of number, called algebra at the beginning undergraduate level, extended to abstract algebra at a more advanced level; and the study of functions, called calculus at the Freshman level and analysis at a more advanced level. Each of these branches of abstract mathematics have many sub-specialties, and there are many connections between pure mathematics and applied mathematics. Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ... Applied mathematics is a branch of mathematics that concerns itself with the application of mathematical knowledge to other domains. ... In geometry, objects are of the same shape if one can be transformed to another by dilating (that is, by multiplying all distances by the same factor) and then, if necessary, rotating and translating. ... Geometry (from the Greek words Geo = earth and metro = measure) is the branch of mathematics first popularized in ancient Greek culture by Thales (circa 624-547 BC) dealing with spatial relationships. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... A number is an abstract entity used originally to describe quantity. ... Algebra is a branch of mathematics which studies structure and quantity. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In general, a function is part of an answer to a question about why some object or process occurred in a system that evolved or was designed with some goal. ... For other uses of the term calculus see calculus (disambiguation) Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ... An analysis is a critical evaluation, usually made by breaking a subject (either material or intellectual) down into its constituent parts, then describing the parts and their relationship to the whole. ...