ABSTRACT: In the study called non-Newtonian Calculus, Grossman and Katz introduced a new type of calculus that includes branches such as geometric and bi-geometric calculus. The aim of this thesis is to examine the basic features of the geometric and bigeometric calculus. This thesis is divided into five parts. In the first part, the literature on non-Newtonian calculus is summarized. In the second part, a sub-branch of non-Newtonian calculus called geometric arithmetic is introduced, along with the properties of geometric real numbers which is called 𝛼-arithmetic. In the third part, the applications of the 𝛼- arithmetic of non-Newtonian calculus is studied. In chapter four the arithmetic, differentiation and integration are applied to a specific example which is hyperbolic tangent. In chapter 5 the application of arithmetic to the cubic calculi is given and according to this geometric and bi-geometric differentiation and integrations are defined. Apart from this some other concepts are given such as absolute value, commutativity, associativity and distributivity properties and q-limit. In conclusion part a discussion about quadratic Calculi is given. Keywords: Non-Newtonian Calculus, 𝛼-Arithmetic, Geometric Calculus, BiGeometric Calculus, Cubic Calculi and Quadratic Calculi.
