The fuzzy set and intuitionistic fuzzy set are two useful mathematical tool for dealing with impression and uncertainty. However sometimes these theories may not suffice to model indeterminate and inconsistent information encountered in real world. To overcome this insufficiency, neutrosophic set theory and single-valued neutrosophic set (SVNS) theory which is useful in practical applications, were proposed. Many researchers have studied on single-valued triangular neutrosophic numbers and single-valued trapezoidal neutrosophic numbers. In this paper, concepts of Gaussian single-valued neutrosophic number (GSVNN), α-cut of a GSVNN and parametric form of a GSVNN are defined, and based on α-cuts of GSVNNs, arithmetic operations for GSVNNs are defined. Also, some results are obtained related to arithmetic operations of GSVNNs. Furthermore, a decision making algorithm is developed by using GSVNNs operations, and its an application in medical diagnosis is given.