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Arithmetic is strictly dealing with real numbers and operators, (addition, subtraction etc.) Mathermatics covers a much broader description including arithmetic, calculus, algebra, etc... You can say that Arithmetic is a subset of Mathematics.
Definition of arithmetic. 1  1a : a branch of mathematics that deals usually with the nonnegative real numbers including sometimes the transfinite cardinals and with the application of the operations of addition, subtraction, multiplication, and division to themb : a treatise on arithmetic. 2  2 : computation, calculation.