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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.
Math uses formal proof and definitions, which is entirely different. No, math is not a science. Math is actually more a language than anything else: a completely formalized quantitative language with rigorous dynamics. Science is a method more than anything else, and it is a method not used in mathematics.