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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.
Arithmetic. a mathematical operation, symbolized by a Ã b, a â b, a â b, or ab, and signifying, when a and b are positive integers, that a is to be added to itself as many times as there are units in b; the addition of a number to itself as often as is indicated by another number, as in 2Ã3 or 5Ã10.