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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.
What is the 'Arithmetic Mean' The arithmetic mean is a mathematical representation of the typical value of a series of numbers, computed as the sum of all the numbers in the series divided by the count of all numbers in the series. The arithmetic mean is sometimes referred to as the average or simply as the mean. Some mathematicians and scientists prefer to use the term arithmetic mean to distinguish it from other measures of averaging, such as the geometric mean and the harmonic mean.