Arithmetic vs Mathematics  Math vs Arithmetic
Many people think that the words ‘arithmetic’ and ‘mathematics’ mean the same. What is mathematics? Mathematics is a difficult term to define as it covers many areas. Mathematics can be defined as the study of measurements and properties of quantities using numbers and symbols. Mathematics also includes proofs of theorems, other than the numbers and symbols. Arithmetic is a branch of mathematics that deals with properties of numbers.
Arithmetic
Arithmetic is the oldest, most basic and fundamental category in mathematics, that involves basic calculations with numbers. The four elementary operations in arithmetic are addition, subtraction, multiplication and division. Therefore, arithmetic can also be defined as the mathematics of numbers (real numbers, integers, fractions, decimals and complex numbers) under the operation of addition, subtraction, multiplication and division. The order of operation is given by BODMAS rule (or PEMDAS rule).
For many years, arithmetic has been a part of human life. For example, they use it in their day today life activities such as counting, purchasing and preparing their accounts and budgets. It is often used in some higher level scientific or mathematical calculation as well.
Mathematics
Mathematics is a very broad field, which is used as an essential tool in many fields. It is not specific. There are two main branches of mathematics; applied mathematics and pure mathematics. Also, it can be categorized as arithmetic, algebra, calculus, geometry and trigonometry.
What is the difference between Arithmetic and Mathematics? Arithmetic: • use numbers for calculation. • deals with four basic operations; addition, subtraction, multiplication and division. Whereas, Mathematics: • is the study of measurements and properties of quantities. • use numbers, symbols and proofs for explanations

David Goloff, PhD says
There are many areas of mathematics that fall outside your definition: Logic, Set Theory, Abstract Algebra, Geometric Ideas transcend measurement. I agree that – for the person familiar with arithmetic – your definition of mathematics would be comprehensible and move their mind in the right direction. Perhaps that was your intention. So what definition would I propose? In my mind it is a focus on the structure apart from the thing having the structure. But I am not sure that is correct or all inclusive. I would be very interested in what others have to say in general or propose as a definition of “mathematics” as distinct from any particular mathematical subject. I will check back.