The set of real numbers is closed under addition, subtraction, multiplication, and division (except that you cannot divide by zero). By closed, this means that if the two numbers in the operation are both real numbers, the result of the operation will always be a real number. Dividing by zero is undefined (for all practical purposes)
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There are different rules for different operations.
There are different rules for different operations.
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There are many. There are those that deal with the four basic binary operations, then there are rules governing exponents and logarithms.
additive inverse and associative property and if one is involved, then also identity