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They do not apply to all expressions - only to those expressions whose elements are either real numerical constants or variables which can only take real values.
The set of real numbers is closed under the operations of arithmetic. As a result each term in an expression, which will be made up of real constants or real variables, will also be a real constant or variable. And since each term in the expression is real, the closure implies that their combination is also real.
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How can the properties of real numbers be used to simplify expressions?
which mixed number or improper fraction is closest to the decimal 5.27?
Is axiom or properties of real numbers the same?
No, they are not the same. Axioms cannot be proved, most properties can.
Properties of real number?
Real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that if a nonempty set of real numbers has an upper bound, then it has a least upper bound. These two together define the real numbers completely, and allow its other properties to be deduced.
How are a numerator and denominator similar?
They are elements of of a set which may consist of integers, real or complex numbers, polynomial expressions, matrices.
How are addition subtraction multiplication and division of real numbers performed?
That really depends how the numbers are expressed - you have to learn separately how to calculate with decimals, with fractions, with expressions involving square roots, etc.
