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Real Numbers

A number which is either rational or irrational is called a real number.

Set R of real numbers

The set R of real numbers is the union of the set of rational numbers and the set of irrational numbers.

We shall take real numbers as undefined objects, satisfying certain properties known as axioms. We shall state these axioms of R which all the other properties of real numbers can be proved. We shall, henceforth, assume the existence of real numbers and the two binary compositions, viz. addition and multiplication which determine for each element (x, y) R × R, a unique element x + y or x . y R called the sum or product of two real numbers x and y satisfying certain axioms. We shall also state some properties of order relation > (greater than) or < (less than) in R as a complete ordered field. By an algebraic structure we mean a non-empty set together with one or more than one binary composition defined in it.