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The set of real numbers, denoted by R, encompasses all rational and irrational numbers. Rational numbers are those that can be expressed as a fraction p/q, where p and q are integers and q is not zero. Examples include 1/2, -3, and 0. Irrational numbers, on the other hand, cannot be expressed as such a fraction. Famous examples include pi (π) and the square root of 2 (√2).

The real number line is a geometric representation of the set of real numbers, where each point on the line corresponds to a unique real number. This set is fundamental in mathematics, serving as the foundation for calculus, analysis, and many other advanced fields. The real numbers possess several important properties, such as completeness, meaning that every non-empty set of real numbers that is bounded above has a least upper bound within the set.

The set of real numbers is infinite and uncountable, meaning that there are more real numbers than there are natural numbers. Arithmetic operations like addition, subtraction, multiplication, and division (excluding division by zero) are well-defined for real numbers, and they follow standard algebraic laws. The ordering of real numbers is also crucial, allowing for concepts like inequalities and intervals.