Final answer:
The Real Number System consists of rational and irrational numbers. Pi (π), although often approximated, is an irrational number because its decimal form is non-terminating and non-repeating. Rounding is a common practice for simplifying numbers in real-world applications where exact precision is not always necessary.
Step-by-step explanation:
The Real Number System is a comprehensive set of numbers composed of the rational and irrational numbers. Rational numbers include integers, fractions, and terminating or repeating decimals; they can be written as the ratio of two integers. Irregular numbers, on the other hand, cannot be expressed as a simple fraction, and their decimal expansions are non-terminating and non-repeating. Examples include numbers like π (pi) and the square root of 2.
π (pi) is an archetypal irrational number, despite common usage of its approximate value (like 3.14) for convenience in calculations and everyday measurements. The shortened version of pi still represents the same irrational number, pi itself. The full decimal expansion of pi cannot be fully determined; pi does not repeat or terminate. This property showcases the difference between exactness in mathematics and the estimation often necessary in the real world.
When rounding numbers, we often simplify irrational numbers to make calculations easier, particularly in practical contexts where perfect precision is unattainable or unnecessary. Rounding can be up (to the nearest higher number) or down (to the nearest lower number) depending on the digit that follows the last digit we wish to keep. Before rounding a number, one should consider the degree of precision required. Rounding is significant because we deal with estimations rather than exact numbers in most real-life situations.