Problem 142 - Algebra and Trigonometry

Question

Can a real number be both rational and irrational? Explain your answer.

Step-by-Step Solution

Answer

No, a real number cannot be both rational and irrational. These two sets of numbers are mutually exclusive, meaning a number can be either rational or irrational, but not both.

Step 1: Definitions Review

A rational number can be defined as a number that can be expressed as the quotient or fraction \( p/q \) of two integers, with the denominator \( q \) not equal to zero. Since the set of rational numbers includes the integers, the rational numbers are countable. On the other hand, an irrational number cannot be expressed as a ratio. That's why it is called irrational. It cannot be expressed as a fraction, and it's decimal goes on forever without repeating. For example, \(\pi\) is an irrational number.

Step 2: Reflection on the Question

With the above understanding, it's clear that the defining properties of rational and irrational numbers are mutually exclusive. A rational number, by definition, has a finite or repeating decimal expansion. An irrational number, by definition, has a non-repeating decimal expansion. There is no overlap in these two sets, meaning a number can either be rational (finite or repeating decimal expansion) or irrational (non-repeating decimal expansion), but not both.