Problem 142 - College Algebra

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 at the same time. This is due to the definitions of rational and irrational numbers being mutually exclusive.

Step 1: Understand the Definitions

A rational number is defined as any number which can be expressed as the fraction (\( \frac{p}{q} \)) of two integers, with denominator (\(q\)) not zero. On the other hand, an irrational number is a number that cannot be expressed as a ratio of two integers. It cannot be represented as a simple fraction and its decimal representation never ends or repeats. Examples of irrational numbers are \(\sqrt{2}\) or \(\pi\).

Step 2: Compare Definitions

A real number must be either rational or irrational. If a number can be expressed as a simple fraction, then it's rational. If it can't, it's irrational. A number can't meet the definitions of both at the same time. As soon as a number can be represented as the ratio of two integers, it can no longer be considered non-representable as a ratio or irrational.

Step 3: Conclusion

Given that a number cannot simultaneously fulfill the conditions of both being a ratio of two integers (rational number) and not being capable of represented as a ratio of two integers (irrational number), it is concluded that a real number cannot be both rational and irrational at the same time.