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Chapter 0: Problem 142

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

Short Answer

Expert verified
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 by step solution

01

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 \(\pi\).
02

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.
03

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.

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