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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.

Step by step solution

01

- Understanding Rational numbers

Rational numbers are real numbers that can be written as the quotient of two integers, with a non-zero denominator. In other words, if a number can be expressed as \(\frac{p}{q}\) where p and q are integers and \(q \neq 0\), it is a rational number.
02

- Understanding Irrational numbers

Irrational numbers are real numbers that cannot be written as the quotient of two integers. They are non-repeating, non-terminating decimal numbers. Simply said, if a number cannot be written as \(\frac{p}{q}\) where p and q are integers, it is an irrational number.
03

- Evaluating if a number can be both Rational and Irrational

Based on definitions, a rational number can be written as a fraction of two integers, while an irrational number cannot. Hence, a real number cannot be both rational and irrational at the same time. These are mutually exclusive categories.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{2 x-1}{x-7}+\frac{3 x-1}{x-7}-\frac{5 x-2}{x-7}=0$$

If you are given a quadratic equation, how do you determine which method to use to solve it?

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. On two examinations, you have grades of 86 and \(88 .\) There is an optional final examination, which counts as one grade. You decide to take the final in order to get a course grade of A, meaning a final average of at least 90 . a. What must you get on the final to earn an A in the course? b. By taking the final, if you do poorly, you might risk the B that you have in the course based on the first two exam grades. If your final average is less than \(80,\) you will lose your \(\mathrm{B}\) in the course. Describe the grades on the final that will cause this to happen.

How is the quadratic formula derived?

Explain how to add rational expressions having no common factors in their denominators. Use \(\frac{3}{x+5}+\frac{7}{x+2}\) in your explanation.
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