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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. They are mutually exclusive categories.

Step by step solution

01

Understand the definitions of rational and irrational numbers

A rational number is any real number that can be expressed as the ratio of two integers, usually written as a/b, where a and b are integers and b does not equal 0. An irrational number is a real number that cannot be expressed as a ratio of two integers.
02

Compare the definitions of rational and irrational numbers

By comparing the definitions, it can be seen that the two types of numbers are mutually exclusive: if a number can be expressed as the ratio of two integers, it is rational, if it cannot, it is irrational.
03

Apply logical deduction

It follows from the definitions and the basis of logic that it is not possible for a number to be both rational and irrational, as these are distinct and mutually exclusive categories of numbers.

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Most popular questions from this chapter

Fill in each box to make the statement true. Place the correct symbol, \(>\) or \(<,\) in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator. a. \(3^{\frac{1}{2}} 3^{\frac{1}{3}}\) b. \(\sqrt{7}+\sqrt{18} \quad \sqrt{7+18}\)

Evaluate each algebraic expression for the given value or values of the variable(s). $$x^{2}-7 x+4, \text { for } x=8$$

Find the union of the sets. $$\\{1,3,7,8\\} \cup\\{2,3,8\\}$$

A football was kicked vertically upward from a height of 4 feet with an initial speed of 60 feet per second. The formula $$h=4+60 t-16 t^{2}$$ describes the ball's height above the ground, \(h,\) in feet, \(t\) seconds after it was kicked. Use this formula to solve Exercises \(19-20 .\) What was the ball鈥檚 height 2 seconds after it was kicked?

$$\text { Factor completely.}$$ $$x^{4}-5 x^{2} y^{2}+4 y^{4}$$
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