Problem 142
Question
Can a real number be both rational and irrational? Explain your answer.
Step-by-Step Solution
Verified Answer
No, a real number cannot be both rational and irrational. They are mutually exclusive categories.
1Step 1: 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.
2Step 2: 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.
3Step 3: 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.
Other exercises in this chapter
Problem 142
Find all integers \(b\) so that the trinomial can be factored. $$x^{2}+b x+15$$
View solution Problem 142
Multiply: \(\quad\left(2 x^{3} y^{2}\right)\left(5 x^{4} y^{7}\right)\)
View solution Problem 143
Use the distributive property to multiply: $$2 x^{4}\left(8 x^{4}+3 x\right)$$
View solution Problem 143
Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\left(4 \times 10^{3}\
View solution