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Chapter 0: Problem 6

Explain why the sum of a rational number and an irrational number is an irrational number.

Short Answer

Expert verified
The sum of a rational number and an irrational number is an irrational number because assuming their sum is rational leads to a contradiction. If the sum is rational, we can represent it as a fraction. However, rearranging the equation shows that the irrational number is equal to the difference of two rational numbers, which would mean it is rational as well. This contradicts the initial claim that the number is irrational, proving that the sum must be irrational.

Step by step solution

01

Definition of Rational and Irrational Numbers

A rational number is a number that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\) is not equal to zero. Examples of rational numbers are \(2.5, -\frac{3}{4}\), and \(0\). An irrational number is a number that cannot be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\) is not equal to zero. Examples of irrational numbers are \(\pi\) and \(\sqrt{2}\).
02

Assume the Sum is Rational

Let's suppose that the sum of a rational number and an irrational number is indeed a rational number. Let the rational number be denoted as \(r\) and the irrational number as \(i\). If their sum is rational, we can represent it as the fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\) is not equal to zero.
03

Rearrange the Equation

Now, we'll rearrange the equation to isolate the irrational number. \(\frac{p}{q} = r + i\) \(i = \frac{p}{q} - r\)
04

Evaluate the Result

The equation we just derived states that \(i\) (the irrational number) is equal to the difference of two rational numbers, \(\frac{p}{q}\) and \(r\). However, we know that the sum or difference of two rational numbers is always a rational number. Since \(i\) is an irrational number, this creates a contradiction.
05

Conclusion

We've reached a contradiction, which means our initial assumption that the sum of a rational number and an irrational number is a rational number is false. Therefore, the sum of a rational number and an irrational number must be an irrational number.

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