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Chapter 0: Problem 8

Suppose \(t\) is an irrational number. Explain why \(\frac{1}{t}\) is also an irrational number.

Short Answer

Expert verified
By assuming that \(t\) is irrational, we supposed that \(\frac{1}{t}\) is rational, which means it can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers, and \(b \neq 0\). However, this led to the contradiction that \(t = \frac{b}{a}\), which means \(t\) would be a rational number. This contradiction implies that our assumption was incorrect, and as a result, \(\frac{1}{t}\) must also be an irrational number.

Step by step solution

01

Rational Numbers Definition

A rational number is a number that can be represented as the fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\).
02

Irrational Numbers Definition

An irrational number is a number that cannot be represented as the fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\). Now let's prove that if \(t\) is an irrational number, then \(\frac{1}{t}\) is also an irrational number:
03

Assume t is Irrational

We are given that \(t\) is an irrational number. Therefore, we cannot express \(t\) as the fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q\neq0\).
04

Assume 1/t is Rational

Suppose, for the sake of contradiction, that \(\frac{1}{t}\) is a rational number. Then we can express \(\frac{1}{t}\) as the fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b\neq0\).
05

Reciprocal Relationship

Knowing that \(t = \frac{1}{\left(\frac{1}{t}\right)}\), we can now plug in our assumption that \(\frac{1}{t} = \frac{a}{b}\). Therefore, \(t = \frac{1}{\left(\frac{a}{b}\right)} = \frac{b}{a}\).
06

Contradiction

We have now expressed \(t\) as the fraction \(\frac{b}{a}\), where \(a\) and \(b\) are integers, and \(a\neq0\). This is a contradiction because we are given that \(t\) is an irrational number, and thus cannot be represented as a fraction of two integers.
07

Conclusion

Since our assumption that \(\frac{1}{t}\) is rational led to a contradiction, we can conclude that \(\frac{1}{t}\) is not rational and therefore must be irrational. This proves that if \(t\) is an irrational number, then \(\frac{1}{t}\) is also an irrational number.

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