# Math Help - Proof by Contradiction help, no clue, lost

1. ## Proof by Contradiction help, no clue, lost

Proof by contradictionTheorem: The reciprocal of any irrational number is irrational.

2. ## Re: Proof by Contradiction help, no clue, lost

Originally Posted by zhengcl86
Proof by contradictionTheorem: The reciprocal of any irrational number is irrational.
Suppose that $\gamma$ is an irrational number. You know that $\gamma\ne 0$. WHY?

Now suppose that $\frac{1}{\gamma}$ is rational.

What can you do with that?

3. ## Re: Proof by Contradiction help, no clue, lost

Lemma: If $\alpha \ne 0$ is rational, then $\frac{1}{\alpha}$ is rational.

Proof: Let $\alpha \ne 0$ be rational. Then $\alpha = \frac{p}{q}$ where $p, q$ are integers with $gcd(p, q) = 1$ and $q \ne 0$. Then $\frac{1}{\alpha} = \frac{1}{\frac{p}{q}} = \frac{q}{p}$. Thus, $gcd(p, q) = gcd(q, p) = 1$ and since $\alpha \ne 0$ we see that $p \ne 0$. Thus, $\frac{1}{\alpha}$ is rational.

Now let $\beta$ be irrational and assume by contradiction that $\frac{1}{\beta}$ is rational. Then by the Lemma, $\frac{1}{\frac{1}{\beta}}$ is rational. But, $\beta = \frac{1}{\frac{1}{\beta}}$ which means a rational number equals an irrational number, a contradiction. Thus, $\frac{1}{\beta}$ is irrational.