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Please note before I ask the actual question, the proof's already been done in a previous question, e.g. here, which has also been mentioned below. What I don't understand is a comment on the same question suggesting the user to state $b^5≠0$ in the proof. But doesn't the proposition $b≠0$ itself imply $b^5≠0$. Why mention it explicitly?

Theorem: If $r$ is irrational, then ${ r }^{ \frac { 1 }{ 5 } }$ is irrational

Proof: We prove the contrapositive: if ${ r }^{ \frac { 1 }{ 5 } }$ is rational, then $r$ is rational

  1. Assume that $r^{1/5}$ is rational, then there exists $a,b∈Z$ such that:

    ${ r }^{ \frac { 1 }{ 5 } }=\frac { a }{ b }$ where $a,b$ are coprime and $b≠0$

  1. Therefore, ${ r }=\frac { a^{ 5 } }{ b^{ 5 } }$

  2. If $a,b∈Z$, then $a^5,b^5∈Z$ as well.

  3. Therefore $r∈Q$

Is this proof incomplete? Are proofs supposed to have redundant arguments or am I missing something?

J.S
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  • Doesn't $b\neq 0$ imply that $b^5\neq 0$? Yes, however it should still be stated and not left to the reader to infer. It is not redundant so much as it is trivial, but being trivial does not mean that it should be excluded. We wanted to show $r=\dfrac{a^5}{b^5}$ is actually rational. To do so, we go through the entire list of necessary properties for the definition of what is needed to be rational, namely that there exists a representation (we use this representation) where the numerator is an integer and the denominator is not only an integer but a nonzero one at that. – JMoravitz Jul 06 '21 at 13:18
  • Now... in some contexts it would have been fine to leave off the part of the proof showing that $b\neq 0\implies b^5\neq 0$. As always, you should write your proof with your audience in mind. If you are writing the proof as homework for your professor to read where the professor is wanting you to prove everything "from first principles" then it is absolutely a necessary part of the proof. If you are writing it with the target audience being people who all clearly know this fact, then it can be omitted for brevity. – JMoravitz Jul 06 '21 at 13:21
  • @JMoravitz Okay so if I understand it correctly, each fact no matter how trivial must be stated explicitly in order to make the proof rigorous and error free. By the way your comments could very well have been a complete answer. Thanks a lot for the explanation. – J.S Jul 06 '21 at 14:05

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