Hw 1

\begin{document}

\maketitle

\section{Problem 5}

Theorem: There exist irrational numbers a,b such that $a^b$ is rational.

Proof:
We know $\sqrt{2}$ is irrational if $a=\sqrt{2}$ and $b=\sqrt{2}$ then $a^b=(\sqrt{2})^\sqrt{2}$, then there are two cases, $a^b$ is either rational or irrational.

If $(\sqrt{2})^\sqrt{2}$ is rational then there exist irrational numbers a,b such that $a^b$ is rational.

If $(\sqrt{2})^\sqrt{2}$ is irrational then let $a=(\sqrt{2})^\sqrt{2}$ and $b=\sqrt{2}$ then $a^b=((\sqrt{2})^\sqrt{2})^\sqrt{2}$ $=(\sqrt{2})^2=2$ and $a^b$. Since we know that 2 is rational, this shows that there exist irrational numbers a,b such that $a^b$ is rational.

\end{document}

page revision: 1, last edited: 25 Jan 2018 23:08
The text is a proof of a theorem stating that there exist irrational numbers \(a\) and \(b\) such that \(a^b\) is rational. The proof considers two cases to demonstrate this.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Hw 1

\begin{document}

\maketitle

\section{Problem 5}

Theorem: There exist irrational numbers a,b such that $a^b$ is rational.

Proof:
We know $\sqrt{2}$ is irrational if $a=\sqrt{2}$ and $b=\sqrt{2}$ then $a^b=(\sqrt{2})^\sqrt{2}$, then there are two cases, $a^b$ is either rational or irrational.

If $(\sqrt{2})^\sqrt{2}$ is rational then there exist irrational numbers a,b such that $a^b$ is rational.

If $(\sqrt{2})^\sqrt{2}$ is irrational then let $a=(\sqrt{2})^\sqrt{2}$ and $b=\sqrt{2}$ then $a^b=((\sqrt{2})^\sqrt{2})^\sqrt{2}$ $=(\sqrt{2})^2=2$ and $a^b$. Since we know that 2 is rational, this shows that there exist irrational numbers a,b such that $a^b$ is rational.

\end{document}

page revision: 1, last edited: 25 Jan 2018 23:08
user
The text is a proof of a theorem stating that there exist irrational numbers \(a\) and \(b\) such that \(a^b\) is rational. The proof considers two cases to demonstrate this.

TYPE: NO_QUESTION