# Tagged Questions

Questions about numbers expressible as the quotient of two integers. For questions on determining whether a number is rational, use the (rationality-testing) tag instead.

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### Examining the nature of mapping of $f(x) = \frac{x}{x^2 - 2}$.

Let f: $\mathbb{Q} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{x}{x^2 - 2}$, $x \in \mathbb{Q}$. Examine the nature of mapping Attempt: I think f(x) is not ...
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### Interesting facts/ proofs about rational and irrational numbers

We got set some work to find some interesting facts or proofs regarding rational and irrational numbers. I wonder if anyone could offer some insight or recommend a good book/ website to look at.
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### What Will Happen Without Decimal Expansion?

After a discussion on the complexity of decimal expansion (such as $0.\bar{9}=1$), some of my students (middle school) decided to throw away the decimal expansion of some numbers! Namely, the numbers ...
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### Numerical polynomials by means of prime powers

Let $\mathbb{E}$ be the set of prime powers (except $1$). Let $f \in \mathbb{Q}[x]$ be a rational polynomial with $f(\mathbb{E}) \subseteq \mathbb{Z}$. Does it follow that $f$ is numerical, i.e. ...
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### Integral of rationals

Define $f(x)$ as $$f(x)=\begin{cases}0,&\text{if }x\in \mathbb{Q}\\ 1,&\text{if }x\notin \mathbb{Q}\;. \end{cases}$$ Considering the fact that there is a countable infinity of rationals yet an ...
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### Prove that the product of an irrational number and a rational number is irrational.

If $x$ is an irrational number and $r$ is a rational number then $xr$ is an irrational number. Proof. Suppose that $xr$ is a rational number. By defintion of a rational number $xr= m/n$ where $m,n$ ...
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### Odd divided by even is a fraction

How can we prove that an odd number divided by an even number is a fraction? I started with odd $=2m+1$ and even $=2n$ and get left with with $(m+2)/n$.
Two rational numbers $\frac{a}{b}$ < $\frac{c}{d}$ will be called neighbors if $\frac{c}{d}$ - $\frac{a}{b}$ = $\frac{bc-ad}{bd}$ = $\frac{1}{bd}$. Suppose $\frac{a}{b}$ and $\frac{c}{d}$ are ...