# 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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### Convert this into fractional number step by step?

3.41287548754875... Convert the above number to a rational number? I was reviewing some pre calculus on my own but couldn't figure this out.
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### The dilemma of Pi [closed]

Is Pi rational or irrational ? Pi can be represented as 22/7 which is a rational number. Whereas 3.14 is a non terminating and non recurring number which is a irrational number
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### Does an analytic $f$ need be polynomial to close $\mathbb{Q}$

If an analytic function $f : \mathbb{R}\to\mathbb{R}$ satisfies $f(\mathbb{Q}) \subseteq \mathbb{Q}$, can we conclude that $f$ is a polynomial?
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### Creating a periodic sequence from a given subsequence

You are given the odd elements of an infinite binary sequence: $$a_1, a_3, a_5, \dots$$ You have to add even elements $a_2,a_4,a_6,\dots$ such that the resulting sequence is periodic (i.e, a ...
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### Proof that 1/x + 1/y is distinct for distinct unordered pairs of (x,y), xy = k.

Take xy = k, for nonzero k. There are many (x,y) that can satisfy this. However, how do I prove that the sums of the members of any two distinct, unordered pairs, is distinct? (This is an equivalent ...
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### Function that maps the “pureness” of a rational number?

By pureness I mean a number that shows how much the numerator and denominator are small. E.g. $\frac{1}{1}$ is purest, $\frac{1}{2}$ is less pure (but the same as $\frac{2}{1}$), $\frac{2}{3}$ is ...
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### What is the name of the set obtained by multiplying a given number by any rational?

Given a number, is there a name for the set where each element results of multiplying this number by a rational? For a given $n \in \mathbb N$: $$\{ r \cdot n \mid r \in \mathbb Q \}$$
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### show that this statement is false (counterexample) if $a,b \in \mathbb R \backslash \mathbb Q$ then $a \cdot b \in \mathbb R \backslash \mathbb Q$

if $a,b \in \mathbb R \backslash \mathbb Q$ then $a \cdot b \in \mathbb R \backslash \mathbb Q$ Okay so the question asks to show, with a counter example, that the above statement is false. Here ...
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### Show that $x=y+z$ for all $x \in S$

We are given a set $S$ as a subset of the rational numbers defined by: $0 \notin S$ If $s_1 , s_2 \in S$, then $\frac {s_1}{s_2} \in S$ There exists a nonzero rational number $q \notin S$ such ...
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### Reversing the digits of an infinite decimal

Let $x$ be a real number in $[0,1)$, with decimal expansion $$x = 0.d_1 d_2 d_3 \cdots d_i \cdots \;.$$ If the decimal expansion is finite, ending at $d_i$, then extend with zeros: $d_k = 0$ for all ...
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### What subsets of rationals have been defined where each element equals the sum of a sequence of numbers?

In particular, I'm interested in the rationals that result from adding a finite sequence of consecutive integer powers of two. Has it been studied somewhere? Update This formalized example might ...
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### Have humans proved Schinzel's conjecture for one specific rational number?

I asked the Tooth Fairy about Schinzel's conjecture that if $x$ is a positive rational number, then it can be represented as $$\frac{p + 1}{q + 1}$$ where $p$ and $q$ are primes, for infinitely many ...
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### Rationality of $a^2+b^2$

I have looked into this topic lately and have not found an answer to the following question. Is the following true: If $a,b\in\mathbb{R}$ and $a + b$ is rational, then $a^2 + b^2$ is rational
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### $\{a x^{z}: a\in \Bbb{Q}, z \in \Bbb{Z}\} \approx \Bbb{Q}^{\times} \otimes_{\Bbb{Z}} \Bbb{Z}^+ \implies$? what about $\Bbb{Q}$-linear sums?

Consider all functions $f: \Bbb{Q} \to \Bbb{Q}$ of the form $f(x) = a x^z$ where $a \in \Bbb{Q}, z \in \Bbb{Z}$, call it $G$. It forms an abelian group under usual multiplication. I think it's ...
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### When is $X^n-a$ irreducible over $\mathbb{Q}$?

Is there a general criterion, when $X^n-a$, $a\in\mathbb{Q}$ is irreducible? Clearly, this is the case if there is a prime $p$ such that $\nu_p(a)=\pm 1$: For $\nu_p(a)=1$, this follows from ...
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### Countable dense subset claim in Arzela Ascoli proof

In every Arzela Ascoli proof you see the following: Let $S = \mathbb{Q} \cap [a,b]$, where $[a,b]$ is an interval in $\mathbb{R}$, then $S$ is a countable dense subset and there exists a ...
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### If $x$ is rational and $xy$ is irrational, then $y$ is irrational. [closed]

This is a statement that I need to prove. Let $x$ and $y$ be real numbers. If $x$ is rational and $x\times y$ is irrational, then $y$ is irrational. I believe you have to prove this using ...
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### Proving that rational equivalence is an equivalence relation on any set.

I seek to prove that the rational equivalence relation is an equivalence relation, in that it is reflexive, symmetric, and transitive. The rational equivalence relation is as follows "Two numbers in ...
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### Prove that $\frac{p}{q}$ is a rational number with a finite decimal expression if $p$ is an integer and $q=(2^n)(5^m)$

Let $p,q$ be two integers and $q=(2^n)(5^m)$. Then $\frac pq$ is a rational number with a finite decimal expression. Any ideas how to do this? I've been thinking about it all day but I have no idea ...
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### Basel problem over $\mathbb{Q}_{\geq 1}$

Let $\mathbb{Q}_{\geq 1}=\{r\in\mathbb{Q}\,|\,r\geq 1\}$. Once $\mathbb{Q}$ is enumerable, $\mathbb{Q}_{\geq 1}$ is also enumerable. Let $\{r_1,r_2,\ldots\}$ be such an enumeration. What can we say ...
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### What is a good way to show that $[0,1]$ is not complete in $\mathbb{Q}$

To show a set is not complete, the best way is always produce a Cauchy sequence that does not converge in the set. I wish to show $[0,1]$ is not complete in $\mathbb{Q}$ I am a little stucking ...
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### How many rational values of x are not integers and satisfy the following equation?

How many rational values of x are not integers and satisfy the following equation: $$x^7 - 6x^6 + 5x^5 - 4x^4 + 3x^3 - 2x^2 + 1 = 0 ?$$ Well, I got this question from one of the Mathcounts ...
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### Onto function with domain of rational numbers and co-domain of natural numbers

I'm trying to find an onto function $f: \mathbb{Q} \to\mathbb{N}$ I'm somewhere along the lines of $f(q) = |(1 - q)| + q$ for non integers, but I'm not sure where to go from there.
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### (H.W question) In Mathemaical Analysis of Rudin example 1.1 Pg 2

The author went on and proved that 1) There exists no rational $p$ such that $p^2=2$ 2) He defined two sets $A$ and $B$ such that if $p\in A$ then $p^2 <2$ and if $p\in B$ then $p^2>2$ and ...
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### Proof of part of properties of exponentiation Tao proposition $4.3.12$
If you let $x,y$ be non-zero rational numbers, and let $n, m$ be integers, I need to prove that if $x \geq y>0$, then $x^{n} \geq y^{n}>0$ if n is positive, and $0< x^{n} \leq y^{n}$ if $n$ ...
How to show that there are continuum many subsets of $\mathbb Q$, no two of which are similar? Two sets are called similar if there is an order-preserving bijection between them.