Questions tagged [rational-numbers]

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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50 views

A sequence of rational numbers having a limit that is an irrational number

Let $T = \{ t \in \mathbb{Q}: 0 \leq t \leq 1 \} $. With the parameterization of the unit circle given by $x(t) = \frac{1-t^2}{1+t^2}$ and $y(t) = \frac{2t}{1+t^2}$. With an indexing of the set $T$ by ...
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Construction in Ciesliski's proof of Sierpinski’s Topological Characterization of Q

An elementary proof of the fact that any countable metric space $(X,d),$ without isolated points is homeomorphic to $\mathbb Q$ uses a construction that I thought I was familiar with, but I'm having ...
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2answers
52 views

Why the proof of given two rational numbers their sum is rational involves the sum of both numbers. Wouldn't this be a contradiction?

In this question why doesn't proof of sum of two rational number is rational not proving the irreducibility of fraction $\frac{ad+bc}{bd}$?, the author gives a proof of why given two rationals the ...
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1answer
34 views

Simple function for enumerating bisections of $[0,1]$

Is there a simple function $f : \mathbb{N}\rightarrow\mathbb{Q}$ that returns further and further bisections of the segment $[0,1]$? For example, we could have $f(0)=0$, $f(1)=(1)$, $f(2)=1/2$, $f(3)=...
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Is there a rational point in a given open set such that the distancse from given rational points to it are all rational numbers on $R^2$?

On $R^2$ there is a nonempty open set $A$ and n rational points $a_1,...,a_n$. Is there a rational point $a$ in $A$ such that $\forall n\in\{1,..,n\},~|a-a_i|$ is a rational number. My idea is to find ...
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4answers
117 views

$a_{n+1} = a_n/2 + 1/a_n$ is Cauchy but has no limit in $\mathbb{Q}$

I want to show that the sequence recursively defined by $a_{n+1} = \frac{a_n}{2} + \frac{1}{a_n}, \:\: a_1=1$ is a Cauchy sequence that does not converge in $\mathbb{Q}$. My idea was to show that ...
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Proof that square root of any non-perfect square is irrational [duplicate]

Hi guys it would be a great help if you could look through my proof and point out any errors. I also want to know whether my proof is coherent, and whether the amount of detail is not too much or ...
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2answers
165 views

Where are the irrational numbers?

I’m new to real analysis and have recently encountered the fact that there are more irrational numbers than rational. However, I can’t seem to reconcile the fact that I can easily think of rational ...
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83 views

Rational number between any rationals, *irrational* doubt of rational numbers

This might be very odd question but hope be still worthy... $\mathbb{Q}$ is known as countable, as it is a union of countable sets $\mathbb Q_n, n\in \mathbb N$, where $$ \mathbb Q_n = \left\{{a\over ...
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1answer
84 views

Does it relate to the uncountability of irrationals?

Recently I am reading Measure, Integration & Real Analysis by Sheldon Axler, and the author claimed that the outer measure is non-additive by constructing a set $\tilde{a}$ generated by $a \in [-1,...
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A question about $\mathbb{Q} \cap [0,1]$? [duplicate]

The question says like this: Let $S \subset [0,1]$ be a set satisfying the following two properties.(1)$0,1 \in S$;(2)For any $n \in \mathbb{N}$ and pairwise distinct numbers $s_{1},...,s_{n} \in S$ ...
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1answer
28 views

Continuity of piecewise function involving rationals/irrationals and Cantor set

I'm struggling to determine the continuity of the following function: $$ f(x)=\begin{cases}0 \quad \text{if $x \in \mathbb{Q}\cap D,$}\\ x^3 \quad \text{if $x \notin \mathbb{Q}\cap D$;} \end{cases}$$ ...
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1answer
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Simplify $A=\frac{y^\frac12}{y^\frac12-2}+\frac{y^\frac12}{y^\frac12+2}-2$

Simplify $$A=\dfrac{y^\frac12}{y^\frac12-2}+\dfrac{y^\frac12}{y^\frac12+2}-2$$ So $$A=\dfrac{y^\frac12\left(y^\frac12+2\right)+y^\frac12\left(y^\frac12-2\right)}{y-4}-2=\dfrac{2y^\frac14}{y-4}-2=\...
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1answer
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Cancel out $\dfrac{a\left(a-b^{\frac12}\right)}{a+a^{\frac12}\cdot b^{\frac14}}$ [duplicate]

Cancel out $$\dfrac{2-54b}{2-6b^\frac13}$$ I really don't see what we are supposed to do. This is what I have tried to do with the numerator $$2-54b=2-9\cdot6b=2-3^3\cdot2b=2(1-3^3b)$$
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1answer
26 views

Cancel out $\frac{a\left(a-b^{\frac12}\right)}{a+a^{\frac12}\cdot b^{\frac14}}$

Cancel out $$\dfrac{a\left(a-b^{\frac12}\right)}{a+a^{\frac12}\cdot b^{\frac14}}$$ The given expression is equal to $$\dfrac{a\left(a-b^{\frac12}\right)}{\left(a^\frac12\right)^2+a^\frac12\cdot b^\...
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23 views

Generalize square roots using matrices

For curiosity reasons, I'm trying to see wether or not it is possible to define square roots of rational numbers using (maybe infinite) matrices, without real numbers, ie. using only integers or ...
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1answer
29 views

Perfect sets and topological vs. limit closure

Given a topological space $(X,\tau)$ and some subset $Y\subseteq X$, the subset $Y$ is perfect when it is closed and dense-in-itself. Consider $X = \mathbb{Q}\cap[0,1]$ equipped with the topology ...
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1answer
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Is it possible to permutate a periodic sequence in such a way that any infinite substring will always be non-periodic?

In particular, I'm interested to know whether there is a shuffling method which can guarantee that the sequence of digits of a repeating decimal is turned into one of an irrational number. Here, ...
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48 views

How to prove that $(a^b)^c = a^{(bc)}$ for rational and negative exponents?

I have this question about exponentiation. How can one prove that $(a^b)^c$ = $a^{(bc)}$ for rational and negative exponents? Firstly, we need a basis of what a rational and negative exponent are. So, ...
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1answer
40 views

Does a rational number to the n-th power exists between a and b integers ? If yes, how to find one easily?

Let $a,b,n\in\mathbb N^*$, $a<b$, Does a rational $p/q$ such that $p,q\in\mathbb N$ and $a<p^n/q^n<b$ exist ? If that's the case, Is there a way to find one ? I know that $\mathbb {\bar Q} = \...
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Is $\mathbb{Q}^\omega$ (a set of all rational sequences) a homogeneous space?

Is $\mathbb{Q}^\omega$ a homogeneous space? This space is defined as a set of all rational sequences, also sometimes denoted $\mathbb{Q}^\mathbb{N}$. It is the set of all functions from the naturals ...
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29 views

Infinite sum of reciprocals of positive integers convergent to some rational number in a given range

I am looking for an example of some infinite sum of reciprocals of positive integers convergent to some rational number in the range $(1,2)$. I am also interested in methods to construct such infinite ...
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1answer
106 views

Is there a single equation describing a triangle in the Cartesian plane, using only arithmetic operations? (Yes.)

Is there a single equation describing a triangle in the Cartesian plane which includes only arithmetic operations? The below uses the square root---but, in this case, the square root is itself may be ...
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Why is $\sum_{n\ge1}\frac{\text B_\frac n2(-n,n)}{n^a b^n }=-\sum_{n\ge1}\frac{\left(\frac 2n-1\right)^n}{n^{a+1}b^n}$ close to (reciprocal) integers?

Here is a possible closed form of a sum with tetration in it made by equating coefficients of the Incomplete Beta function. This question is inspired by: Closed form of $$\sum\limits_{n=1}^\infty \...
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0answers
59 views

Is $f: \mathbb Q \to \mathbb Q$ strictly increasing if locally strictly increasing everywhere?

(At the outset of this question, I will note that my background in analysis is introductory, derived from the first 11 chapters of Spivak's Calculus.) I recently proved the following theorem: If the ...
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61 views

Rigorous book developing number systems and set theory

Baby Rudins chapter 1 assumes familiarity with the arithmetic of natural numbers and rational numbers. Rudin constructs the reals and complex numbers from the rational numbers. For curiosity's sake, I ...
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1answer
41 views

Linear Diophantine equations with fractional coefficients

I would like to find an expression for an equation of the following form $$a n+bm$$ where $a,b\in\mathbb Q$ are rationals and $n,m\in\mathbb Z$ are integers, such that there is only one integer ...
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27 views

How to evaluate rational numbers to rational powers

Is there a method besides a Taylor series to calculate $x^y$, where $x$ and $y$ are rational numbers? So for example, I require a method to calculate $2.128321\dots^{5.3212\dots}$ to a certain amount ...
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1answer
103 views

Find minimum $n$ for $x^2+3x+6=f_1(x)^2+f_2(x)^2+ \cdots +f_n(x)^2$

The problem Find minimum $n$ for $x^2+3x+6=f_1(x)^2+f_2(x)^2+ \cdots +f_n(x)^2$Find the minimum value of $n$ such that there exists $f_i \in \mathbb{Q[i]}, i=1,2,...,n$ such that, \begin{align}x^2+3x+...
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1answer
71 views

Let $n,m\in N0$ and $\sqrt{n}\not\in Q.$ Show that $\sqrt{n} + \sqrt{m}\not\in Q$

Let $n,m\in N0$ and $\sqrt{n}\not\in Q.$ Show that $\sqrt{n} + \sqrt{m}\not\in Q$ My first thought was to prove it via Contradiction. So, suppose $\sqrt{n} + \sqrt{m}\in Q$ And since $\sqrt{n}\not\...
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2answers
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Are the rational numbers in order of size a sequence?

I am struggling through Abbott's Understanding Analysis and have been asked if there is a sequence that contains subsequences that converge, severally, to each term of the harmonic sequence. The ...
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1answer
67 views

Infinite irreducible polynomial over Q[x] using Eisenstein

Like the title describes, I know that over Q for each number n ≥ 1, one can easily construct infinitely many irreducible polynomials of degree n. But I want to prove using Eisenstein's criterion this ...
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2answers
98 views

For which $f: \mathbb{Q}\rightarrow\mathbb{Q}^+$ does the sum $\sum_{q\in\mathbb{Q}} f(q)$ converge?

BACKGROUND: This is not a homework question -- I am not even in school. I am purely interested in the question itself. I actually asked this question in my second semester of real analysis during my ...
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2answers
73 views

How do you find "good" rational approximations to a decimal number?

When presented with real number as a decimal, are there any methods to finding "good" rational approximations $a/b$ to that number? By "good" I mean that $a$ and $b$ are reasonably ...
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1answer
48 views

Is there a proof that logarithm whose anti-logarithm is not a natural power of the base is not a rational number?

To write it simplier (at least for me): Is there a proof that $\forall a,b\in\mathbb{N}_+\backslash{\{1\}},b>a, \text{a is prime}, \forall k\in\mathbb{N}, b \neq a^k: \\ \log_a b\notin \mathbb{Q}$ ...
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1answer
81 views

Homomorphism of $𝜑: \mathbb{Q}^{*}/⟨-1⟩\rightarrow \mathbb{Q}^+$

There is a definition from Group Theory about isomorphisms and homomorphisms. For example, if $G$ and $H$ are groups and $\varphi: G \rightarrow H$ is a homomorphism, it must satisfy that $\forall x,y ...
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1answer
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Determine whether a set is dense in $\mathbb{R}$

we just learned in class about dense sets in $\Bbb R$ . We learned how to prove that a set is dense in $\Bbb R$ but not how to disprove. I got the following question: Determine whether the set is ...
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1answer
49 views

The image set of the function $f(x) = \dfrac{x -1}{x + 1}$ generates $\Bbb{Q}$ multiplicatively?

Define $f(x) = \dfrac{x - 1}{x+1}$. Compute the composition $f\circ f(x) = \dfrac{\dfrac{x-1}{x+1}-1}{\dfrac{x - 1}{x+1} +1} =\dfrac{\dfrac{x - 1 -x -1}{x+1}}{\dfrac{x-1 + x + 1}{x+1}} =\dfrac{-2}{2x} ...
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1answer
62 views

Prove that between any two positive real numbers there is a rational number

I am having a hard time coming to grips with one assumption of this proof Let $x$ and $y$ $\in\mathbb{R+}$ and without loss of generality x<y. We pick a natural number large enough to make $\frac{...
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1answer
21 views

What could be an element that fulfills the given criteria? [closed]

Given that $A=\left \{ x^2|x\in \mathbb{Q}, x^2<2 \right \}$ I want to find an element $y\in A$, with $|y-2| < 0.001$ There must be a square of a rational number(which thus, is also a rational ...
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0answers
59 views

Ratios of integers in Bezout's Identity

Bezout's Identity is a classic of elementary number theory: let $m,n\in\mathbb{N}^+$ with $\gcd(m,n)=1$. Then there are $a,b\in\mathbb{Z}$ with $$ am+bn=1 $$With no loss of generality we can assume $m&...
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5answers
253 views

Prove or disprove a claim regarding irrational numbers

I am trying to prove the following claim: Let $ 0\leq n \in \Bbb Z$ and suppose that there exists a $k \in \Bbb Z$ such that $n=4k+3$. Prove or disprove: $\sqrt n \notin \Bbb Q$ . The problem I am ...
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1answer
16 views

Does the range of the correlation coefficient or the covariance belongs to $\mathbb{Q}$

I have a question that seems to me logic, but I haven't seen anywhere such a claim. Could we claim that the range of the correlation coefficient or the covariance belongs to $\mathbb{Q}$, namely in ...
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1answer
104 views

What is the reason for the existence of real numbers? Is it an artifact/side effect of our thought process?

The observation of lengths that can not be represented by rational numbers was noticed if I recall correctly by some Pythagorean disciple over applying the Pythagorean theorem on a triangle with side ...
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1answer
98 views

Is $\frac{0}{\pi}$ rational?

In math lesson, our teacher showed this formula. $Q = \{\frac{a}b\}\land (a\land b\in Q)\land (b\neq 0)$ According to this formula, $\frac{0}\pi$ is... strange. You know, $\frac{0}\pi$ is 0 and 0 is ...
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3answers
80 views

How to prove that $(x^a)^b=x^{ab}$

Let $a$ and $b$ be rational numbers. How can I prove that $(x^a)^b=x^{ab}$. I just don't know where to begin, can someone tell me a hint? I know that if $a=c/d$ and $b=e/f$ then $(x^a)^b$ is ${(x^{c/d}...
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1answer
46 views

Are sets of rational sequences open, or closed in $\mathbb{Q}^{\omega}$?

Are sets of rational sequences open, or closed in $\mathbb{Q}^{\omega}$? I think that any subset of $\mathbb{Q}^{\omega}$ is both open and closed in $\mathbb{Q}^{\omega}$, since: Any subset is closed ...
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1answer
44 views

Prove or disprove a claim regarding rational numbers [closed]

I need to prove or disprove the following claim. Let $ x \notin \Bbb Q $ such that $ x^3 \in \Bbb Q $. Then $x^2+x+1 \notin \Bbb Q$. I tried to find a lot of counter examples in order to disprove it, ...
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0answers
62 views

Are clopen sets of rational sequences $F_{\sigma\delta}$?

I am studying properties of clopen subsets of $\mathbb{Q}^{\omega}$ (subsets of the space of all rational sequences, endowed with the product topology) and for verifying some properties, I need the ...
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0answers
44 views

Proof that $\mathbb{Q} \subseteq \{\frac{3m}{7n}\| m\in\mathbb{Z},n\in \mathbb{N}\}$

It seems to me that the statement is true and I thought about prooving it by showing that if you take any rational number of form $\frac{a}{b}$ you can cross multiply it with $\frac{3m}{7n}$ and get $...

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