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.

221 questions with no upvoted or accepted answers
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18
votes
1answer
344 views

Do all rational numbers repeat in Fibonacci coding?

In a regular, positional number system (like our decimal numbers), every rational number ends in a repeating sequence (even if it's just 0). For example, $\frac{1}{3}$ in base 10 is $0.33333...$, in ...
16
votes
1answer
244 views

Existence of rational sequence such that a polynomial is split over $\Bbb{Q}$

Does there exist a sequence $(a_n)_{n\in \Bbb{N}}$ of rationals such that for all $n\in \Bbb{N}$, $a_n\neq 0$ and the polynomial $a_0+a_1X+\cdots+a_nX^n$ is split over $\Bbb{Q}$? I was asked this ...
9
votes
1answer
110 views

Does there exist a generating function for the rational numbers?

Since the rationals are countable, you can list them in a sequence $(a_n)_{n\geq 0}$ such that each rational appears at least once in the sequence. Is there such a listing $(a_n)_{n \geq 0}$ for which ...
8
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0answers
157 views

Is $z=e^{\frac{1}{\log(x)}}$ a solution to a polynomial?

Is $z=e^{\frac{1}{\log(x)}}$ a solution to a polynomial? $x\in\Bbb Q~\cap(0,1).$ Wolfram is telling me $z$ is algebraic but I'm not sure that I believe this. I believe it would follow from Schanuel'...
8
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0answers
218 views

What is the best rational approximation of $\frac{1}{x}$

Let $x \in \mathbb{R}$, $x \notin \mathbb{Q}$ and let the function $f:\mathbb{R}\setminus \mathbb{Q} \,\times \mathbb{N}\rightarrow \mathbb{Q}$ provide the best rational approximation for $x$ where ...
7
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0answers
77 views

Visualizing rational numbers as multiplication graphs

It's an interesting fact, that there's a straight forward way to visualize rational numbers. To each rational number – given as two integers $n<m$ – there corresponds a multiplication ...
7
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0answers
215 views

Is there any continuous function that is only differentiable on $\mathbb{Q}$?

I am looking for a continuous function $f: \mathbb R \rightarrow \mathbb R$ so that $f$ is differentiable in $x$, if and only if $x \in \mathbb Q$. I already know there is no function that is ...
7
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0answers
115 views

Best rational approximation with a twist: denominator must be a square

I'm looking to form a series of better and better rational number approximations to a known value $x$. This is a classic problem elegantly solved by the use of a continued fraction representation. For ...
7
votes
0answers
104 views

The period of decimal expansion of $1/m$

I am trying to prove that for a composite, positive integer $m$ such that $2 \nmid m$ and $5 \nmid m$, the period of the decimal expansion of $1/m$ is equal to $\text{ord}_{m}(10)$, where $\text{ord}_{...
6
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0answers
96 views

Proof that $(1+\frac{1}{n})^n$ can't converge to a rational number

One of my colleagues challenged me (and his students) with the following: "Assume you don't know that $\lim_{n\to +\infty}(1+\frac{1}{n})^n=e$. Prove the sequence $u_n=(1+\frac{1}{n})^n$ ...
6
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0answers
94 views

Rational Trig Solutions for $n\ge 3$

Are there solutions to $$\sin(x+y)\sin(x-y)=n\ \sin(x)\sin(y)$$ for $n\ge 3$ where $x$ and $y$ are rational multiples of $\pi$? (excluding the trivial solutions when both sides are $0$). Known ...
6
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0answers
542 views

Rational analysis

I found myself thinking about how much of real analysis that can also be developed within the rational numbers. Of course, $\Bbb Q$ is lacking what is perhaps the most important property of the real ...
6
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0answers
285 views

Rational multiples of $\pi/2$ whose sines are also rational

Let $f(x)=\sin(x\frac{\pi}{2})$. Let $R$ the set of $x$ such that : $0\le x\le 1$ $x \in \mathbb Q$ $f(x) \in \mathbb Q$ Hence, $0\in R$ as $f(0)=0$. $1\in R$ as f(1)=1. And $\frac{1}{3}\in R$ as $f(...
6
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0answers
163 views

Is there always a telescopic series associated with a rational number?

Here is something I thought up while I was bored and my, erm, fish were busy: Given a rational number $p\in(0,1)$, are there always positive integers $n$, $c_m$, and $w_m$ such that $$p=\sum_{k=1}^\...
5
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0answers
155 views

Rational numbers as angles - where do irrationals fit in?

If we make a rectangular grid with integer coordinates, it's possible to assign a unique angle to any rational number, using the definition $\tan \phi=y/x$ for $\phi \in (-\pi/2, \pi/2)$. For ...
4
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0answers
141 views

Representation of rationals as finite continued fractions with restricted coefficients

This question and its answer incidentally show that every non-zero rational number $q$ can be written as a finite generalised continued fraction of the form: $$ \dfrac{2^{n_0}}{1- \dfrac{2^{n_1}}{1 -...
4
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0answers
57 views

Rational integral conjecture of $(x^{2k} - \sin(\pi x))^2$

I was messing around, and came up with the following conjecture: $$\int_{-1}^1(x^{2k} - \sin(\pi x))^2dx = \frac{4k+3}{4k+1}$$ More generally, it seems that for any polynomial $P$ with rational ...
4
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0answers
226 views

Rational length between Fermat point and vertices in a triangle with rational sides

I was working on a quite challenging problem (thought by myself): Find all triangles (with rational sides), such that the lengths between the Fermat Point and its vertices are rational. ...
4
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0answers
371 views

Is there a definition of a “pseudo period” for $f(x)=\sin(3x)+\sin(\pi x)$?

Sums of trigonometric functions may or may not be periodic functions; in particular, $\sin(ax)+\sin(bx)$ is periodic if $a/b$ is rational. If we consider the function \begin{equation} f(x) = \sin(3x) ...
4
votes
1answer
161 views

A rational orbit that's provably dense in the reals?

Iterating the map $\ \ x\ \mapsto\ x-\frac{1}{x},\ \ $ the orbit of initial point $2$ is "probably" dense in $\mathbb{R}$. Is there an explicit rational mapping together with an initial rational ...
4
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0answers
1k views

How to estimate the number of decimal places required for a division?

Given two decimal numbers, is it possible to estimate the number of decimal places required to fit the result of their division? Provided that the division yields a finite number of decimals, of ...
4
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0answers
289 views

Irrational numbers to the power of other irrational numbers: A beautiful proof question

The following theorem has a very beautiful proof. Theorem: There exist two irrational numbers $x$ and $y$ such that $x^y$ is rational. Proof: If $\sqrt{2}^{\sqrt{2}}$ is rational then we are ...
4
votes
1answer
394 views

Linear independence over rationals

I am trying to figure out for what values of $n$, the numbers $\sin\left(\frac{2\pi k}{n}\right)$, for $k = 1,\dots,n-1$, are linearly independent over the rationals. Any thoughts on how I may want ...
4
votes
1answer
812 views

Counting Real Numbers

Forgive me if this is a novice question. I'm not a mathematics student, but I'm interested in mathematical philosophy. Georg Cantor made an argument that the set of rational numbers is countable by ...
3
votes
0answers
73 views

What is the correct name for a non-whole real number?

I apologise for the simple nature of this question but I can't find the answer. I know that a whole number is an integer. I also know that a number that can be expressed as the quotient of two ...
3
votes
1answer
72 views

How are the Periods of the Decimal Expansions of $\frac{p}{q}$ and $\frac{q}{p}$ Related?

In an excellent post several years ago, we learn that the period of the decimal expansion of a rational number $\frac{p}{q}$ must divide the multiplicative order of $10\pmod q$ assuming that there are ...
3
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0answers
49 views

Finding a subspace of dimension $3$ which does not intersect a rational subspace of dimension $2$

Some context. By rational subspace, I mean a subspace of $\mathbb R^5$ which admits a rational basis. In other words, a basis formed with vectors of $\mathbb Q^5$. For instance, the vector $v=(0,\pi,...
3
votes
2answers
38 views

Help proving there is a sequence of rational numbers

I'm trying to prove the following: Let $\Bbb Q$ be the countable set of rational numbers and $\{x_n\}_{n=1}^\infty$ be a sequence such that for every q $\in$ $\Bbb Q$ there is a $n \in \Bbb N$ with $...
3
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0answers
97 views

Is there a well-behaved finitely additive “measure” on $\mathbb{Q }$?

There exists no nontrivial measure on the set of rational numbers for which the measure of singletons is zero. That’s because the rational numbers are countable, so any set of rational numbers is a ...
3
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0answers
32 views

Number of continous functions from set of all rationals to itself

Since Rational numbers is countable it is equivalent to Natural numbers. Now the number sequences forming by more than 2 numbers is uncountable..Also here the open sets are singleton sets. So ...
3
votes
0answers
45 views

Cumulative distribution function for discrete distribution on $\mathbb{Q} \subseteq \mathbb{R}$.

Consider a function $F : \mathbb{R} \rightarrow [0,1]$ with the following $3$ properties: $F$ is monotonically increasing, $F$ is right-continuous and $\lim_{x \rightarrow -\infty} F(x) = 0$ and $\...
3
votes
0answers
172 views

Factor the two polynomials into a product of irreducible elements of $\mathbb{Q}[x]$

I need to find a factorization of both $f_{1}=2x^{2}+4x+6$ and $f_{2}=2x^{2}+4x-6$ into a product of irreducible elements of $\mathbb{Q}[x]$. I already was able to do so in the case of $\mathbb{Z}[x]$...
3
votes
1answer
42 views

Showing a Sequence of Subsets of $\Bbb Q$ Tends to a Dense Subset of $\left[0,\infty\right)$

For $N\in\mathbb{N}$, let: $S_{N}=\left\{ \left(\frac{2}{9}\right)^{N},\left(\frac{2}{9}\right)^{N}\times6,\left(\frac{2}{9}\right)^{N}\times6^{2},...,\left(\frac{2}{9}\right)^{N}\times6^{2N}\right\}$...
3
votes
1answer
223 views

Proving addition of rational numbers is well defined

I am trying to show that the addition of rational numbers is well defined. Does anyone know if this is a legitimate strategy? Also, I am quite unfamiliar with coding on here. Q is defined as $Q = \{m/...
3
votes
1answer
124 views

Show that $S=\mathbb{Q} \cap [0,1]$.

Can you give a hint to show the following exercise? Let $S \subset [0,1]$ such that $0, 1\in S$ and for all $n\in \mathbb{N}$, $s_1,...,s_n \in S$ distintc, then $\dfrac{s_1+...+s_n}{n}\in S$. Show ...
3
votes
0answers
412 views

Generalizing the Rational Roots Theorem to complex rational roots

Is there a generalization of the Rational Roots Theorem to complex rational roots? This answer on MSE shows that it can be applied to purely imaginary rational roots. I'm curious about roots of the ...
3
votes
0answers
147 views

Why Hardy-Littlewood method is called circle method?

I read the phrase Hardy-Littlewood circle method in many places in Analytic number theory books and papers. I would like to know why it is called circle method, what is the idea behind it? Is it ...
3
votes
0answers
79 views

Non-constructive proofs for the rationality of a number

One of the key ideas in transcedental number theory is proving that a number is transcedental (i.e. not the root of any polynomial with integer coefficients) by showing a sequence of rational numbers ...
3
votes
1answer
169 views

Nonlinear regular bijection from $\mathbb Q$ to itself

Is there a bijection $\phi\colon \mathbb Q \to \mathbb Q$ such that $\phi$ is nonlinear (i.e., different from $x\mapsto ax+b$), $\phi$ is regular: the extension $\hat{\phi}$ of $\phi$ over $\mathbb R$...
3
votes
0answers
199 views

Is there a stable probability distribution on the rational numbers?

Does there exist a (non-trivial) probability distribution on the rational numbers $$\sum_{r\in\mathbb{Q}}p_r=1$$ with $0\leq p_r$, which is stable, meaning that the sum of two i.i.d. random variables ...
3
votes
0answers
199 views

What is the Best Introduction to Dedekind Cuts?

I'm looking for a clear, thorough, and easy-to-follow introduction to Dedekind cuts that is specifically geared towards those with an interest in foundational issues. So far, the discussions that I ...
3
votes
1answer
104 views

Real numbers and rationals - Decimal Expansion

How would one endeavor to show that A real number is rational if and only if its decimal expression ends in recurring digits?
3
votes
1answer
61 views

Is a subset of $\mathbb{Q}\times\mathbb{Q}$ that all variants of an exponentiation equation have answers in it, infinite?

Note that we have: $$A=\{(a,b)\in\mathbb{Q}\times\mathbb{Q}~|~\text{Both equations}~a+x=b, b+y=a~\text{have answers in }~\mathbb{Q}\}=\mathbb{Q}\times\mathbb{Q}$$ $$B=\{(a,b)\in\mathbb{Q}\times\...
3
votes
0answers
36 views

Properties of digit functions for numbers in $[0,1]$

Consider a function $g(n): \mathbb N \to \{0,1,2,3,4,5,6,7,8,9\}$, ie. $g$ maps the natural numbers to natural numbers between $0$ and $9$. Then, no matter what $g(n), \ n\in \mathbb N$ is, the sum $$\...
3
votes
0answers
191 views

“Rational grids” on manifolds.

Here is something which is bothering me a bit. You have rationals on a line. You can define a rational grid on R^n by taking points with all coordinates having rational values. Is there a ...
3
votes
0answers
199 views

Must be rational number

Let $a$, $b$ positive rational number. Suppose that there exist two odd positive integers $p$, $q$ such that $\sqrt[p]{a}+\sqrt[q]{b}$ is rational. Prove that both $\sqrt[p]{a}$ and $\sqrt[q]{b}$ are ...
2
votes
0answers
37 views

There is, up to isomorphism, a unique smallest field $\mathbb{Q}$, which contains $\mathbb{Z}$ as a subring

I am reading Section 9. The Rational Numbers from textbook Analysis I by Amann/Escher, where there is a theorem: I would like to confirm if my understanding about the proof (which leaves ...
2
votes
1answer
78 views

Define the set $E(\mathbb{Q})$ of $\mathbb{Q}$-rational points on an elliptic curve

I'm a bit struggling with defining the set of $\mathbb{Q}$-rational points on an elliptic curve $E:\;y^2=x^3+ax^2+bx+c$ with $a,b,c\in\mathbb{Q}$. I'm actually guessing that If we let $K$ and $L$ be ...
2
votes
0answers
95 views

Vector field in $\Bbb Q^2$ and $\Bbb R^2$

I'd like to define a weak vector field in $\Bbb R^2$ that is tangential to a family of sine curves at each point. I define the family of sine curves as: $A_n\sin(x)$; $n=1,2,3,...$ and $A_n$ is a ...
2
votes
0answers
42 views

Prove or refute that $\sum_{n=1}^\infty\frac{1}{n^2+A\varphi(n)+B}$ will be irrational for some choice of integers $A,B\geq 1$

We denote the Euler's totient function as $\varphi(n)$ for integers $n\geq 1$, that is a multiplicative function that counts the number of integers $1\leq k\leq n$ up to the given integer n that ...