# 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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'...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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) ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\mathbb R$...
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### 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 ...
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### 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?
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### “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 ...
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### 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 ...
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### 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 ...
### 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 ...
### 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 ...