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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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Discretized Distributions on Rationals?

Consider the measure space $(\mathbb{Q}, 2^{\mathbb{Q}}, \nu)$, with $\nu$ being the counting measure. The space is then $\sigma$-finite. Is there any attempts made to define analogues of continuous ...
温泽海's user avatar
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1 answer
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Closed form for the area under $ f(x):=\lim_{N \to \infty}\frac{\pi(Nx)}{\pi(N)} $

Define a function $f:\Bbb Q \to \Bbb Q$ by the following $$ f(x):=\lim_{N \to \infty}\frac{\pi(Nx)}{\pi(N)} $$ where $\pi(\cdot)$ is the prime counting function and $N\in \Bbb N.$ I would like to find ...
zeta space's user avatar
13 votes
6 answers
2k views

Constructing the interval [0, 1) via inverse powers of 2

If $x$ is rational and in the interval ${[0,1)}$, is it always possible to find constants $a_1, a_2, ..., a_n\in\{-1, 0, 1\}$ such that for some integer $n\geq{1}$, $x = a_1\cdot2^{-1} + a_2\cdot{2^{-...
Garrett's user avatar
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2 votes
4 answers
152 views

Prove $\sum_{i=1}^{30}\frac{1}{10+i} >\frac{13}{12}$

Prove $$\sum_{i=1}^{30}\frac{1}{10+i} >\frac{13}{12}$$ My effort: $$\begin{aligned} & \frac{1}{11}>\frac{1}{42} \\ & \frac{1}{12}>\frac{1}{42} \\ & \frac{1}{13}>\frac{1}{42} \\...
LifeIsMath's user avatar
0 votes
3 answers
73 views

Is $(\mathbb{Q},0,+,<)$ isomorphic to $(\mathbb{Q}^{+},1,\cdot,<)$? ("Logic in Mathematics and Set Theory" by Kazuyuki Tanaka and Toshio Suzuki)

I am reading "Logic in Mathematics and Set Theory" by Kazuyuki Tanaka and Toshio Suzuki. Problem 1.17 Is $(\mathbb{Q},0,+,<)$ isomorphic to $(\mathbb{Q}^{+},1,\cdot,<)$? My attempt: ...
佐武五郎's user avatar
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1 vote
1 answer
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Proof that Dedekind Cuts are isomorphic to decimal expansions?

The real numbers $\mathbb{R}$ are defined as the superset of the $\mathbb{Q}$ with additional elements to yield Dedekind Completeness. Specifically, every subset with an upper bound in $\mathbb{R}$ ...
SarcasticSully's user avatar
0 votes
1 answer
104 views

Rational point on a pythagorean rectangle

Consider the rectangle shown in the above diagram with vertices $(0,0)$, $(a,0)$, $(0,b)$ and $(a,b)$. The sides and diagonals of this rectangle are integers i.e., $a, b, c$ are integers such that $a^...
Shiva Kintali's user avatar
1 vote
0 answers
54 views

Injective monotonic mapping from rationals $\mathbb Q^2$ to $\mathbb R$

Exercise: $f: \mathbb Q^2\to\mathbb R$. Where $\mathbb Q$ is the set of rational numbers. $f$ is strictly increasing in both arguments. Can $f$ be one-to-one? This question is related to many ...
High GPA's user avatar
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1 vote
2 answers
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Rationals are incomplete and naturals are complete

Why are rationals incomplete and natural complete? I have been going through Analysis textbooks but I don't totally get the reason why. So, naturals are complete because you can divide them into two ...
pdaranda661's user avatar
2 votes
1 answer
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Are there more than two rational solutions to a certain system $abcd=a+b+c+d=K$ ($K$ a given constant)?

This question is a follow-on of a previous question asked some days ago which has been deleted due to its lack of precision. In fact, I found it well explained here But, implicitly, the domain of ...
Jean Marie's user avatar
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2 votes
1 answer
124 views

Rational solutions to Rational Equation

I am looking for rational solutions to $$x\frac{x^2-1}{(x^2+1)^2}+y\frac{y^2-1}{(y^2+1)^2}=2z\frac{z^2-1}{(z^2+1)^2}$$ besides $(x, x, x)$, $(-x, 1/x, 0)$, and other similar non interesting solutions. ...
Xander's user avatar
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12 votes
1 answer
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Is there a $f: \mathbb{N} \to \mathbb{N}$ such that $\sum_{n=1}^{\infty} \frac{1}{n^2f(n)} \in \mathbb{Q}$?

Take by convention $0 \not \in \mathbb{N}$, and let $f: \mathbb{N} \to \mathbb{N}$. Define the real number $N(f)$ by $$N(f) = \sum_{n=1}^{\infty} \frac{1}{n^2f(n)}.$$ $N(f)$ is well-defined because, ...
Robin's user avatar
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1 answer
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Prove that the number $0.a_1a_2a_3\ldots$ is a rational number.

Let $a_1$ be any number from the set {$0, 1, 2, \ldots, 9$}. For each $n \in \mathbb{N}$, denote by $a_{n+1}$ the last digit of the number $19a_n + 98$ in decimal notation. Prove that the number $0....
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1 answer
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are there 2 or more irrational numbers between any 2 rationals?

… in general, but also related to a calculus problem I have before me which is about continuity. The question regards continuity wrt the function $$ f(x) = \begin{cases} x, x \in \mathbb{Q} \\ 0, ...
El Jfe's user avatar
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3 votes
1 answer
88 views

Simplifying Gauss’s Lemma

Hello Math StackExchange Community, I am revisiting Gauss's Lemma in my lecture notes and considering a simplification in its proof. I am proposing to remove the necessity of proving that $\lambda$ is ...
Martin Geller's user avatar
2 votes
1 answer
36 views

Find $m\in Q$ and $n\in Z$, for which the equality $m(19m+10)=143^n-2$ occurs.

The problem Find $m\in Q$ and $n\in Z$, for which the equality $m(19m+10)=143^n-2$ occurs. my idea All I could think off is that we can note $m=\frac{a}{b}, (a,b)=1$ and get that $m(19m+10)=\frac{a^2*...
IONELA BUCIU's user avatar
0 votes
2 answers
58 views

Proving, using Dedekind cuts, that $C(0)$ is an additive identity for addition on cuts.

My source is Franck Ayres' Modern Algebra. The author states the fact under discussion (Chapter 7, "Real numbers") but does not provide a proof. My question is about the second part of the ...
Vince Vickler's user avatar
1 vote
1 answer
45 views

Inquiries Regarding the Submodule Structure of $ _{\mathbb{Z}}\mathbb{Q} $

As is well known, any cyclic submodules of $ _{\mathbb{Z}}\mathbb{Q} $ are superfluous. Here is a proof. This raises the following questions: Are all proper submodules of $ _{\mathbb{Z}}\mathbb{Q} $ ...
Liang Chen's user avatar
0 votes
1 answer
61 views

Are over half of the positive rational numbers in the interval [0, 1]?

There is a problem in my calculus textbook that wants me to draw/look at the following function $f:\Bbb{R}\to\Bbb{R}$ $$f(x)=\begin{cases} 1 & : x \in \Bbb{Q} \\ 0 & : x \in \Bbb{P} \\ \end{...
volticus's user avatar
0 votes
0 answers
83 views

Proof that a base 2 logarithm of a rational number is irrational

How can I prove that if $a = \log_2 b, b \in \Bbb Q, b \neq 2^c$ and $c \in \Bbb Z$ then $a \notin \Bbb Q$ ? And could the proof be easily adapted to differently-based logarithms? I am familiar with ...
fedsavi's user avatar
  • 35
0 votes
0 answers
17 views

Define this step function over the rational numbers

In desmos I plotted a step function (I only plotted 30% of it). Here is my graph: This function is a function from $\Bbb Q\cap (0,1) \to \Bbb Q\cap(0,1).$ The step function is generated by counting ...
zeta space's user avatar
2 votes
5 answers
203 views

approximate square roots of fractions with rationals

How to compute the rational approximation of square root of a fraction, i.e. I'd like to find $ \frac{a}{b} \approx \sqrt{\frac{m}{n}}$, where $a$, $b$, $m$ and $n$ are integers. Ideally, the ...
chaohuang's user avatar
  • 6,399
6 votes
1 answer
297 views

How to construct a nonzero real number between two given nonzero real numbers?

Statement: Let $$X=$$ $$\{(a,b) \in \mathbb{R} \setminus \{0\} \times \mathbb{R}\setminus \{0\}:a<b\}$$ There exists a function $f:X \rightarrow \mathbb{R} \setminus \{0\}$ such that for all $(a,b) ...
Mohammad tahmasbi zade's user avatar
14 votes
5 answers
397 views

Is there a dense subset of the rationals (between 0 and 1) that doesn't include its midpoints?

Is there a subset $S$ of the rationals between 0 and 1 $S \subset \mathbb{Q} \cap[0,1] \stackrel{\text{def}}= [0,1]_\mathbb{Q}$ such that is dense in $[0,1]_\mathbb{Q}$, in the sense that $\forall ...
Nicola Sap's user avatar
0 votes
1 answer
50 views

When is this rational function of exponentials actually rational-valued?

This has come up in my research, and I am sorry if it is obvious. I am looking at the following expression $$ m\frac{\tanh(xm)}{\tanh(x)} = m\frac{e^{2xm}-1}{e^{2xm}+1} \frac{e^{2x}+1}{e^{2x}-1}, $$ ...
Croc2Alpha's user avatar
  • 3,847
1 vote
0 answers
51 views

Show that a given procedure generates all rationals greater than 1

Question without Context Let be given a sequence of natural numbers $(A_1,A_2,...)$, such that $A_i\ge A_{i-1}+1$, and consider the set generated by dividing $(1,...,A_1)$ by $1$, $(A_1,...,A_2)$ by $...
Stamatis's user avatar
  • 490
2 votes
1 answer
39 views

Any manifold with fundamental group isomorphic to $\mathbb{Q}$ is orientable

I have enjoyed the construction of the rational circle found here, but I still have no clue on how to show that such a space, or any other with fundamental group $\mathbb{Q}$, must be orientable. In ...
danimalabares's user avatar
4 votes
2 answers
252 views

Prove that if $\forall n\in\Bbb N\quad a_{n+1}^2-a_{n+1}=a_n\in\Bbb Q$, then the sequence is constant

This question was posted, downvoted and closed today (2022 Thailand Olympiad problem) and 8 days ago ($f(x+1)^{2} - f(x+1) = f(x)$. What values of $f(1)$ allow $f(x)$ to be always rational if $x$ is ...
Anne Bauval's user avatar
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2 votes
0 answers
49 views

Prove that $(f+g)(x)<t$ ($t\in\mathbb{R}$) holds if and only if there is a rational number $r$ such that $f(x)<r$ and $g(x)<t-r$.

I got stuck on this question: Prove that $(f+g)(x)<t$, $t\in\mathbb{R}$, holds if and only if there is a rational number $r$ such that $f(x)<r$ and $g(x)<t-r$. I think one direction is ...
Beerus's user avatar
  • 2,493
0 votes
1 answer
42 views

Is this proof of the density of the rationals in the reals correct?

Theorem: For any 2 non-equal positive real numbers, there exists a rational number "between" them. $ x , y \in \mathbb{R} $, $ 0 \lt x \lt y \implies \exists q \in \mathbb{Q} | x \lt q \lt ...
Tim's user avatar
  • 53
4 votes
1 answer
55 views

Using Eisenstein's Criterion with a transformation

I'm trying to prove that the following polynomial is irreducible in $\mathbb{Q}$ :$$14x^{10} + 18x^9 + 4x^3 + 1$$ Obviously, we can't apply Eisenstein's Criterion here so I tried setting $y = \frac{1}{...
user avatar
1 vote
0 answers
37 views

Calculate the simplest dyadic rational in half-open interval

Given a half-open interval $[a, b)$ can one calculate, without brute force or looping of any kind, the simplest dyadic rational within said range? Context: I'm reading about arithmetic coding which ...
tsujp's user avatar
  • 111
-3 votes
2 answers
83 views

What is the exact mathematical definition of a fraction? Can all real numbers be expressed as fractions?

As per some sources fractions have been defined as a quotient of two numbers whereas some sources restrict the numerator to be a whole number and denominator to be a positive integer. What is the ...
Anvi Mahajan's user avatar
0 votes
0 answers
39 views

Approximating a rational number in a subset of Q defined by limited prime factors

I'm wondering if there is an efficient (or good enough for small numbers) algorithm for the following problem: Suppose I have a rational number in the form of its prime factorization: $k = p_0^{x_0}...
retooth's user avatar
3 votes
1 answer
68 views

Closed expression of a specific splitting of a continued fraction

Recently in my research I stumbled upon this splitting of a periodic continued fraction. I wondered whether there is any closed expression or literature on this topic. Visualizing continued fractions ...
Bindajoba's user avatar
1 vote
0 answers
40 views

Find all field homomorphisms $f:\mathbb{Q}(\sqrt{2}) \rightarrow \mathbb{Q}(\sqrt{2})$ [duplicate]

Find all field homomorphisms $f:\mathbb{Q}(\sqrt{2}) \rightarrow \mathbb{Q}(\sqrt{2})$. We know that: $ \mathbb{Q}(\sqrt{2}) = \text{span}_\mathbb{Q} \{ 1, \sqrt{2} \} = \{ a+\sqrt{2}b \,|\, a,b\in \...
haifisch123's user avatar
0 votes
2 answers
80 views

Sets as numbers through Dedekind's cuts

I am having some issues understanding the definition of real numbers as a Dedekind Cut. Firstly, a Dedekind cut is a set, not a number (namely, a set in which all its values are less than the value of ...
pdaranda661's user avatar
0 votes
1 answer
29 views

Show that $\mu$ is the Lebesgue measure on $\mathbb{R}$, $\mu$ being invariant by translation and $\mu(]0,1])=1$

$\mu$ is a measure on $(\mathbb{R}, B(\mathbb{R}))$ which is invariant by translation ($\forall a,b,h\in\mathbb{R}:\,\mu(]a+h, b+h]) = \mu]a, b]$) and such that $\mu(]0,1])=1$. What I need to show is ...
Alex's user avatar
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0 votes
0 answers
24 views

Mordell-Weil theorem for rational points on $y^2=x^2+1$

The set $R$ of rational points on the curve $y^2=x^2+1$ can be viewed as a group under the following operation: if $X_1=(x_1,y_1),\ X_2=(x_2,y_2)\in R$, define $X_1*X_2:=(x_1y_2+x_2y_1,x_1x_2+y_1y_2)$....
aleph0's user avatar
  • 103
0 votes
0 answers
35 views

Proof verification: The relation $<_{\mathbb{Q}}$ is well-defined on $\mathbb{Q} := (\mathbb{Z}\times(\mathbb{Z}-\{0\})) / \asymp$.

today I tried to prove that $<_{\mathbb{Q}}$ is well-defined on $\mathbb{Q} := (\mathbb{Z}\times(\mathbb{Z}-\{0\})) / \asymp$. I'd like to know if the logical structure of the proof is correct ...
J. J. Gonzalez's user avatar
4 votes
1 answer
231 views

The simplest curve which is never straight and has a rational arc length.

This tweet claims to give an explanation for why one should expect the perimeter of a circle with a rational radius to be irrational. It doesn't strike me as that convincing (although feel free to ...
Davis Yoshida's user avatar
0 votes
0 answers
93 views

As I understand it, it is possible to define division by zero in a consistent way. The problem is that such a definition is worthless.

I was reading this answer to the question of the division by zero which proposes the idea of "warped" numbers. So, am I correct, the following rules could be a definition of division by zero:...
Larry Freeman's user avatar
1 vote
1 answer
39 views

Is the following proof of equality complete?

Say we have only defined the rational numbers. If we have two rational numbers, s and r, and we can show that $|s-r|<\epsilon$, for all positive rational $\epsilon$, is that enough to show that $s =...
Vector's user avatar
  • 377
0 votes
0 answers
16 views

Nontrivial rational solutions for an anticommutative functional equation

I'm trying to find possible solutions for the following equation: $${ f\!\left(\matrix{x_1&x_2\\y_1+z_1&y_2+y_1z_1+z_2}\right) = f\!\left(\matrix{x_1&x_2\\y_1&y_2}\right) + f\!\left(\...
Aberone's user avatar
  • 212
0 votes
2 answers
80 views

Does the set ${\displaystyle A =\{q\in \mathbb {Q} |q<a\}} $ have a maximum element?

Does the set A have a maximum element? ${\displaystyle A =\{q\in \mathbb {Q} |q<a\}} $ Thoughts: I don't think it does. We can suppose that it has a maximum element, q. Then q is a real number and ...
Hjm's user avatar
  • 101
0 votes
1 answer
144 views

Show some proofs in the set $M=\{a+x+\sqrt{a-x+x^2}|x\in N\}$

The question Let $a \in (0, \infty)$ and the set $$M=\{a+x+\sqrt{a-x+x^2}|x\in N*\}$$ Show that: a)if $a=1$, then the set $M$ contains exactly $2$ rational numbers b) if the set $M$ contains at least $...
IONELA BUCIU's user avatar
-1 votes
2 answers
50 views

My question is whether given function is periodic or not [closed]

F(x) = 1, if x is rational 0, if x is irrational So what i am really confused about is because all professors are telling me its periodic whose period is not defined, i mean it doesn't make any ...
MultiUniverseExplorer's user avatar
8 votes
3 answers
219 views

Determine all the integers $x$ that have the property that $\sqrt{x^2+7x+21}$ is a rational number.

The Question Determine all the integers $x$ that have the property that $\sqrt{x^2+7x+21}$ is a rational number. The Idea The number would be rational only if $x^2+7x+21$ would be a square number ...
IONELA BUCIU's user avatar
0 votes
0 answers
52 views

Show the values of $x$ such that $g$ is continuous.

Let $g(x)$ be: \begin{cases} 0 \space\space \text{if}\space\space x \in \mathbb Q \\ x\space\space \text{if} \space\space x \in \mathbb R - \mathbb Q \\ \end{cases} That's what I've ...
LightL96's user avatar
3 votes
1 answer
62 views

Prove that there are infinitely many distinct natural numbers $a,b$ for which $\sqrt{a+b}, \sqrt{a-b}$ are simultaneously rational

the question Prove that there are infinitely many distinct natural numbers $a,b$ for which $\sqrt{a+b}, \sqrt{a-b}$ are simultaneously rational. the idea A radical is rational only if the number below ...
IONELA BUCIU's user avatar

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