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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.

1
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1answer
27 views

“The order of a torus link can be understood as a rational number”

The order of a torus link consists of a pair of integers $(m,n)$, with at least one of them nonzero, and it is such that if the two integers are not coprime, i.e. of the form $(km, kn)$, the link ...
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2answers
37 views

Integer solutions of $ X+Y+Z=X\cdot Y\cdot Z $ [closed]

An integer solution of above equation is $(X,Y,Z)=(1,2,3)$. But I am wondering: are there other natural solutions? And what about rational or irrational solutions, where $X,Y,Z$ are different ...
8
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2answers
235 views

Why do we run in diagonals when proving that $\mathbb{Q}$ is countable?

Why do we index the elements like this but not finishing the 1/x elements and then going through 2/x then 3/x...
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0answers
34 views

Construction of Rational Numbers without quotients

The context is Intensional Type Theory, where quotients are unavailable. I managed to construct Integers in this way: $\mathbb{Z}:=(\mathbb{N}^+\times\{{+,-\}})+\{{0\}}$, but I can't see a way to ...
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1answer
66 views

If $x^2$ and $x^3$ are rational, does it imply that $x$ is rational? [closed]

It is given that $x^2$ is rational and $x^3$ is rational. Is $x$ rational for all cases satisfying these conditions or is there are case where $x$ won't be rational? If so, then what other condition(...
1
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1answer
83 views

Questions about 0.999… equals 1 [closed]

Being 0.999... = 1, I expect that they have the same behaviour when applying the same algorithm/operation, but: If we define >, <, =, as checking digit by digit two number, we have that 0 < 1 ...
4
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1answer
59 views

Does $x^3 - \frac{m}{n}\sqrt{5}x - 1$ has rational root?

I am trying to show whether $p(x) = x^3 - \frac{m}{n}\sqrt{5}x - 1$ has a rational root or not, where $\frac{m}{n}$ is rational. My attempt so far is to turn $p(x)$ into another polynomial $q(x) = ( - ...
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1answer
39 views

Proving piecewise function is not continuous [closed]

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = \left\{ \begin{array}{ll} 2x & \quad x \text{ is rational} \\ -2x & \quad x \text{ is irrational }...
9
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1answer
95 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 ...
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2answers
45 views

Is the additive group of integers a rational group?

A group $\mathbb{G}$ is called rational (https://groupprops.subwiki.org/wiki/Rational_group) if $g,g' \in \mathbb{G}, \langle g \rangle = \langle g' \rangle \Rightarrow \exists x \in \mathbb{G}: xgx^{-...
0
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1answer
34 views

Two different rational numbers to the power irrational both rational

I'm trying to generalize the question Is there such an $x$ that both $2^{\frac{x}{3}}$ and $3^{\frac{x}{2}}$ are simultaneously rational?. Do there exist $a,b \in \mathbb{Q}^+ \setminus \{1\}$ and $x ...
1
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1answer
37 views

Density of $\mathbb{Q}$ in $\mathbb{R}$

Let $X$ be a topological space and $Y\subseteq X$ a subset. $Y$ is said to be dense in $X$ if $\overline{Y}=X$ (where $\overline{Y}$ denotes the closure of $Y$). Now consider $X=\mathbb{R}$ (with ...
1
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1answer
45 views

Countability of Sets with rational and real numbers [closed]

Determine whether it is finite, countably infinite, or uncountably infinite. Justify $$\Big\{\Big(\frac{m}{2}, \frac{n}{3}\Big) \in \mathbb{R}^2 \mid m,n \in \mathbb{Z}\Big\}$$ The set is ...
0
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1answer
16 views

Difference between rationals in a certain set is at least a certain amount

Define $A = \{\frac{p}{q} \in \mathbb{Q} \mid q \in \mathbb{N}, q < n, gcd(p,q) = 1\}$. I am trying to prove that the difference of any 2 distinct elements of this set is greater than $\frac{1}{n}$....
1
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1answer
47 views

Definition of the supremum in $\mathbb{Q}$

If $M=\sup \left( A \right)$ and $A\subseteq \mathbb{Q}$ (rational numbers) $\forall\ \varepsilon>0$ of $\mathbb{Q}, \exists\ x\in A$ s.t. $x > M–\varepsilon$. Is this property true for ...
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0answers
48 views

Mapping $\Bbb N\to\Bbb Q$

I want to set up a map from $\Bbb N\to\Bbb Q.$ Take $\Phi_S(x)=e^{(S/\ln(1-x))}$ and $M_T(1-x)=\Phi_S(x); S,T\in\Bbb N.$ Set $\Phi_S(x)=M_T(x)$ to obtain algebraic $x$ coordinates. If $x$ happens ...
0
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1answer
23 views

True or False: If $x\notin \mathbb{Q}$ then $\sum_{m\geq 0} mx^{m-1}\notin \mathbb{Q}$

True or False: If $x\notin \mathbb{Q}$ then $\sum_{m\geq 0} mx^{m-1}\notin \mathbb{Q}, $ where $|x|<1.$ So I considered the contra-positive of the above statement: If $\sum_{m\geq 0} mx^{m-1}\in \...
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0answers
5 views

Confirmation of domain notation, rational expression multiplication with 4 variables

Just wanted to confirm that my notation is ok down the bottom. I've never stated the domain for more than 1 variable, so a bit unsure.
17
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1answer
496 views

Is there such an $x$ that both $2^{\frac{x}{3}}$ and $3^{\frac{x}{2}}$ are simultaneously rational?

I'm having difficulty with the following problem: Is there such an $x$ that both $2^{\frac{x}{3}}$ and $3^{\frac{x}{2}}$ are simultaneously rational, for $0<x<1$? I've tried proving by ...
1
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1answer
106 views

Finding a rational root on this particular two variables polynomial

Some context. While working on a larger proof, I needed to show that a particular homogeneous system of polynomial equations had no rational solution except for the trivial one. I have reduced this ...
0
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1answer
62 views

Does the Pythagorean formula $a^2+b^2=c^2$ hold in the plane $\mathbb{Q} \times \mathbb{Q}$? [closed]

Does the Pythagorean formula $a^2+b^2=c^2$ hold in the plane $\mathbb{Q} \times \mathbb{Q}$ ? For example, The triangle with vertices $(0,0), \ (1,0), \ (0,1) \in \mathbb{Q} \times \mathbb{Q}$ and ...
6
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4answers
650 views

Rational with finite decimals values for sine, cosine, and tangent

What are the possible combinations of sine, cosine, and tangent values such that all three are simultaneously rational with finite decimals? I am aware of the below two cases. $\sin(x) = 0, \cos(x) =...
0
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2answers
22 views

Conditions for simplified rational expressions

$$\frac{x^2+6x+5}{x^2-x-2}$$ $$\frac{(x+5)(x+1)}{(x-2)(x+1)}$$ $$\frac{x+5}{x-2}$, $ x \ne -1$$ My question is when it comes to specifying that $ x \ne -1$, the end result is also undefined where $...
0
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1answer
60 views

The proof that $\sqrt{q}$ is a rational number iff $q$ is a perfect square

I have a proof of that if $q\in \mathbb{Q}$ then $ \sqrt{q}$ is rational if and only if $q$ is a perfect square (it can be written in the form $q={p_1}^{a_1}...{p_n}^{a_n}$ where integers $a_j$, which ...
0
votes
1answer
21 views

How to determine the kernel $\ker \varepsilon_{\sqrt 2}$ and the image $\varepsilon_{\sqrt 2}(\mathbb Q[X])?$

Consider the $\mathbb Q$-algebra homomorphism $\varepsilon_{\sqrt 2}:\mathbb Q[X]\rightarrow \mathbb C$ defined by $\varepsilon(X)=\sqrt 2$. How to determine the kernel $\ker \varepsilon_{\sqrt 2}$ ...
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2answers
27 views

Rational Numbers Proof

Apologies for the vagueness before, I'm new here. I hope this clears it up: Show that, for all non zero $b\in \Bbb Z$, $${(0,b)}=((a',b')\in F:a'=0)$$ $$F=((a,b)\in \Bbb Z*\Bbb Z: b\ne 0))$$ where F ...
0
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1answer
107 views

Why doesn't the construction of $\mathbb{N}$ through ordinals in ZFC violate Gödel's Incompleteness Theorem?

The title kind of says it all. I've been working through Axiomatic Set Theory, Suppes and Mathematical Logic, Kleene. And I haven't thoroughly studied ordinals and incompleteness yet. But, ...
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1answer
46 views

Writing Short Equations/Equivalents For A Group Of Numbers.

I have a series of numbers between 0 to 16,000,000. It's certainly possible to describe some (if not all) of these numbers with some equations. For example, it's possible to write 720 as 6! and 5040 ...
0
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0answers
33 views

What does $\omega/\mathbb{Q}$ mean?

I have been reading a paper titled Applied Koopmanism several times but cannot find what does this mean (given on page 21): If $\lambda = e^{i2\pi \omega}$ is such that $\omega/\mathbb{Q}$ I am ...
2
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2answers
73 views

Is $ (3+\sqrt{2})^{2/3} $ an irrational number?

I am supposed to find out whether $ (3+\sqrt{2})^{2/3} $ is an irrational number and prove it, but I have no idea how to begin. Thanks
0
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1answer
32 views

Finding the rationalizing factor of rational numbers with denominator 1

I have a question which I could not solve after hours of research. It goes like this: Find the rationalizing factor of $$\sqrt[3]{16} - \sqrt[3]{4} + 1$$ I can rationalize the denominator but can’...
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0answers
24 views

A set containing the mean of every nonempty finite subset contains all rationals.

Let $S \subset \mathbb{R}$ be the smallest set satisfying $$\text{(i) } 0 \in S, 1 \in S \\ \text{(ii) } S \text{ contains the mean of every nonempty finite subset of } S$$ Prove that $S = [0, 1] \cap ...
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3answers
42 views

Finding rational numbers in an equation with two variables

How should we find two rational numbers $\alpha$, $\beta$ such that $\sqrt[3]{7+5\sqrt{2}}=\alpha+\beta\sqrt{2}$? The answer I got alpha = 1 and betta = 1. If I'm wrong, please correct me. Thank you
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3answers
23 views

Some numbers represented by symbols

I am trying to find some numbers that are represented by symbols, such as π, e, i, φ. I couldn't find more. Can you guys help me? (English is not my main language and it is for school project.)
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2answers
57 views

Rational as series?

I was checking out a few things in the geometric series and realized all rational numbers can be shown as a geometric series.I was pretty sure I read something like that somewhere.Can anyone tell me ...
1
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1answer
39 views

Well-Ordering Irrationality

Let $D$ be a positive integer and the let the square root of $D$ be a real number. Assuming that the square root of D is not an integer (i.e. $D$ is not a perfect square), use Well-Ordering to prove ...
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2answers
42 views

Sum of two irrational numbers being rational or irrational

I am currently doing a project on irrational and transcendental numbers and part of this project requires me to look at sums and products of irrational numbers. I am aware that the sum of 2 ...
0
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2answers
63 views

For all integers $w, x, y, z$ with $w\neq{y}$ and $wz-xy\neq0$, prove that there exists a unique rational number $r$ such that $(wr+x)\div(yr+z)=1$

For all integers $w, x, y, z$ with $w\neq{y}$ and $wz-xy\neq0$, prove that there exists a unique rational number $r$ such that $(wr+x)\div(yr+z)=1$ How do I prove uniqueness? I know to show that ...
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1answer
26 views

Symbol for dyadic rationals

Set of integers is denoted by the symbol $\mathbb Z$, $\mathbb Q[x]$ stands for univariate polynomials over rationals, etc. Is there a symbol which indicates the set of dyadic rationals?
0
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1answer
51 views

Abstract Algebra Square Roots Are Irrational

For part (a), I begin by trying to prove $S$ is empty implies the square root of $D$ is irrational. If we take the contrapositive of this implication, this is equivalent to proving that if the square ...
0
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0answers
64 views

Sums and Products of Algebraic and Transcendental numbers

I am doing a project about irrational and transcendental numbers and part of this project involves looking at sums and products of various combinations of rational, irrational, algebraic and ...
0
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1answer
19 views

simple question : sequential criterion for continuity

Let $A$ be a subset of $\mathbb{R}$, and let $f:A\to\mathbb{R}$ be a function. Now, let $\{r_{n}\}$ be any rational sequence in $A$, and let $\{s_{n}\}$ be any irrational sequence in $A$. Suppose ...
0
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2answers
55 views

A well-defined map from rational numbers to integers

I am trying to come up with a well-defined map from $\mathbb{Q}$$\to$ $\mathbb{Z}$ i.e. find a map $f$ such that it maps $\frac{a}{b}$ $\epsilon$ $\mathbb{Q}$ to an integer in $\mathbb{Z}$. I tried a ...
3
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3answers
81 views

If x,y and z are positive integers and $\frac 1x + \frac 1y = \frac 1z$ then $\sqrt{x^2+y^2+z^2}$ is rational.

To solve this problem I first started off by factoring to get $z^2=(x-z)(y-z)$ only to realise that this does nothing so I then tried squaring both sides to get the reciprocals of $x,y$ and $z$ ...
2
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2answers
36 views

bijection between $\mathbb{Q}$ and $\mathbb{N}$ that preserve the order.

I know there is a bijection between $\mathbb{Q}$ and $\mathbb{N}$. But is there a bijection $\mathbb{Q}\xrightarrow{f}\mathbb{N}$ that preserves the order? Intuitively I think this is not possible. ...
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2answers
88 views

Contrapositive proof: If $H$ and $K$ are nontrivial subgroups of $\Bbb Q$, then $H\cap K$ is also nontrivial.

I'm reading "Contemporary Abstract Algebra," by Gallian. This is (part of) Exercise 26 of the supplementary exercises for chapters 1-4 ibid., although I am requesting a proof of the contrapositive ...
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1answer
54 views

Why do we consider $\pi$ as a irrational number?

Why do we consider $\pi$ as a irrational number? Why is that? We all know that $\pi$ is the solution of circumference / diameter of a circle and there could be infinite amount of circles which can ...
1
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1answer
32 views

How can I prove that no limit exists for this function over this particular interval?

I was given the following function: $$ f(x) = \begin{cases} \frac{1}{q} &\text{if }x = \frac{p}{q} \text{ is rational, reduced to lowest terms} \\ 0 &\text{if }x \text{ is irrational}\end{...
0
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0answers
53 views

Alternative solution to a problem involving an enumeration of rationals

I'm working on the same problem as in this post. I understand the solutions provided in the answers. The question basically asks us to find an enumeration of the rationals $\{r_n\}_{n≥1}$ such that ...
-1
votes
3answers
62 views

Can the product of two rational numbers be an irrational number? (Kindly see the example in description)

I checked in many sources and I saw "Multiplication is closed under Rational Numbers Q". But consider $$ a = \frac{1}{7} ; \;\;\; b = \frac{22}{1} ;$$ both a, b are individually rational (either ...