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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4
votes
1answer
71 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) = ( - ...
-2
votes
1answer
173 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
votes
1answer
111 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 ...
5
votes
2answers
83 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
votes
1answer
51 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
vote
1answer
52 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
vote
1answer
86 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
votes
1answer
20 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
vote
1answer
76 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 ...
0
votes
0answers
55 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
30 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 \...
0
votes
0answers
10 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
525 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
vote
1answer
147 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 ...
-1
votes
1answer
64 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
votes
4answers
682 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
votes
2answers
30 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
votes
1answer
67 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
50 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 $\ker \varepsilon_{\sqrt 2}$ ...
-3
votes
2answers
29 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
votes
1answer
121 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, ...
-1
votes
1answer
51 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 ...
2
votes
2answers
79 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
votes
1answer
303 views

Finding the rationalizing factor of real 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’...
0
votes
0answers
25 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 ...
1
vote
3answers
50 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
0
votes
3answers
28 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.)
1
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2answers
62 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
vote
1answer
65 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 ...
0
votes
2answers
60 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
votes
2answers
155 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 ...
1
vote
1answer
44 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?
1
vote
1answer
61 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
votes
0answers
86 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
votes
1answer
65 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
votes
2answers
232 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
votes
3answers
88 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
votes
2answers
73 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. ...
-1
votes
2answers
110 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 ...
-2
votes
1answer
61 views

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

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
vote
1answer
40 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
votes
0answers
62 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
76 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 ...
2
votes
1answer
94 views

How to show that $\mathbb{Q}$ and $\mathbb{Q}^2$ are elementarily equivalent?

I try to prove that $(\mathbb{Q},P)$ and $(\mathbb{Q}^2,P)$ are elementarily equivalent using Ehrenfeucht–Fraïssé games $P(a,b,c)=True$ when $a+b=c$ $+$ on $\mathbb{Q}^2$ is defined as $(x,x')+(y,y'...
0
votes
2answers
276 views

Prove or disprove that there is a rational number $x$ and an irrational number $y$ such that $x^y$ is irrational.

In solving the following problem: Prove or disprove that there is a rational number $x$ and an irrational number $y$ such that $x^y$ is irrational. I let $x=2$ and $y = \sqrt 2$, so that $x^y = 2^\...
0
votes
4answers
43 views

Cardinality of set of sequences from $\mathbb{Q}$ that converge to $0$

I'm looking for cardinality of a set of sequences from $\mathbb{Q}$ which are convergent to $0$. I think the answer is continuum, but I don't know how to prove it.
0
votes
0answers
55 views

Can nth root of 2 be a rational number for any natural number n > 1?

We know square and cube root of 2 are irrational numbers, but is it possible for any a rational number to be multiplied by itself finite times and it results in 2.
-2
votes
1answer
39 views

How can an convergent series of rational numbers result in a irrational number?

In mathematics, nearly all significant irrational numbers can be expressed as a sum of an infinite convergent series, but according to law of addition of rational numbers, adding any to rational ...
2
votes
1answer
236 views

Sixth grade math (number related) problem

We have this statement (about rational numbers, btw): If $m-n+p = p$ and $ m \neq n \neq 0$ then $ m = -n$ Is this true? a) always b) never c) sometimes The given answer is b) but: ...
0
votes
2answers
41 views

Roots of polynomial irreducible over the rationals

If a polynomial is "irreducible over the rationals", does it mean that it has no rational roots? I would say yes because otherwise I could divide out the linear factors (i.e. rational roots) but ...