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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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1answer
39 views

Alternation of Rationals and Irrationals?

I'm in a lunch group at work of recreational math geeks and we came up with a question which we need help to resolve. I apologize in advance, if my explanation is not perfectly rigorous. Given these ...
0
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0answers
33 views

What numbers can be approximated by ratios of numbers containing only specified digits?

Let $D$ be a subset of the decimal digits $ \{ 0, 1, 2, \ldots, 9\}$, with $D \neq \{0\}$ or $\emptyset$. Let $N$ be the set of positive integers whose decimal representations (without leading $0$'s) ...
1
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2answers
88 views

Are Rationals constructed from infinite Naturals valid? [closed]

I can construct a Rational like $3/4$. And then I can construct anther one like $31/41$, and then another like $311/411$. I can envisage a Rational whose numerator is $31111111...$ and denominator ...
0
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1answer
16 views

Numerator and denominator of rational number

I have to prove that $\sqrt5$ is irrational. I prove it by contradiction. I assume that $p$ is an integer and $q$ is a positive integer such that $gcd(p,q)=1$ and $(\frac{p}{q})^2 = 5$. And then ...
2
votes
2answers
26 views

Prove that $a$ and $b$ are rational numbers [closed]

If $a+b$, $a^2+b$ and $b^2+a$ are rational numbers and $a+b\neq 1$ then $a$ and $b$ are rational. I try, sum the expresions but I only got that $a^2+b^2$ and $ab$ are rational. Any suggestion?
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0answers
23 views

Can we fill the plane with a certain operation?

Background: Paint the origin $(0,0)$ black in $\mathbb{R}^2$. Let $S$ be a set $\{ (x,y) \in \mathbb{R}^2 ~|~ x^2 + y^2 =1 \}$. Paint $S$ black. Paint $(u,v) +S$ black for all $(u,v) \in S$. (...
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2answers
36 views

Recurring Decimal in base 3

I have this decimal $y=0.012012012012....$ I was wondering how I could put this into the form of a rational number in base 3. So far I have $y= \frac{0}{3}+\frac{1}{3^2}+\frac{2}{3^3}+\frac{0}{3^4}......
4
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3answers
73 views

Famous fractions: Can any “special” numbers be approximated by simple ratios like $3.14\ldots$ as $22/7$?

The ratio $22/7$ dates back to antiquity as an approximation of $3.14\ldots$. I'm wondering whether there are any other "famous" numbers with a similar situation. That is, something like $e$ or $\phi$ ...
1
vote
1answer
52 views

Are there $2^{\aleph_{0} }$ sets of natural numbers such that each two have finite intersection [duplicate]

Question: Are there $2^{\aleph_{0} }$ sets of natural numbers such that each two have finite intersection. From what I've read about infinite families, I need to ignore those who have the properpty $...
6
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2answers
72 views

Prob. 1, Sec. 29, in Munkres' TOPOLOGY, 2nd ed: The subspace $\mathbb{Q}$ of $\mathbb{R}$ is not locally compact

Here is Prob. 1, Sec. 29, in the book Topology by James R. Munkres, 2nd edition: Show that the rationals $\mathbb{Q}$ are not locally compact. My Attempt: Here the topology on the set $\mathbb{...
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3answers
148 views

What is $\mathbb{Q}$?

When we say set of rationals $\mathbb{Q}$, which of the following does it refer to? $$\left\{\frac{p}{q}~|~p,q\in\mathbb{Z},q\neq 0\right\}$$ or $$\left\{\left[\frac{p}{q}\right]~|~p,q\in\mathbb{Z},...
2
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3answers
107 views

Suppose that $x$ and $y$ are irrational, but $x + y$ is rational. Prove that $x -y$ is irrational.

i was wondering if someone could check my proof $Q= \{a/b , c,d : a,c ∈ \mathbb Z , b,d ∈ N>0\}$ $a/b =x+y$ $a/b -y=x$ proof by contradiction. Let $x-y$ is rational $c/d = x-y$ sub $a/b -y = ...
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2answers
33 views

Proving the negation of a conditional using proof by contradiction

CONTEXT: Question made up by uni maths lecturer Prove the following statement using a proof by contradiction: For all nonzero rational numbers $x$, if $y$ is irrational then $\frac{x}{y}-3$ is ...
2
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0answers
62 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 ...
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2answers
59 views

Dedekind cut/additional property

Consider the following lemma and its proof. My question follows. Let $(P,<)$ be a dense unbounded linearly ordered set. Then there is a complete unbounded linearly ordered set $(C,\prec)$ such ...
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0answers
70 views

Does $x^x$ exists for certain rational numbers at $x<0$?

There have been debates in mathematics that $x^x$ cannot exist for any rational numbers for $x<0$ since $${x}^{x}=e^{x\ln(x)} \ \text{for all}\ x$$ and for $x=-1/1=-2/2$ $$(-1)^{-1/1}\neq(-1)^{-...
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2answers
36 views

Analytic functions on $\mathbb{Q}$

$\mathbb{Q}$ has the topology induced from $\mathbb{R}$, therefore it is in principle possible to talk about power series and define analytic functions on $\mathbb{Q}$ to be power series (with ...
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1answer
74 views

Does there exist a bijection $f$ from $\mathbb{N}$ to $\mathbb{Q}^+$ such that $\lim_{n \to \infty} \frac{f(n+1)}{f(n)}$ exists?

Does there exist a bijection $f$ from $\mathbb{N}$ to $\mathbb{Q}^+$ such that $$\lim_{n \to \infty} \frac{f(n+1)}{f(n)}$$ exists? My guess that no such $f$ exists.
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2answers
33 views

Numbers between decimal number

So there is a question that says I can choose a number between $0.00001$ and $0.1$, but the problem is what exactly comes after $0.00001$? Would it be $0.0001$, continued by $0.001$, $0.01$, $0.1$? ...
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2answers
29 views

Rainbow numbers: Can mapping digits to different bases produce different varieties of irrationality?

This is a follow-up to the question, "Irrationality of 0.123456789101112 … and similar numbers." There I took some decimal number, in one case Champernowne's constant, $$ n_{10} = 0....
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2answers
34 views

If $n \in \mathbb{Z}^+$ and $x \in \mathbb{R}$, show that $x \in \mathbb{Q}$ if $n x \in \mathbb{Q}$

For this question would I be correct in stating the following If $n$ is a positive integer and $n x$ is a rational number then $n x = n (a / b)$. Simplifying this = $x = (a / b)$ so $x$ must also ...
2
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1answer
56 views

De-rationalisation of a surd expression $\sqrt p - \sqrt {pq} + q$

Consider two dissimilar surds $\sqrt p$ and $\sqrt q$. Then the problem asks to find rational numbers $a,b,c$ and $d$ such that for $x=\sqrt p + \sqrt q$ we can write, $$ \sqrt p - \sqrt {pq} + q = \...
3
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3answers
40 views

Cauchy sequences of rationals with limit irrational: natural, or geometric examples

As we know, real numbers are constructed by filling up gaps between rationals by the limits of all Cauchy sequences of rationals. Q. What are examples of sequence of rationals such that its easy ...
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0answers
16 views

How to express that for $\beta \in (0,1)$, $\bigg(\frac{|x_i|}{i} \bigg)^2 \leq \beta^2 |x_i|^2$?

How to express that for $\beta \in (0,1)$, $\bigg(\frac{|x_i|}{i} \bigg)^2 \leq \beta^2 |x_i|^2$? What I want to say is that in R.H.S. the $|x_i|^2$ is divided by something smaller than $i$. and thus ...
2
votes
1answer
47 views

Abelian groups about rationals [duplicate]

Is the set $\mathbb{Q}$ under $×$ an abelian group? It is sure for $\mathbb{Q} - {0}$, but i think the whole set of rationals is not an abelian group as $0 × a = a × 0 = 0$, but the identity element ...
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1answer
29 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
46 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
243 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
43 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
70 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(...
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1answer
87 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
61 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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votes
1answer
62 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
98 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
46 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^{-...
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1answer
45 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
41 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 ...
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1answer
58 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 ...
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1answer
19 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
49 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
53 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 ...
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1answer
25 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
512 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
131 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 ...
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votes
1answer
63 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
666 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
23 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
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
24 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 $\ker \varepsilon_{\sqrt 2}$ ...