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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6
votes
0answers
163 views

Is there always a telescopic series associated with a rational number?

Here is something I thought up while I was bored and my, erm, fish were busy: Given a rational number $p\in(0,1)$, are there always positive integers $n$, $c_m$, and $w_m$ such that $$p=\sum_{k=1}^\...
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2answers
1k views

Ratio GRE question

Cashews cost 4.75 per pound and hazelnuts cost 4.50 per pound. What is larger, the number of pounds of cashews in a mixture of cashews and hazelnuts that costs $5.50 per pound, or 1.25? ...
2
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4answers
251 views

float result for two smallest integer division

I want to know the two integer number that division of them is this float. for example x / y = 1.333333333.... $x$ and $y$ can be ...
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2answers
1k views

Can dividing two rational numbers yield an integer?

I wanted to know if two non-int numbers (non-zeroes) when divided with each other can give an integer or not.I believe that's a NO. However I know they can only yield an integer $1$ provided both are ...
2
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2answers
108 views

for what value of $a$ has equation rational roots?

Suppose that we have following quadratic equation containing some constant $a$ $$ax^2-(1-2a)x+a-2=0.$$ We have to find all integers $a$,for which this equation has rational roots. First I have ...
8
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2answers
3k views

Is there a sequence that contains every rational number once, but with the “simplest” fractions first?

The Calkin-Wilf sequence contains every positive rational number exactly once: 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, …. I'd consider 5/1 to be a "simpler" ratio than ...
4
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1answer
97 views

How do I scale 3 fractions to 3 natural numbers?

Disclaimer: I'm an engineer, not a mathematician I have a set of three fractions (a/b, c/d, e/f). I can multiply them all by another fraction, so that their mutual ratios remain the same. I want to ...
6
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3answers
7k views

Dedekind Cut Proof

I am greatly confused with Dedekind cuts... I am trying to prove that this is a Dedekind cut: If $D$ and $E$ are in $\mathbb{Q}$ and are Dedekind cuts, then prove that $$D*E=(-\infty, 0] \cup \{...
15
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3answers
3k views

How can we find and categorize the subgroups of $\mathbb{R}$?

$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}\newcommand{\Z}{\Bbb Z}$ What are all the subgroups of R = $(\R, +)$ and how can we categorize them? I started thinking about this question last night ...
20
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4answers
1k views

What do the cosets of $\mathbb{R} / \mathbb{Q}$ look like?

$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}$ Looking at the group of real numbers under addition $(\R, +)$ it contains the (normal) subgroup of rational numbers $(\Q, +)$. I am wondering how to ...
39
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3answers
7k views

GCD of rationals

Disclaimer: I'm an engineer, not a mathematician Somebody claimed that $\gcd$ only is applicable for integers, but it seems I'm perfectly able to apply it to rationals also: $$ \gcd\left(\frac{13}{...
2
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1answer
143 views

Is there a rational univariat polynomial of degree 3 with 3 irrational roots?

The title pretty much asks my question: Does $f\in\mathbb{Q}[x]$ such that $$ f(x)=(x-\alpha_1)(x-\alpha_2)(x-\alpha_3),$$ where $\alpha_1, \alpha_2, \alpha_3\in\mathbb{R}\setminus\mathbb{Q}\ $ exist?...
4
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3answers
997 views

Compute the period of a decimal number a priori

I noticed that WolframAlpha, given an operation like $\frac{n}{m},\;n,m \in N$ that result in a periodic decimal number, computes really fast the length of the period. E.g. $\frac{3923}{6173}$ has a ...
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3answers
5k views

Show that the ring of all rational numbers, which when written in simplest form has an odd denominator, is a principal ideal domain. [closed]

Show that the ring of all rational numbers $m/n$ with $n$ an odd integer is a principal ideal domain. We haven't really discussed principal ideal domains. I've heard that this is easy, but I just ...
1
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1answer
68 views

How well do we need to do in the remaining months to meet an annual percentage goal?

I am required to have a employment team meet a yearly percentage rate of 50% for a process. The team is currently short of the goal with a rate of 43%. My boss wants to know what percentage is ...
1
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3answers
88 views

If $\log_{b}N$ is rational, what are the limitations on the possible values of $b$ and $N$?

If $\log_{b}N$ is rational, is there a set of values to which $b$ and $N$ must belong? Is there a set of values to which $b$ and $N$ cannot belong? Further, if it is presupposed that $b$ and $N$ are ...
2
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1answer
273 views

Why is the group of rational numbers with odd denominators residually finite?

I want to prove that the additive group of rational numbers with odd denominators is residually finite. However, despite pondering for quite some time, I can't even find a single subgroup of finite ...
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2answers
190 views

Narrowing a Stern-Brocot tree

Say I only wanted to enumerate the rational numbers between 0 and $a$. Is there a way to "narrow" a Stern-Brocot tree to provide this? I tried keeping my left bound at "$\frac{0}{1}$" and setting my ...
4
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3answers
480 views

Determine if $(p/q)^{a/b}$ is rational

I know, in general, that it isn't true. ${\frac{2}{1}}^{1/2}$ is irrational. I'm only interested in this where $\frac{p}{q}$ and $\frac{a}{b}$ are positive, but to make this even simpler, lets just ...
12
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1answer
130 views

Combinatorial question about sets of rational numbers

The following question came up in my research. Since lots of clever people post here, I thought I'd ask it. Recall that the group ring of a group $G$ is the abelian group $\mathbb{Z}[G]$ consisting ...
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4answers
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Enumeration of rationals from Stein-Shakarchi's Real Analysis (Chapter 1, Exercise 24)

The exercise is from Stein-Shakarchi's Real Analysis (Chapter 1, ex. 24). Does there exist an enumeration $\{r_{n}\}_{n=1}^\infty$ of the rationals such that the complement of $\bigcup_{n=1}^{\...
2
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3answers
417 views

What condition that if imposed on $\alpha$ imply that $\cos^{-1} \alpha$ is a rational multiple of $\pi$?

It is well known that if $x$ is a rational multiple of $\pi$ then $\cos x$, $\sin x$, etc, are algebraic numbers. What is known about the inverse problem? That is, is there a set of conditions that ...
-1
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1answer
128 views

Properties of lines in the Pixel + Zoom geometry

Problems Prove that in PZ geometry, every PZ line has an equation of the form ax+by=c, where a, b, c are all rational numbers and a and b are not both zero. Prove that every equation of the form ax+...
2
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2answers
985 views

Rational numbers- sticks and stones

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. For all rational numbers, we will have a stick of variable length ...
14
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2answers
615 views

Solutions of $q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$ s.t. $q \geq 3$

Is it true that for every rational $q \geq 3$ , the following equation has a solution over $\mathbb N$ ? $$q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$$
1
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1answer
74 views

How many sieves are there on a given rational number $q$?

Consider the poset category $\mathbb{Q}^{op}$, i.e. where $p \rightarrow q$ iff $p \geq q$. Take any $q \in \mathbb{Q}$. Then how many sieves are there for $q$ in $\mathbb{Q}^{op}$? Supposedly, the ...
3
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1answer
562 views

If a finite set of rational numbers sums to one, does one of the rationals have a denominator equal to the LCM of all the denominators?

I was experimenting with an algorithm for generating random numbers from a discrete distribution and came across an interesting observation. Suppose that you have any finite set of rational numbers ...
6
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3answers
3k views

What is the ratio of rational to irrational real numbers?

There exists an infinite amount of rational and irrational numbers. But is there more irrational numbers than rational? And if so can a ratio of one to the other be calculated?
5
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1answer
2k views

How to find all rational points on the elliptic curves like $y^2=x^3-2$

Reading the book by Diophantus, one may be led to consider the curves like: $y^2=x^3+1$, $y^2=x^3-1$, $y^2=x^3-2$, the first two of which are easy (after calculating some eight curves to be solved ...
0
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1answer
494 views

When are $\theta$ and $\sin\theta^\circ$ both rational? [duplicate]

Possible Duplicate: Sine values being rational I'm guessing that if I look in Ivan Niven's elementary book on irrational numbers, I'll find the answer to this quickly, but I'm posting it here in ...
18
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4answers
22k views

How can I prove that all rational numbers are either terminating decimal or repeating decimal numerals?

I am trying to figure out how to prove that all rational numbers are either terminating decimal or repeating decimal numerals, but I am having a great difficulty in doing so. Any help will be greatly ...
70
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7answers
38k views

Produce an explicit bijection between rationals and naturals?

I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but ...