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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.

64
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7answers
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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 ...
16
votes
4answers
20k 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 ...
17
votes
3answers
2k views

Double limit of $\cos^{2n}(m! \pi x)$ at rationals and irrationals

I stumbled upon this "relation" (is the name correct?): $$ \lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases} 1,&x\text{ is rational}\\ 0,&x\text{ is irrational}\end{...
32
votes
9answers
23k views

Proof that every repeating decimal is rational

Wikipedia claims that every repeating decimal represents a rational number. According to the following definition, how can we prove that fact? Definition: A number is rational if it can be written ...
38
votes
11answers
4k views

What are the Laws of Rational Exponents?

On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number ...
8
votes
1answer
2k views

System of linear equations having a real solution has also a rational solution.

I saw this question Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $b ∈ \mathbb{Q}^m$. Suppose that the system of linear equations $Ax = b$ has a solution in $\mathbb{R}^n$. Does it necessarily have a ...
38
votes
3answers
6k 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}{...
32
votes
5answers
37k views

Is there a rational number between any two irrationals?

Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 < i_2$. Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$? How would you construct such ...
1
vote
2answers
990 views

Show that $X = \{ (x,y) \in\mathbb{R}^2\mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\}$ is path connected. [duplicate]

How do I show that $X = \left\{ (x,y) \in \mathbb{R}^2 \mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\right\}$ is path connected? Note that $X$ is a topological space with subspace topology $\tau =...
37
votes
9answers
6k views

Can sum of a rational number and its reciprocal be an integer?

Can sum of a rational number and its reciprocal be an integer? My brother asked me this question and I was unable to answer it. The only trivial solutions which I could think of are $1$ and $-1$. ...
7
votes
3answers
1k views

Why (directly!) does every number divide 9, 99, 999, … or 10, 100, 1000, …, or their product?

A curiosity that's been bugging me. More precisely: Given any integers $b\geq 1$ and $n\geq 2$, there exist integers $0\leq k, l\leq b-1$ such that $b$ divides $n^l(n^k - 1)$ exactly. The question ...
4
votes
5answers
1k views

H0w t0 prove that periodic decimal numbers are rational? $a_1…a_k(b_1b_2..b_l)={m \over n}$

Given $a_1...a_k(b_1b_2..b_l)={m \over n}$ how can I prove that periodic decimal numbers are rational? Where do I even begin?
69
votes
2answers
2k views

Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

About a month ago, I got the following : For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that $$r=\frac{a^\color{red}{3}+b^\color{red}{3}}{...
27
votes
2answers
8k views

Is sin(x) necessarily irrational where x is rational?

My friend and I were discussing this and we couldn't figure out how to prove it one way or another. The only rational values I can figure out for $\sin(x)$ (or $\cos(x)$, etc...) come about when $x$ ...
5
votes
1answer
201 views

Multiplying and adding fractions

Why multiplying fractions is equal to multiply the tops, multiply the bottoms? $$\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b \times d},$$ And why $$\frac{a}{b}\times \frac{c}{c}=\frac{a}{b},$$ ...
3
votes
3answers
4k 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 ...
16
votes
4answers
7k views

Show that the curve $x^2+y^2-3=0$ has no rational points

Show that the curve $x^2+y^2-3=0$ has no rational points, that is, no points $(x,y)$ with $x,y\in \mathbb{Q}$. Update: Thanks for all the input! I've done my best to incorporate your suggestions and ...
4
votes
1answer
707 views

Prove that any two nontrivial subgroups of $\mathbb{Q}$ have nontrivial intersection

I need to prove that any two nontrivial subgroups of $\mathbb{Q}$ have a nontrivial intersection as part of a larger proof that $\mathbb{Q}$ cannot be represented as a nontrivial direct product. (...
11
votes
4answers
9k views

Proving the rationals are dense in R

I know this is a common proof. I'm following Rudin's proof and I'm following everything except for one step. Suppose $x, y \in \Bbb R$ and $x < y$. Then there exists an $n \in \Bbb N$ such that $n(...
5
votes
2answers
265 views

Proving that $x$ is irrational if $x-\lfloor x \rfloor + \frac1x - \left\lfloor \frac1x \right\rfloor = 1$

Prove : $$ \text{If } \; x-\lfloor x \rfloor + \frac{1}{x} - \left\lfloor \frac{1}{x} \right\rfloor = 1 \text{, then } x \text{ is irrational.}$$ I think the way to go here is to falsely assume that $...
0
votes
4answers
532 views

How can I explain $0.999\ldots=1$? [duplicate]

Possible Duplicate: Does .99999… = 1? I have to explain $0.999\ldots=1$ to people who don't know limit. How can I explain $0.999\ldots=1$? The common procedure is as follows \begin{align} x&...
80
votes
7answers
27k views

Why is $\frac{987654321}{123456789} = 8.0000000729?!$

Many years ago, I noticed that $987654321/123456789 = 8.0000000729\ldots$. I sent it in to Martin Gardner at Scientific American and he published it in his column!!! My life has gone downhill since ...
15
votes
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 ...
12
votes
4answers
2k views

Additive group of rationals has no minimal generating set

In a comment to Arturo Magidin's answer to this question, Jack Schmidt says that the additive group of the rationals has no minimal generating set. Why does $(\mathbb{Q},+)$ have no minimal ...
6
votes
3answers
144 views

How to show that if $x, y, z$ are rational numbers satisfying $(x + y + z)^3 = 9(x^2y + y^2z +z^2x)$, then $x = y = z$ [closed]

Let $x,y,z$ rationals Show that if $(x+y+z)^3=9(x^2y+y^2z+z^2x)$ then $x=y=z$ I tried this : Let $x$ be the smallest variable Write $y=a+x$ and $z=b+x$ Prove $a=b=0$ by factoring the equation as ...
1
vote
1answer
391 views

About the continuity of the function $f(x) = \sum\limits_k2^{-k}\mathbf 1_{q_k \leq x}$

Let $q: \mathbb{N} \to \mathbb{Q}$ be a bijection and denote the image of $k \in \mathbb{N}$ by $q_k$. Let $f: \mathbb{R} \to (0,1)$, $$ f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k} $$...
5
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2answers
19k views

Can we ever get an irrational number by dividing two rational numbers?

If we try to divide any two random arbitrarily long rational numbers like 103850.2387209375029375092730958297836958623986868349693868398659825528365... and <...
6
votes
4answers
388 views

Are there any natural proofs of irrationality using the decimal characterization?

Mathematicians typically define rational number to mean quotient of two integers. It is not hard to show that a number is rational by that definition if and only if its decimal expansion terminates ...
4
votes
3answers
882 views

Compute the period of a decimal number a priori [duplicate]

Possible Duplicate: Upper bound/exact length of decimal expansion of simple fraction I noticed that WolframAlpha given an operation like $\frac{n}{m},\;n,m \in N$ that result in a periodic ...
1
vote
1answer
2k views

Show that the set of polynomials with rational coefficients is countable.

Problem: Show that the set of polynomials with rational coefficients is countable. Idea: We know that the set of rational numbers is denumerable. This implies that the set of rational numbers is ...
90
votes
12answers
12k views

Why can't calculus be done on the rational numbers?

I was once told that one must have a notion of the reals to take limits of functions. I don't see how this is true since it can be written for all functions from the rationals to the rationals, which ...
54
votes
3answers
4k views

Is it possible to find an infinite set of points in the plane where the distance between any pair is rational?

The question is written like this: Is it possible to find an infinite set of points in the plane, not all on the same straight line, such that the distance between EVERY pair of points is rational? ...
7
votes
3answers
848 views

Rational solutions of Pell's equation

1) $D$ is a positive integer, find all rational solutions of Pell's equation $$x^2-Dy^2=1$$ 2) What about $D\in\Bbb Q$ ?
19
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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 ...
12
votes
6answers
18k views

Infinite number of rationals between any two reals.

Let $a$ and $b$ be reals with $a<b$. Show that there are infinitely many rationals $x$ such that $a<x<b$. My plan of action was to assume that $x$ is the smallest such rational and find ...
10
votes
1answer
916 views

Predicting the number of decimal digits needed to express a rational number

The number $1/6$ can be expressed with only two digits (and a repeat sign denoted as $^\overline{}$), $$ \frac{1}{6} = \,.1\overline{6}$$ Meanwhile, it takes 49 digits to express the number $1/221$, ...
8
votes
1answer
679 views

Group $\mathbb Q^*$ as direct product/sum

Is the group $\mathbb Q^*$ (rationals without $0$ under multiplication) a direct product or a direct sum of nontrivial subgroups? My thoughts: Consider subgroups $\langle p\rangle=\{p^k\mid k\in \...
4
votes
6answers
4k views

Why is $[0, 1] \cap \mathbb{Q}$ not compact in $\mathbb{Q}$? [duplicate]

Statement: $[a, b] \cap \mathbb{Q}$ in $\mathbb{Q}$ is not compact. Thus the interior of all compact subsets of $\mathbb{Q}$ is $\emptyset$. I am trying to understand the first sentence. I read that ...
3
votes
1answer
145 views

Proof that Epicycloids are Algebraic Curves?

Epicycloids are most commonly described by the parametric equations, $x(t) = (R + a)\cos(t) – a \cos \left(\frac{R + a}{a} t \right),$ $y(t) = (R + a)\sin(t) – a \sin \left(\frac{R + a}{a} t \right)...
3
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3answers
532 views

Equality of positive rational numbers.

I am reading the second article Rational Numbers of the book "A Treatise on Advanced Calculus" by Philip Franklin. I have mainly 3 questions regarding this article. I am writing all these $3$ ...
2
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1answer
6k views

Prove that the difference between two rational numbers is rational

This is a terribly simple question I'm sure, but I can't find a work-around in my proof. I must prove that the difference between two rational numbers is thus rational. Here is my attempt: Let $a$ ...
1
vote
1answer
64 views

Another polynomial equation

Let $r$ be a root of the polynomial $p(x)=(\sqrt{3}-\sqrt{2})x^3 + \sqrt{2}x-\sqrt{3}+1$. Find another polynomial $q(x)$, with all integer coefficients, such that $q(r)=0$.
11
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1answer
3k views

Show that $\{1, \sqrt{2}, \sqrt{3}\}$ is linearly independent over $\mathbb{Q}$.

My apologies if this question has been asked before, but a quick search gave no results. This is not homework, but I would just like a hint please. The question asks Show that $\{1, \sqrt{2}, \...
2
votes
3answers
220 views

proving $ \sqrt 2 + \sqrt 3 $ is irrational [duplicate]

I need to proof that $\sqrt{3} + \sqrt{2}$ is irrational, without using the fact that an irrational number plus a rational number equals irrational. also, i can't use the rational root theorem. that's ...
12
votes
2answers
419 views

Given dividend and divisor, can we know the length of nonrepeating part and repeating part?

$13/92=0.14\overline{1304347826086956521739}$ In this example, the length of nonrepeating part is $3$. The length of repeating part (repeating period) is $21$. I collected some properties related to ...
12
votes
4answers
10k views

Length of period of decimal expansion of a fraction

Each rational number (fraction) can be written as a decimal periodic number. Is there a method or hint to derive the length of the period of an arbitrary fraction? For example $1/3=0.3333...=0.(3)$ ...
12
votes
3answers
3k views

Show that it is impossible to list the rational numbers in increasing order

This is problem #6 from Section 1.2 of Ash's Real Variables With Basic Metric Space Topology. I am asked to show that it is impossible to list the rational numbers in increasing order. While I ...
9
votes
4answers
1k views

How to obtain all the rational numbers without repetitions?

Some days ago I've seen Cantor's diagonal argument, and it presented a table similar to the following one: $$\begin{matrix} {\frac{1}{1}}&{\frac{1}{2}}&{\frac{1}{3}}&{\frac{1}{4}}&{\...
8
votes
5answers
5k views

Definition(s) of rational numbers

The definitions of rational numbers are somewhat confusing for me. The definition of rational numbers on wikipedia and most other sites is: In mathematics, a rational number is any number that can ...
8
votes
1answer
441 views

For each irrational number $b$, does there exist an irrational number $a$ such that $a^b$ is rational?

It is well known that there exist two irrational numbers $a$ and $b$ such that $a^b$ is rational. By the way, I've been interested in the following two propositions. Proposition 1 : For each ...