For questions on rational numbers, numbers that can be expressed as the quotient or fraction $\frac pq$ of two integers.

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12
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8answers
4k 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 ...
6
votes
3answers
570 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 ...
4
votes
4answers
708 views

How to prove this is a rational number

I'm not sure how to prove this is a rational number $\frac{q}{m}$, can some one show me? $$\frac{q}{m}=\frac{(\frac{1+\sqrt5}{2})^n - (\frac{1-\sqrt5}{2})^n}{\sqrt5}$$
6
votes
1answer
372 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 ...
55
votes
5answers
12k views

Why is 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 ...
9
votes
4answers
374 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 ...
8
votes
1answer
284 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 $1/221$, ...
2
votes
3answers
220 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
votes
3answers
174 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 ...
18
votes
3answers
1k 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: $$ ...
12
votes
4answers
4k 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 ...
17
votes
4answers
452 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 ...
16
votes
7answers
5k views

What rational numbers have rational square roots?

All rational numbers have the fraction form $$\frac a b,$$ where a and b are integers($b\neq0$). My question is: for what $a$ and $b$ does the fraction have rational square root? The simple answer ...
1
vote
1answer
145 views

Properties of homomorphisms of the additive group of rationals

Let $f : (\mathbb{Q},+) \longrightarrow (\mathbb{Q},+)$ be a non-zero homomorphism. Can we conclude that $f$ is bijective (or, if that fails, that $f$ is injective or surjective)? Context The ...
13
votes
2answers
1k 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 $x$ ...
2
votes
2answers
777 views

In every interval there is a rational and an irrational number.

When the interval is between two rational numbers it is easy. But things get complicated when the interval is between two irrational numbers. I couldn't prove that.
12
votes
5answers
1k views

Are there infinitely many rational outputs for sin(x) and cos(x)?

I know this may be a dumb question but I know that it is possible for $\sin(x)$ to take on rational values like $0$, $1$, and $\frac {1}{2}$ and so forth, but can it equal any other rational values? ...
5
votes
4answers
219 views

What is a suitable name for numbers like $a + b\sqrt{c}$

The motivation for this is to find a succinct name for a data type in a Python module. Suppose I choose an integer $c$ and I want to talk about the set of numbers of the form $a + b\sqrt{c}$, where ...
5
votes
4answers
2k 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 ...
2
votes
2answers
159 views

Let a, b, c, d be rational numbers… [closed]

Let $a, b, c, d$ be rational numbers, where $\sqrt{b}$ and $\sqrt{d}$ exist and are irrational. If $a + \sqrt{b} = c + \sqrt{d}$, prove that $a=c$ and $b=d$.
1
vote
1answer
85 views

For what values of $b\in \mathbb R$ is $\pi-b\in \mathbb Q$ true?

Just a simple short question. I'm looking for values $b$ such that $\pi-b$ is a rational number. Obviously $\pi$ is such a number, but are there more? Edit: $b$ is in $\mathbb R$
0
votes
3answers
140 views

Finding Rational numbers

Please help with the following question: Find rational numbers a and b such that: $$\left(7 + 5\sqrt2\right)^{\frac13} = a + b \sqrt2$$ Thank you
0
votes
4answers
247 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 ...
-2
votes
1answer
672 views

Rational Number Proof [duplicate]

Stuck on a tutorial question trying to study for a test. The question is : Consider the following statement: "Between any two different rational numbers, there are at least two different rational ...
-3
votes
2answers
178 views

Proof Involving Rational Numbers [duplicate]

I asked this same question last night and got some answers but still can't make sense of this, normally I'd move on but since I know how to do everything else for the test I'm going to try to get this ...
60
votes
21answers
12k views

Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical. Numerical computations, to my understanding, never deal ...
6
votes
4answers
305 views

Is there a bijection $f: \mathbb{Q} \to \mathbb{Q}_{>0}$?

For $\mathbb{R}$, we have the exponential function. Is there also a bijection $f: \mathbb{Q} \to \mathbb{Q}_{>0}$ or to $\mathbb{Q}_{\geq 0}$?
14
votes
2answers
342 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 $\mathbb N$ ? $$q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$$
6
votes
2answers
244 views

For what algebraic curves do rational points form a group?

For what real algebraic curves do rational points form a group ? How does this relate to Jacobian Varieties ?
14
votes
1answer
269 views

Why is $x^3-5x$ injective on the rationals?

I've found the statement on the internet that the polynomial $x^3-5x$ is injective on the rational numbers, but without any comments on how to prove it. I think it means it must be easy, but I don't ...
7
votes
4answers
484 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} ...
18
votes
2answers
514 views

Order preserving bijection from $\mathbb Q\times \mathbb Q$ to $\mathbb Q$

Do you have a simple example of order preserving bijection from $\mathbb Q\times \mathbb Q$ to $\mathbb Q$? On $\mathbb Q$, I use the usual order and on $\mathbb Q\times\mathbb Q$, I define the ...
5
votes
1answer
138 views

What are the positive rational solutions of $x^{(x+y)} = (x+y)^y$?

I saw this problem in the Problem-Solving through Problems book by Larson (# 3.3.25b). I got to here: $$x \log(x) = y\log\left(1+ \frac yx\right)$$ But I can't seem to find a way to reduce this ...
5
votes
0answers
105 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 ...
4
votes
1answer
146 views

Is $\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$ a rational number for $m,n\ge 2\in\mathbb N$?

Question : Is $$\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$$ a rational number for $m,n\ge 2\in\mathbb N$ where $\zeta (s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$? Motivation : We know that $$\zeta ...
11
votes
6answers
2k views

Proof that there are infinitely many positive rational numbers smaller than any given positive rational number.

I'm trying to prove this statement:- "Let $x$ be a positive rational number. There are infinitely many positive rational numbers less than $x$." This is my attempt of proving it:- Assume that ...
4
votes
5answers
345 views

Given a rational number $x$ and $x^2 < 2$, is there a general way to find another rational number $y$ that such that $x^2<y^2<2$?

Suppose I have a rational number $a$ and $a^2 < 2$. Can I find another rational number $B$ such that $a^2<B^2<2$? Based on the answer to this question, I thought of doing the following: ...
4
votes
3answers
397 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 ...
1
vote
4answers
443 views

How can I expand mathematical induction to rational numbers?

I know mathematical induction can be used to prove that a statements is true for all natural numbers (or those belonging to a certain subset of N). However, it is pretty obvious, unless I'm terribly ...
1
vote
1answer
435 views

Prove that given any rational number there exists another greater than or equal to it that differs by less than $\frac 1n$

I am currently attempting to prove a claim in Hardy's Course of Pure Mathematics and am currently stuck. I was hoping that someone would be able to provide some assistance on how to go about this. ...
6
votes
1answer
573 views

Deciding whether $2^{\sqrt2}$ is irrational/transcendental

Is $2^\sqrt{2}$ irrational? Is it transcendental?
5
votes
6answers
387 views

Prove that - for every positive $x \in \mathbb{Q}$, there exists positive $y \in \mathbb{Q}$ for which $y \lt x$

First my apologies if this question has been asked before. Exposition I'm new at learning how to prove theorems and among the given exercises from my reference material it is asked to prove the ...
4
votes
2answers
141 views

Rational solutions of $x^3+y^3=2$

I came along the problem of finding three perfect cubes that are consecutive numbers of an arithmetic progression, i.e: $a^3-b^3=b^3-c^3$, where $a>b>c$ (to avoid trivial solutions). Clearly it ...
3
votes
3answers
64 views

The supremum of rationals that are less than a given number is equal to that number

I have the following theorem to prove. Given a real number $a$, define the set $S$ such that $S = \{x \in \mathbb Q: x < a\}$. Show that $a = \sup S$. My attempt at a proof is as follows ...
3
votes
2answers
144 views

Define two rational numbers $\alpha$ and $x$ such that $\sin( { \alpha }) =x$

Of course for $x\neq 0 $ and $\alpha$ in radians. Can you define them?
3
votes
1answer
225 views

length of period

Each rational number (fraction) can be written as decimal periodic number. Does exists a method or hint that show how long will be the period of arbitrary fraction. For example $1/3=0.3333...=0.(3)$ ...
3
votes
1answer
276 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 ...
2
votes
2answers
125 views

A question about rational.

Is that true : Every positive rational number $q$ can be written as $q = \sum_{i=0}^{k}1/n_i$ , where $n_i,k$ are positive intergers and $n_i\not=n_j$ if $i\not=j$.
2
votes
4answers
125 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 ...
2
votes
1answer
119 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}\ $ ...