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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2
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3answers
82 views

Definition of rational numbers from real numbers

Usually the set of numbers are introduced starting from integers, from wich the rational numbers are defined using equivalence classes of couples of integer numbers. Than, from these rational numbers, ...
5
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3answers
136 views

Showing the irreducibility of $x^8 - 60 x^6 + 1160 x^4 - 7800 x^2 + 8836$ in $\mathbb{Q}[x]$

I would like to show the irreducibility of $x^8 - 60 x^6 + 1160 x^4 - 7800 x^2 + 8836$ and $x^8 - 120 x^6 + 4360 x^4 - 45600 x^2 + 15376$ in $\mathbb{Q}[x]$. In both cases Eisenstein criterion fails. ...
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2answers
345 views

$\frac{7x+1}2, \frac{7x+2}3, \frac{7x+3}4, \ldots ,\frac{7x+2016}{2017}$ are reduced fractions for integers $x\in(0,301)$. [closed]

BdMO 2017 junior catagory Question 7. $$\dfrac{7x+1}2, \dfrac{7x+2}3, \dfrac{7x+3}4, \ldots ,\dfrac{7x+2016}{2017}$$ Here $x$ is a positive integer and $x < 301$. For some values of $x$ it is ...
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2answers
251 views

Finding irrational entries such that the determinant will never be zero

Context. The main goal is to find whether or not a subspace of $\mathbb R^5$ of dimension $3$ intersects a rational subspace of dimension $2$. By rational subspace, we mean a subspace of $\mathbb R^5$...
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37 views

For which x the following sequences converges

If $q_n$ be an enumeration of rational numbers, for which $x$ the following sequence converges? $$\sum_{n=1}^{\infty}e^{-n^2|x-q_n|}.$$ I guess that for no $x$ the sequence converges. I tried to ...
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1answer
253 views

How is it that there are 'gaps' in rational numbers and yet between any two rational numbers, there exists another rational number?

If there are gaps in rational numbers then lets assume we have a gap between a and b, both being rational. Then we have $\frac{a+b}{2}$ which is inside the gap which essentially makes it a non-gap. ...
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2answers
105 views

Apostol proof for $\mathbb{Q}$ being countable.

I am trying to understand a proof from Mathematical Analysis by Apostol for the following theorem: The set of rationals $\mathbb{Q}$ is countable. Here is the proof (I rewrote a few things): ...
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1answer
96 views

Proving the Set of Rational Numbers is the collection of equivalence classes of ratios of integers with nonzero denominators.

I read in a textbook that the set of Rational Numbers is the collection of equivalence classes of ratios of integers with nonzero denominators. I found it interesting and tried to prove it but ended ...
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1answer
110 views

Recurring Decimal Expansion

For any natural number $n>1$, we write the infinite decimal expansion of $\frac 1n$ (for example, $\frac 14$ is written as $0.24999$... instead of $0.25$). We need to determine the length of the ...
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0answers
24 views

Question on proving that the rationals are countably infinite

I just a question on a proof for the rationals being countably infinite from a textbook. We consider the following function, a mapping from $\Bbb Q$ to $\Bbb N$ $$f(x) = \begin{cases} 0, & \text{...
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87 views

Showing $\mathbb{Q} \cap [a,b]$ is an open set in $\mathbb{Q}$ for irrational $a$, $b$.

I came up with this lemma (although not confident enough about it) while solving Baby Rudin. In the chapter "Basic Topology", I attempted to solve question 16, in which $\mathbb{Q}$ is regarded as the ...
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1answer
47 views

Prove at least one of the length, width, height of the box must be rational.

Assume there is a big box be combined by finite small boxes and that the small boxes are not necessarily be the same. The statement in my note is " If there is at least one of length, width, height ...
0
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1answer
61 views

Is this a valid definition of the rationals?

$$\mathbb{Q}=\left\{\sum_{n=1}^k f(n)\mid k,n\in\mathbb{N}\land f\text{ is a finite composition of $+$, $-$, $\div$, $\times$}\right\}$$ Reasoning: Any real number can be described by a (sometimes ...
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1answer
20 views

Handling opposites when adding and subtracting rational expressions

I'm following example 8.47 from the OpenStax book Elementary Algebra. When trying to create a common denominator it is sometimes necessary to handle opposites on either side of an equation. In the ...
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2answers
184 views

Proving fraction is irreducible

Example: The fraction $\frac{4n+7}{3n+5}$ is irreducible for all $n \in \mathbb{N}$, because $3(4n+7) - 4(3n+5) = 1$ and if $d$ is divisor of $4n+7$ and $3n+5$, it divides $1$, so $d=1$. I want to ...
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Sequence of Number System Construction

After constructing the naturals, why construct integers before rationals? Is there a historical explanation? Couldn't ordered pairs of fractions constructed from the naturals be used to represent ...
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0answers
61 views

Roots of polynomials are Gaussian integers

I have got a question. I want to show the following: Let P be a normalized polynomial with integer coefficients and let w be a root of this polynomial (in $\mathbb{Q}[i]$), then w is a Gaussian ...
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49 views

Finding a subspace of dimension $3$ which does not intersect a rational subspace of dimension $2$

Some context. By rational subspace, I mean a subspace of $\mathbb R^5$ which admits a rational basis. In other words, a basis formed with vectors of $\mathbb Q^5$. For instance, the vector $v=(0,\pi,...
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4answers
140 views

Strange sum that always end up with 9

If we have any number, example 4896, and sum all digits sum = 4+8+9+6 = 27 and than substract this number from the original number, we always get a number that is divisible by 9: 4896-27=4869 -> ...
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2answers
38 views

Help proving there is a sequence of rational numbers

I'm trying to prove the following: Let $\Bbb Q$ be the countable set of rational numbers and $\{x_n\}_{n=1}^\infty$ be a sequence such that for every q $\in$ $\Bbb Q$ there is a $n \in \Bbb N$ with $...
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1answer
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Find natural number $0 < n < 30,000$ such that $\sqrt[3]{5n}+\sqrt{10n}$ is rational

I was thinking that I could try to make some sort of substitution to convert $\sqrt[3]{5n}+\sqrt{10n}$ into a polynomial with integer coefficients then use the Rational Roots Theorem to find a ...
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Am I right in this proof of a criterion for the nonsingularity of a conic curve?

$\newcommand{\C}{\mathcal{C}}$ This is an exercise in Silverman and Tate's Rational Points on Elliptic Curves: Let $\C$ be the conic given by the equation $$ F(x,y)=ax^2+bxy+cy^2+dx+ey+f=0.$$ ...
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0answers
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Unique Rational Approximation With “Small” Denominator

Suppose we have some irrational $x > 0$ and some $\epsilon > 0$. I want to show that there is at most one rational approximation $\frac{a}{b}$ such that both $| x - \frac{a}{b}| < \epsilon$ ...
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1answer
61 views

The oscillation of a bounded function at a point

Enumerate the rationals in $[0,1]$ (ie. $\mathbb{Q}\cap[0,1]$) by $q_n$. Define $f:[0,1]\to\mathbb{R}$ by $$ f(x)= \begin{cases} 1/n & \text{if } x=q_n \text{ for some }n\\ 0 & \text{...
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1answer
196 views

$f(a)-f(b)$ is rational iff $f(a-b) $ is rational

Prove that the continuous function $f:\mathbb{R} \to \mathbb{R}$ satisfying $f\left(x\right)-f\left(y\right) \in\mathbb{Q} \iff f\left(x-y\right) \in \mathbb{Q}$ is of the form $ f\left(x\right)=ax+...
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1answer
45 views

For $0 \le \theta \le \pi/2$, When are both $\theta/\pi$ and $\sqrt2\sin\theta$ rational?

For $0 \le \theta \le \pi/2$, when are both $\theta/\pi$ and $\sqrt2\sin\theta$ rational? I think $\theta=0, \pi/4$ is the only cases. This problem seems to be related to Niven's theorem, but I ...
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1answer
56 views

Definition of rational numbers

We can uniquely define the set of real numbers as a complete ordered field. But can we do something similar with the set of rational numbers ? I think we need to change the completness axiom with some ...
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2answers
103 views

Rational solution to a system of equations

Some context. By rational subspace, I mean a subspace of $\mathbb R^5$ which admits a rational basis. In other words, a basis formed with vectors of $\mathbb Q^5$. For instance, the vector $v=(0,\pi,...
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4answers
58 views

Show $(3 + \sqrt{2})^{2/3}$ is irrational using RZT

I am asked to prove that $(3+\sqrt{2})^{2/3}$ is irrational via the rational zeroes theorem. This is what I have so far: $ x = (3+\sqrt{2})^{2/3} $ $ x^3 = (3+\sqrt{2})^{2} $ $ x^3 - 11 - 6\sqrt{2}...
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2answers
91 views

Prove or disprove non-constructively there exist irrationals $a, b, c$ such that $a^{b^c}$ is rational.

Consider the interesting question: Do there exist irrationals $a$ and $b$ such that $a^b$ is a rational? Alternatively, prove or disprove that there exist irrationals $a$ and $b$ such that $a^b$ is ...
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3answers
39 views

Determine all $q \in \mathbb{Q}$, so that $ \sum_{n=1}^{\infty}{\frac{\sqrt{n+1}-\sqrt{n}}{n^q}} $ converges

I already tried the ratio and root criterion, but it didn't get me anywhere. I'd be thrilled if you had any suggestions. (also applied the third binomial formula, so it would "look nicer", maybe it ...
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1answer
333 views

Does this polynomial have a rational value which is the square of a rational number?

I have the following polynomial: $$P(x,y,z):=9y^2z^2-30x^2z+90xyz+54yz-270x+81\in\mathbb Q[x].$$ It came up in a larger proof, and I would need in order to complete the proof to prove the following ...
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3answers
54 views

Proof by Induction Question including Rational Numbers

I just recently covered 'rational numbers' in class and was assigned the following question to solve using induction for n, so that for all $q \in \mathbb{Q}$ \ {1}:...
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0answers
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A question construction of rational numbers

In $\mathbb{Z}$ we can do addition, substraction and multiplication, but not division. For example we cannot divide $2$ by $3$ (i.e. the equation $3x=2$ has no solution in $\mathbb{Z}$, because $\...
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3answers
79 views

Silly Question about $π$ [closed]

In our junior classes, we learnt that $π$ is an irrational number. Now, also we know about rational numbers. So, if I say that I have a thread of 44cm long and we may convert it into a circle. Then ...
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3answers
48 views

Arithmetic laws for rational numbers vs. real numbers

Are there any arithmetic laws that are always true for the set of rational numbers but not always true for the set of real numbers? This came up because I was doing various exercises in different ...
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1answer
23 views

Irreducibility of a polynomial over Rationals with condition given on its coefficients.

Let $f = a_nX^n+\cdots+a_1X\pm p \in \mathbb{Z}[X]$ with $\sum_{i=1}^n |a_i| < p$. Show that $f$ is irreducible in $\mathbb{Q}[X]$. Hint: Show that every root of $f \in \mathbb{C}$ has modulus ...
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3answers
88 views

Given $x^2=2$ prove for any rational number $\frac{p}{q} < x$,there exists $\frac{m}{n}$ such that $\frac{p}{q}<\frac{m}{n}<x$

Without using limits or the definition of irrational numbers, how do you solve this? I was thinking proof by contradiction, but I keep running into problems.
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Visualizing rational numbers as multiplication graphs

It's an interesting fact, that there's a straight forward way to visualize rational numbers. To each rational number – given as two integers $n<m$ – there corresponds a multiplication ...
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4answers
64 views

Square root of 6 proof rationality

I was proving $\sqrt 6 \notin \Bbb Q$, by assuming its negation and stating that: $\exists (p,q) \in \Bbb Z \times \Bbb Z^*/ \gcd(p,q) = 1$, and $\sqrt 6 = (p/q)$. $\implies p^2 = 2 \times 3q^2 \...
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1answer
55 views

I don't understand why this axiom makes the difference between a completely ordered field and an ordered field

I've been reading about ordered fields and completely ordered fields, and I'm stuck on the difference between the rational and real numbers that makes the rational numbers an ordered field and the ...
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2answers
76 views

A course of Pure mathematics incorrect

I thought I'd go through 'A Course Of Pure Mathematics' by G H Hardy And I came to this part: If the reader will mark off on the line all the points corresponding to the rational numbers whose ...
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1answer
24 views

Show that if $x$ has a terminating decimal expansion then $x=p/q$ for integers $p,q$ where the only prime factors of q are 2's and 5's.

I've proven the converse statement but don't know where to start for this statement. Would you suggest doing a proof by contradiction?
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2answers
30 views

The set {${m^m}/{n^n}: m,n\in \mathbb{N}$} density in ${\mathbb{Q}_+}$

I missed this topic (density) , so need help. Is the set {${m^m}/{n^n}: m,n\in \mathbb{N}$} everywhere dense in positive rational numbers set? I think maybe first of all here we must try with, for ...
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4answers
123 views

Evaluating $\sum_{(a,b,c)\in T}\frac{2^a}{3^b 5^c}$, for $T$ the set of all positive integer triples $(a,b,c)$ forming a triangle

Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a$, $b$, $c$. Express $$\sum\limits_{(a,b,c)\,\in\,T}\;\frac{2^a}{3^b 5^c}$$ as ...
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2answers
37 views

Abelian Group of Rationals over addition - axiomatic or derived? [closed]

Sorry if this is a daft question, but if we consider the set of rationals under the 'addition' operator we can form an Abelian Group. I'm curious: in that situation, is the definition of addition (...
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2answers
53 views

Show that rational numbers have the Archimedean property

By Archimedean property I mean: For any positive rational numbers $x = \frac{a}b$ and $y = \frac{c}d$, there is an integer $n$ such that $nx > y$, namely, $nx \equiv (x+x+ \ldots+x)$ with $n$ ...
0
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1answer
53 views

$\mathbb{Z}\leq G \leq \mathbb{Q}$ and $\varphi:G\rightarrow H$ a map which behaves nicely on fractions.

Let $(H,+)$ be any abelian group, and let $\mathbb{Z}\leq G\leq \mathbb{Q}$ a subgroup of the rational numbers with respect to addition. Let $\varphi:G\rightarrow H$ be a map with the following ...
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0answers
31 views

Prove or disprove $aB_m-bA_m=[q_n,\cdots,q_m]$

Some definitions and notation: Let the natural numbers $q_0,q_1,\cdots ,q_n$ be the $n$ terms in the continued fraction expansion of the rational number $\frac{a}{b}$, that is $$\frac{a}{b}=q_0 +\...
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1answer
62 views

Decimal expansions of rational numbers.

One can use modular arithmetic to find the decimal expansion of a rational number. (see 1: https://i.stack.imgur.com/kw4Gk.png). Using the same method I have run into a couple of problems. $x = \...