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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relation between rational and irrational, non-transcendental numbers?

a long time ago, when watching a video about continued fractions, I saw something interesting, all continued fractions in that video (all that were non-transcendental) had a rational-looking fraction. ...
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3answers
734 views

Probability of a number being rational

If $x \in [0, 1]$, what is $\text{P}(x\in \mathbb Q)$? In other words, what is the probability that $x$ is rational? This is what I tried: $$\begin{array}{rcl}\text{P}(x \in \mathbb Q) &=&...
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33 views

How to express it as a rational number [on hold]

Evaluate ((((3^2)^3)^1/3)^5)^−3/5) and express it as a rational number in lowest terms, as above. $${3^{{{2^3}^\frac13}^ 5}}^{-\frac35}$$
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Intuition behind Diophantine approximation: why do we express the bound as function of denominators?

The field $\mathbb Q$ is dense in $\mathbb R$, so we can approximate a real number $\alpha$ by an arbitrarly close rational number $r=\frac{a}{b}$. The purpose of Diophantine approximation is to find ...
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118 views

Proving that the set of rational numbers between 0 and 1 is disconnected.

Show that the set of rational numbers between 0 and 1 $(A = \mathbb{Q} \cap [0,1])$ is disconnected. Note that $A \subseteq \mathbb{R}$ is a subspace topology. Definition of disconnectedness: ...
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1answer
988 views

Rational vs Irrational distribution

Imagine I draw a number line, and I took two points. What's the distribution of rational and irrational numbers between them? If I put it in a diagram where I color rational with a color and ...
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1answer
18 views

Need help solving word problem with Negative integers involving descent - would greatly appreciate it.

This is a question from my son's test. He got (a) correct but (b)wrong. He doesn't understand why. Would appreciate an explanation that can help him understand his mistake. A submarine is at −750 ...
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7answers
18k views

Why is a repeating decimal a rational number?

$$\frac{1}{3}=.33\bar{3}$$ is a rational number, but the $3$ keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ratio? ...
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Why does the elliptic curve for $a+b+c = abc = 6$ involve a solvable nonic?

The curve discussed in this OP's post, $$\color{brown}{-24a+36a^2-12a^3+a^4}=z^2\tag1$$ is birationally equivalent to an elliptic curve. Following E. Delanoy's post, let $G$ be the set of rational ...
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1answer
31 views

Circular definition of rationals.

If we define rational numbers as A rational number is any number that can be fraction $\frac pq$ of two integers $p$ and $q$, with the denominator $q$ not equal to zero. But integers themselves ...
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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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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 ...
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1answer
35 views

Significant figures problems

Between my first assessments I met some exercices that I don't know how to do them because I did not understand well the rules or maybe I applied in a wrong manner. For instance: The following ...
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1answer
59 views

Proving an inequality between the difference of $\sqrt2$ and any rational number. [duplicate]

Let $a = \sqrt{2}$ Prove that for every $m,n\in N$ $|a - \frac{m}{n}| \gt \frac{1}{(2\sqrt2+1)n^2}$ Hint: Consider $|a - \frac{m}{n}|\geq 1$ and $|a - \frac{m}{n}|\leq 1$ as separate cases and ...
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38 views

Absolute value in rational numbers

We define the absolute value in $\mathbb{Q}$ as an application $||\, \cdot \, || : \mathbb{Q}\rightarrow [0,\infty )$ that fulfills the properties: $||x||=0$ if and only if $x=0$. $||xy||=||x||\, ||y|...
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76 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, ...
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117 views

Optimal division on $\mathbb{Z} $

I am trying to understand the construction of $\mathbb{R}$ with slopes / quasi-isomorphism, as shown here at some point, the following property is used : $$\forall p \in \mathbb{Z},\forall q \in \...
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1answer
84 views

For integer $n>1$, can $\sum_{k=1}^{n}\sqrt{k}$ be a rational number? Can it be an integer? [duplicate]

I know that the sum of two (or more) irrational numbers can be rational. For example, both $\sqrt{2}$ and $1-\sqrt{2}$ are irrational numbers, but their sum is rational. Also I know that $\sqrt{m}$ ...
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2answers
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Let the rational number $p/q$ be closest to but not equal to $22/7$ among all rational numbers with denominator $< 100$.

Let the rational number $p/q$ be closest to but not equal to $22/7$ among all rational numbers with denominator $< 100$. What is the value of $p − 3q$ ? My approach: $22/7=3.14$, therefore, $p/q=...
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Prove that $x^3 + x^2 = 1$ has no rational solutions?

Is this enough for a proof?: $$x^3+x^2 = 1$$ I would factor and get: $x^2(x+1) = 1$ I would show that $x = \sqrt1$, which is rational but then what else would I have to show? $x+1=1$ which gives me ...
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4answers
990 views

Proof that there is no rational solutions to the equation $x^3+2x-1=0$

Proof by contradiction: Assuming that there is a rational solution to the equation $x^3+2x-1=0$. Let $x=a/b$ where $a$ and $b$ are coprime with $b$ not equal to zero. Performing a substitution into ...
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Prove the sum of two rational number is equal to $\frac{e}{lcm(b,d)}$ for some integer $e$.

As title state: $\frac{a}{b} + \frac{c}{d}=\frac{e}{lcm(b,d)}$ for some integer $e$. Here is what I tried: $\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}$ Since $gcd(b,d)lcm(b,d)=bd$, so I got $\...
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65 views

Ordering of rationals

Let $(\mathbb{Q},<)$ be the usual ordering of rationals. Show that there is a family $\mathcal F$ of subsets of $\mathbb{Q}$ such that $|\mathcal F|=2^\omega$ and for every $A, B \in \mathcal F, (A,...
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2answers
38 views

Generalising a problem when two fields $F \ncong K$

I am trying to generalize a problem that I came across previously. $\mathbf{ Problem:}$ Are the fields $\mathbb{Q}$ and $\mathbb{Q[\sqrt2]}$ isomorphic? $\mathbf{Generalisation:}$ Let $F$ and $K$ ...
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1answer
38 views

To prove that $\mathbb{Q}$ is the smallest subfield of $\mathbb{C}$

Assumption: There exsits $F$ which is a subfield of $\mathbb{C}$ such that $F\subsetneq \mathbb{Q}$. Claim: $\mathbb{Z}\subset F$. Proof: Let $m \in \mathbb{Z^+ }$. We know, that $1 \in F$. Taking $...
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Function that maps the “pureness” of a rational number?

By pureness I mean a number that shows how much the numerator and denominator are small. E.g. $\frac{1}{1}$ is purest, $\frac{1}{2}$ is less pure (but the same as $\frac{2}{1}$), $\frac{2}{3}$ is ...
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2answers
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Proof that the set of rationals is countable with finite preimages?

I'm working through the proof that the set of Rational numbers is countable and the proof says in order to do this you just have to show every rational number can be mapped to the set of natural ...
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Orbits of vectors under the action of $\mathrm{GL}_n(\mathbb Q)$

Context. While working on a larger proof, I would love to have the following lemma, but I can't even decide if it's true or not. The question. We consider the action of $\mathrm {GL}_n(\mathbb Q)$ ...
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3answers
51 views

Fractions that have interesting, fun or noteworthy decimal expansions

I'm looking to discover more fractions that have interesting* decimal expansions. (I'm asking out of curiosity, there is no particular academic reason as far as I'm concerned). Here are a few ...
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1answer
45 views

Showing that $\mathbb{Q}(\sqrt{2}) \subseteq \mathbb{Q}(\sqrt{2}+\sqrt[3]{2})$

I am doing some early study in field theory and am stuck on the following problem. Show that $\mathbb{Q}(\sqrt{2}) \subseteq \mathbb{Q}(\sqrt{2}+\sqrt[3]{2})$ and that $\mathbb{Q}(\sqrt[3]{2}) \...
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307 views

Collection of intervals covers $[0,1]$?

For each $n=1,2,3,...$ and each $m=0,1,2,...,n-1$, let $$K^n_m = \left[ \frac{3m-n+1}{n} , \frac{3m-n+2}{n} \right] \subset (-1,2). $$ I am struggling these with two questions for quite some time: (...
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2answers
455 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 $2$. The length of repeating part (repeating period) is $22$. I collected some properties related to ...
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1answer
30 views

Why $\mathbb{Q}$-vector space has no $\mathbb{Q}$-torsion?

Why $\mathbb{Q}$-vector space has no $\mathbb{Q}$-torsion ? What does the notion of $\mathbb{Q}$-torsion technically mean ?
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HINT: Prove there does not exist a rational number that satisfies $x^3=p/q$

I would like to request a hint for a problem I am working on form Hardy's a Course of Pure Mathematics. Question Prove generally that a rational fraction $p/q$ in its lowest terms cannot be the cube ...
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Proof that $(1+\frac{1}{n})^n$ can't converge to a rational number

One of my colleagues challenged me (and his students) with the following: "Assume you don't know that $\lim_{n\to +\infty}(1+\frac{1}{n})^n=e$. Prove the sequence $u_n=(1+\frac{1}{n})^n$ ...
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1answer
32 views

Why does the definition for the multiplication of dedkind cuts explicitly include the negative rationals?

If A and B are both dedekind cuts. Then $A \times B=\{ab \mid a \in A, b \in B, a \geq 0, b \geq 0 \} \cup \{q \in \mathbb{Q} \mid q <0 \}$. Can someone explain why this definition doesn't work: $...
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2answers
40 views

The smallest positive integer vector from a positive rational vector

Suppose $\mathbf{q} = \left[\begin{array}{cccc}q_1 & q_2& \dots &q_n\end{array}\right]\in \mathbb{Q}_{>0}^n$ is a vector of positive rational numbers with relatively prime numerator and ...
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1answer
68 views

Constructing a non-empty perfect set of real numbers that does not contain rationals.

Duplicate: Perfect set without rationals My approach: We consider the set $[e, \pi]$. I am trying to "cover" the rationals by enclosing each one of them by open intervals with irrational endpoints, ...
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156 views

Is $z=e^{\frac{1}{\log(x)}}$ a solution to a polynomial?

Is $z=e^{\frac{1}{\log(x)}}$ a solution to a polynomial? $x\in\Bbb Q~\cap(0,1).$ Wolfram is telling me $z$ is algebraic but I'm not sure that I believe this. I believe it would follow from Schanuel'...
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68 views

Is even divided by even a rational or irrational number? [closed]

For any rational number, $\frac{p}{q}$ , $p$ and $q$ should be integers, $q\neq0$ and $p,q$ should not have any common factors. Now, if we have two even numbers, say $2m$ and $2n$ where $m$ and $n$ ...
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1answer
46 views

Rational topological basis for Euclidean topology - topology without tears 2.2.3

Context: self-studying topology without tears, now at question 2.2.3. Question: Let $\mathcal{B}$ be the collection of all open intervals $(a,b) \in \mathbb{R}$ with $a \lt b$ and $a, b$ rational ...
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1answer
39 views

Can a power series with rational cffs. that sum to irrational lim evaluate to rational lim at non-zero rational point?

Assume we have $f(x) = \sum a_n x^n, a_n \in \mathbb{Q}$, and convergent $f(1) \notin \mathbb{Q}$. Assuming $f(x)$ converges at some $f(x \in \mathbb{Q})$, is it possible for $f(x \in \mathbb{Q}) \in \...
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2answers
145 views

Intuition for non-convergence of Cauchy sequence in $\mathbb{Q}$

Suppose we were standing on the rational line at the point 3. Then we took a step to the point 3.1, then to 3.14, etc. (Cauchy sequence of decimal approximations of $\pi$). Suppose, also, that it ...
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2answers
60 views

Is the additive rational group $\mathbb{Q},+$ generated by $\frac{1}{p}$ where p is a prime?

So it is known that the additive group of rationals numbers $\mathbb{Q},+$ is generated by $\frac{1}{n}$ with $n \in \mathbb{N_0}$ so that: $$\mathbb{Q},+ =grp\{\frac{1}{n} | n \in \mathbb{N}\}$$ Now ...
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5answers
509 views

Proof that all real numbers have a rational Cauchy sequence?

I saw in an article that for every real number, there exists a Cauchy sequence of rational numbers that converges to that real number. This was stated without proof, so I'm guessing it is a well-known ...
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1answer
1k views

Is the set of real numbers really uncountably infinite?

The proof that the set of real numbers is uncountably infinite is often concluded with a contradiction. In the following argument I use a similar proof by contradiction to show that the set of ...
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2answers
190 views

Can $\frac{1}{2a}\left(-b+\sqrt{b^2-4ac}\right)$ be rational if $a=3n_1$, $b=-3n_1^2$, $c=n_1^3-n_2^3$, for positive rational $n_i$ with $n_1<n_2$?

Let $n_{1}$ and $n_{2}$ be positive rational numbers such that $n_{1}<n_{2}$. Let $a=3n_{1}$, $b=-3n_{1}^2$, $c=n_{1}^3-n_{2}^3$. Can $$\frac{-b+\sqrt{b^2-4ac}}{2a}$$ be a rational number? In ...
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1answer
58 views

Can the Average of an Infinite Number of Rational Numbers be Irrational?

In game theory, there is something called the Folk Theorem, which basically says that you can create a special strategy for any average of possible payoffs as long as the average payoffs are better ...
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6answers
36k 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 ...