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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43 views

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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1answer
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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3answers
742 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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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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3answers
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
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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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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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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1answer
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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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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43 views

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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Integer multiply by rational without overflow in fixed size

I want to accurately compute a function with the below signature ...
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2answers
65 views

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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1answer
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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2answers
40 views

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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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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2answers
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
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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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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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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3answers
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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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2answers
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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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
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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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
318 views

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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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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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
88 views

Enumerate rational numbers in ascending order [duplicate]

Rational numbers are in 1-1 correspondence with natural numbers. For example, let's consider enumeration mentioned in wikipedia: https://en.wikipedia.org/wiki/Rational_number#Properties (https://en....
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5answers
65 views

Limit approaching 0 of a rational defined function

For $f(x)=\begin{cases} e^{x^2}-1, & x \in \mathbb Q\\[2ex] 0, & x \not\in \mathbb Q \end{cases} $ Evaluate $\lim_{x\to 0}f(x)$. Can someone point me in the right direction? I have no ...
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0answers
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Finding a rational map which creates an independent vector of an irrational plane

By rational subspace, we mean a subspace of $\mathbb R^n$ which admits a rational basis. The question. Let $A$ be a $2$-dimensional subspace of $\mathbb R^n$. We can assume that for all rational ...
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0answers
73 views

What is the correct name for a non-whole real number?

I apologise for the simple nature of this question but I can't find the answer. I know that a whole number is an integer. I also know that a number that can be expressed as the quotient of two ...
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6answers
282 views

In the proof of irrationality of $\sqrt{2}$ or $\sqrt{7}$, why do the numerator and denominator of a rational number have to be in their lowest term?

I have been confused by this problem for a very long period of time, and I think I am personally opposed to this concept and refused to agree with it in my introduction to mathematical proof course. ...
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4answers
103 views

Is $9$ the limit of $\frac{987654321}{123456789}, \frac{998877665544332211}{112233445566778899}\ldots$

I was playing with my calculator and found out $987654321$ $÷$ $123456789$ is very close to $8$. Trying out some more numbers i saw $998877665544332211$ $÷$ $112233445566778899$ gives $8.9$. When when ...
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2answers
97 views

Value of cos 1 (angle is in radians)?

How can we calculate the value of $\cos 1$ where the angle is in radians (and not degrees). If this isn't possible, can we somehow find whether this value would be rational or irrational? P.S: I know ...
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1answer
34 views

Would a drawing of the rationals on a number line create the appearance of a line, nothing, or both?

On a number line, the rationals are dense in R, meaning there are infinite rationals between rationals. However, there are infinite undefined irrationals with a Lebesgue measure of 1. So how do we ...
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0answers
17 views

Rational points in $\mathbb{R}^2$ on a set of lower Hausdorff dimension

Suppose $X \subseteq \mathbb{R}^2$ is a set of Haudorff dimension less than or equal to $1$. Suppose also that $X$ is compact. I was wondering can we obtain a bound for the following quantity? $$ \#\{ ...
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1answer
66 views

How many Unique numbers?

$\require{cancel}$ I saw a meme that fraction $$\frac{163}{326}=\frac{1\cancel6\cancel3}{\cancel{3}\cancel{6}2}=\frac{1}{2}$$ And It means that $$1\leq a_i,b_i \leq 9, a_i,b_i \in \mathbb{N},\\\sum_{i=...
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2answers
46 views

About rational numbers as Dedekind cuts.

I am reading "An Introduction to Calculus" by Kunihiko Kodaira. There is Theorem 1.3 in this book and I am very confused. We identify a rational number $r$ with a Dedekind cut $(R, R')$ where $R = \{q ...
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1answer
18 views

Two queries on triangles whose side lengths are rational

Let us define a 'rational triangle' as one with lengths of all sides rational. We are aware that a positive integer is called 'congruent' only if it is the area of a RIGHT triangle with rational ...
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2answers
88 views

Is there any way to avoid using Axiom of Choice in proving this theorem?

I asked for proof verification of a proof about nest of intervals here, where I appeal to a theorem: Theorem: Let $a,b \in \mathbb R$ such that $a <b$ and $X := \{p \in \mathbb Q \mid a<p<...
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1answer
72 views

How are the Periods of the Decimal Expansions of $\frac{p}{q}$ and $\frac{q}{p}$ Related?

In an excellent post several years ago, we learn that the period of the decimal expansion of a rational number $\frac{p}{q}$ must divide the multiplicative order of $10\pmod q$ assuming that there are ...