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.

Filter by
Sorted by
Tagged with
0
votes
1answer
42 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|...
4
votes
1answer
86 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 $\sqrt{m}$ ...
3
votes
2answers
59 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=...
1
vote
2answers
73 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 $\...
1
vote
1answer
70 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,...
2
votes
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 $K$ ...
10
votes
1answer
223 views

$\mathrm{tr}(A^4)$ is a rational number [closed]

Let $A \in \mathcal{M}_3(\mathbb{C})$ such that $\mathrm{tr}(A^2) = \mathrm{tr}(A^3) \in \mathbb{Q},$ where $\mathrm{tr}(A)$ is the trace of $A.$ It is possible to prove that $\mathrm{tr}(A^4) \in \...
1
vote
1answer
43 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 $...
1
vote
2answers
41 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 ...
2
votes
3answers
53 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 ...
3
votes
2answers
120 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 \...
4
votes
1answer
51 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}) \...
0
votes
1answer
37 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 ?
6
votes
0answers
122 views

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 $u_n=(1+\frac{1}{n})^n$ ...
1
vote
1answer
35 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: $...
2
votes
2answers
41 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 ...
1
vote
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, ...
8
votes
0answers
198 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'...
-3
votes
3answers
85 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 $n$ ...
0
votes
1answer
66 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 ...
8
votes
2answers
324 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: (...
3
votes
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 \...
2
votes
2answers
65 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 ...
5
votes
2answers
157 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 ...
17
votes
2answers
334 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 $\mathrm {GL}_n(\mathbb Q)$ ...
2
votes
1answer
63 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 ...
8
votes
2answers
194 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 ...
0
votes
1answer
101 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....
2
votes
5answers
67 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 ...
0
votes
0answers
5 views

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 ...
3
votes
0answers
76 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 ...
1
vote
6answers
293 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. ...
0
votes
4answers
126 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 ...
3
votes
2answers
115 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 ...
0
votes
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 ...
0
votes
0answers
18 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? $$ \#\{ ...
2
votes
1answer
73 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=...
0
votes
2answers
50 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 ...
0
votes
1answer
20 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 ...
4
votes
2answers
94 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<...
3
votes
1answer
73 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 ...
1
vote
1answer
34 views

What is the total production??

From the total production of pencils of a factory, they sell half of their production plus 300 pencils to a company, then, they sell 1/3 of what is left minus 200 pencils to a second company and then, ...
3
votes
2answers
56 views

What does $\varepsilon \leq 1 \wedge s$ mean?

I'm reading Section. The Rational Numbers in textbook Analysis I by Amann/Escher. Below is Proposition 10.9: and part of its proof: From what the authors write, I guess that $\color{blue}{\...
1
vote
1answer
57 views

Imprecise Chinese Remainder Theorem with Fractions

I am familiar with how to use CRT on integers, but I have a case where I am operating on fractional values. For instance, say I have the equations $$ x \equiv 6.8 \ \mathbf{mod} \ 10.1$$ $$ x \equiv ...
0
votes
2answers
96 views

Does $y^2 = x^6 - 3x^5 + 3x^4 + 10x^3 + 3x^2 - 3x + 1$ have any rational solutions?

Does $y^2 = x^6 - 3x^5 + 3x^4 + 10x^3 + 3x^2 - 3x + 1$ have any rational solutions? I have some reasonable pre/post graduate Math skills but no access to Magma etc. I suspect there are none other ...
0
votes
1answer
56 views

Do distinct infinities exist?

We all know that there are the infinite sets $Q$ and $R$. I have been taught, and also read that there is one to one correspondence between the elements of both sets, and thus both are regarded as ...
1
vote
2answers
60 views

Counter example for Baire's Theorem

Theorem: Let $(X,d)$ be a complete metric space, and let $D_n, n\in \mathbb N$ be open, dense subsets of $X$. Then also $\bigcap_{n\in\mathbb N} D_n$ is dense in $X$. This statement is false if $X$ ...
0
votes
2answers
50 views

Which of the statements are false?

I have this statement: Let $a, b, c, d \in \mathbb{R} - $ {$0$}, with $\quad acd> 0$. If $– 1 < \frac{a}{b} < \frac{b}{c} < \frac{c}{d} < \frac{d}{a} < 1,$ Which of ...
4
votes
1answer
79 views

Can uncountability of reals be proved only from the axioms?

If we define real numbers, as is sometimes done, with field axioms, and order axioms, and completeness (or continuity) axiom, then, rational numbers fulfill field axioms and order axioms, but they do ...
0
votes
0answers
21 views

Rational number sets which grow in cardinality and a related sequence of numbers.

Define the collection of sets $\left\{Q_n\right\}$ as follows: $$Q_1 = \{0\};\quad Q_2 = \left\{\frac{1}{2}\right\} \cup Q_1; \quad Q_3 = \left\{\frac{1}{3}, \frac{2}{3}\right\} \cup Q_2;$$ $$\left\{\...