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

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Is the number 0.2343434343434.. rational? [duplicate]

Consider the following number: $$x=0.23434343434\dots$$ My question is whether this number is rational or irrational, and how can I make sure that a specific number is rational if it was written in ...
-2
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2answers
69 views

If$\ f(n)-g(n)$ tends to a rational number as$\ n \to \infty$, might we say that either both have an “irrational growth” or a “rational” one? [closed]

Let me explain myself.$\ x-y=\frac{a}{b}$ if and only if either$\ x$ and$\ y$ are both rational or irrational. Does this also mean that with$\displaystyle\lim_{n \to +\infty} f(n)=\lim_{n \to +\infty} ...
1
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1answer
31 views

Let$\ \lim_{n\to \infty} \frac{ \ln n}{f(n)}=1$. If$\ a,b,c$ are natural, can we have$\ a^{b+c \ln n}\sim a^{c f(n)}$?

I shall note that$\ n$ as well goes through the natural numbers and that$\ f(n)$ is rational for any$\ n$. Also, I'm obviously excluding$\ a=1$. I'm inclined to think my claim is not possible ...
1
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3answers
72 views

As$\ n \to \infty$, can a transcendental function$\ f\left(1+ \frac{1}{n}\right)$ to the power of$\ n$ tend to a rational power of$\ e$?

Let$\ f(n)$ be a transcendental function$\ \ne e^{g(n)}$, for any function$\ g(n)$. Does$$\ \lim_{n \to \infty} \left(f\left(1+ \frac{1}{n}\right)\right)^n =e^{ -k} = \lim_{n \to \infty} \left(1 - ...
4
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1answer
59 views

Given $\alpha$, can we always find $\beta$ such that both $\sin(\alpha+\beta)$ and $\sin(\alpha-\beta)$ are rational?

Given $\alpha$, can we always find $\beta$ such that both $\sin(\alpha+\beta)$ and $\sin(\alpha-\beta)$ are rational?
0
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1answer
19 views

Quotient of two rational sequences and the nature of its limit

Suppose we have two sequences of rational numbers, $(p_i)_{i=1}^\infty$ and $(q_i)_{i=1}^\infty$, and suppose $$\lim_{i\to\infty}\frac{p_i}{q_i}=c<\infty,$$ where $c$ is known. Are there any ...
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3answers
25 views

Can a limit of form$\ \frac{0}{0}$ be rational if the numerator is the difference of transcendental functions, and the denominator a polynomial one?

Let$\ f_1(x)$ and$\ f_2(x)$ be transcendental functions such that$\ \lim_{x\to 0} f_1(x)-f_2(x)=0$, and$\ f_3(x) $ polynomial, such that$\ f_3(0)=0$. Can$\ \lim_{x\to 0} \frac{f_1(x)-f_2(x)}{f_3(x)}$ ...
2
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3answers
97 views

$\mathbf{Q}$ basis of $\mathbf{R}$.

Could someone give me an explicit basis of $\mathbf{R}$ as a vector space over $\mathbf{Q}$? I no some linearly independent subset, namely $1,e,e^2,\ldots$ but this seems to be a deep result already ...
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2answers
26 views

rational number plane vector space or not?

Two questions: 1. Is $\mathbb{Q}^2$ a vector space over the field $\mathbb{Q}$? 2. Is $\mathbb{Q}^2$ a vector space over the field $\mathbb{R}$? My answer to the first question is yes. Because the ...
2
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2answers
98 views

Is the set of rationals between $\sqrt{2}$ and $\sqrt{3}$ open or closed in $\mathbb Q$?

Consider the set of all rationals, $\mathbb Q$ as a subset of the set of all reals $\mathbb R$. Assign $\mathbb Q$ the subspace topology induced by the standard topology on $\mathbb R$. ...
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4answers
49 views

question about the proof about the square root of natural numbers [duplicate]

Could someone please help me to prove that for $t \in \mathbb{N}$ , $\sqrt{t} \in \mathbb{Q} $ if only if $\sqrt{t} \in \mathbb{N}$
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2answers
22 views

Decimals and Rational numbers

How do you prove: Q1) Why is every rational number (say m/n, where m and n are both positive integers) either a terminating or a repeating decimal? Q2) Why is every repeating decimal (or terminating ...
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2answers
91 views

lowest denominator to lie between to rational numbers.

What's the lowest $ m\in \mathbb {N} $ such that the exists an $ n $ with $1/3 >\frac {n}{m}>33/100$? note that there used to be a typo in the inequality which gives the opposite sign I'm on my ...
0
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1answer
41 views

Irrational power of root

Let $a$ and $b$ be rational numbers, such that $\sqrt{a}$ and $\sqrt{b}$ are irrational. Can $\sqrt{a}^\sqrt{b}$ be rational? I found examples, where the irrational power of an irrational number is ...
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0answers
24 views

Digamma equation identification

I was messing around with the digamma function the other day, and I discovered this identity: $$\psi\left(\frac ...
1
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4answers
63 views

Real Between Rationals

Let $x$ be a real number. Show that, for any $\varepsilon>0$, there exist two rationals $q$ and $q'$ such that $q<x<q'$ and $|q-q'|<\varepsilon$ How should I approach this prove?
9
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4answers
179 views

Show that $({\sqrt{2}\!+\!1})^{1/n} \!+ ({\sqrt{2}\!-\!1})^{1/n}\!\not\in\mathbb Q$

How could we prove that there does not exist a positive integer $n$, so that the number $$({\sqrt{2}+1})^{1/n} + ({\sqrt{2}-1})^{1/n}$$ is rational? I think it could be done inductively from a more ...
0
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1answer
53 views

Monotone increasing sequence of rationals with an irrational limit

I am trying to use rationals in order to approximate irrationals. Is it possible to construct a monotonically increasing sequence of rationals the limit of which is an irrational? If so, how?
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4answers
93 views

Example of a non-trivial function such that $f(2x)=f(x)$

Could you give an example of a non-constant function $f$ such that $$ f(x) = f(2x). $$ The one that I can think of is the trivial one, namely $\chi_{\mathbb{Q}}$, the characteristic function on the ...
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1answer
68 views

What can be said about $\pi+e$ and $\pi e$? Are these numbers rational or irrational? [duplicate]

"homework" What can be said about $\pi+e$ and $\pi e$? Are these numbers rational or irrational? I know that both $\pi$ and $e$ are irrational. What can be said about $\pi+e$, and $\pi e$?
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3answers
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Prove that if $x$ and $y$ are rational numbers and $y\ne 0$, then $x/y$ is a rational number

Prove that if $x$ and $y$ are rational numbers and $y\ne 0$, then $\frac{x}{y}$ is a rational number. How do I prove this, and also which proving method would I use? I'm confused between that and ...
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3answers
55 views

There is at most one way to represent a number as $a+b\sqrt 2$ with rational $a,b$

If $a,b,c,d\in\mathbb Q$ and $a+b\sqrt 2= c + d\sqrt 2$, then prove $a=c$ and $b=d$ ? I don't have any idea to solve this , it's freaking me out.
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3answers
53 views

Find all $n\in \mathbb N$ such that $\sqrt{n+7}+\sqrt{n}$ is rational.

Find all $n\in \mathbb N$ such that $\sqrt{n+7}+\sqrt{n}$ is rational. By inspection it is pretty easy to see that the only $n$ that will work is $n=9$. Because the distance between perfect squares ...
3
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3answers
245 views

Using a given identity to solve for the value of an expression

This problem caught my eye in the book yesterday. Till now I still get stuck. Here it is: If $$\frac{x}{x^2+1}=\frac{1}{3},$$ what is the value of $$\frac{x^3}{x^6+x^5+x^4+x^3+x^2+x+1}?$$ The ...
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4answers
875 views

Can any two irrational numbers NOT of the form (m+A) and (n-A) be added to produce a rational number?

$m$ and $n$ being rational numbers, A being an irrational number. I was wondering if two irrational numbers when added always yield an irrational number. All the counter-examples I could find were of ...
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2answers
80 views

$\mathbb{Q}$ can not be embedded in $\mathbb{Z}$

Show that $\mathbb{Q}$ can not be embedded in $\mathbb{Z}$ (where both has the subspace topology of $\mathbb{R}$) My attempt at a solution Since Z is discrete, {k} is open in $\mathbb{Z}$ with ...
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3answers
77 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 ...
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0answers
13 views

Smallest survey size algorithm

I was inspired to think of this problem when I saw a commercial saying "97.1% of audiences loved this movie!". After fiddling around with some numbers, I realised that at least 34 people must have ...
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1answer
46 views

Usage of decimal expansion

I learned about the rigorous construction of rationals as a set of equivalence classes of ordered integers with operations defined on this set. I understand that the decimal expansion is another way ...
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0answers
22 views

Verify if the following demonstration is correct

THEOREM: "There are $a, b$ irrationals, such that $a^b$ is rational." PROOF: " If $\sqrt{2} ^ \sqrt{2}$ is rational, $a = b = \sqrt{2}$, otherwise, $a = \sqrt{2} ^ \sqrt{2}$ and $b = \sqrt{2}$, so ...
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1answer
26 views

Given that $a>1$, show that the exponential function $a^x$ is increasing for $x\in\mathbb{Q}$

The assumption one can make here is that it is increasing for $x\in\mathbb{Z}$. I have tried to make a proof but I'm not sure if it is valid. Here it goes. Say $x,y\in\mathbb{Q}$. Then they can be ...
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0answers
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How many rationals for a given $n \in \Bbb N \;\backslash \{1\}$?

Fix $n \in \Bbb N, n> 1$. Now choose a two digit base-$n$ number $ab $ say. There's $n^2$ choices for this. Consider the number $0.c_1 c_2 c_3 \ldots$ where the $c_i$ are defined recursively: ...
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9answers
2k views

Can rational numbers have decimals?

I had a question in my exam paper - Which of the following is not a rational number? a) $\sqrt{25}$ b) $\sqrt{45}$ c) $\sqrt\frac{256}{225}$ d) $\frac{3}{4}$ The answer to this is b. Now, ...
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2answers
246 views

What do rationals represent?

While learning about the construction of number systems, I realized that I had many misunderstandings of crucial concepts which I was learning intuitively. I recently learned about the construction of ...
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0answers
14 views

What is a class of equations? Which class would require the rational numbers to guarantee a solution?

What is a class of equations? Which class would require the rational numbers to guarantee a solution? The biggest problem I am having is that I am unsure what exactly a class of equations is. The ...
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3answers
55 views

Is a complex fraction considered part of the rationals?

I have always been taught that $\mathbb{Q}=\{ \frac{a}{b}|\,\,a,b\in \mathbb{Z},\, \,b\neq0\}$. Is this definition of the rationals limited? Could it also be true that a complex fraction, i.e. ...
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2answers
49 views

Cantor diagonal argument; related number

I was reading another question on mse about cantors proof and I'm curious about a number that could be defined from it. Well there could be a whole heap of them, but one for now. Define $A=\Bbb{Q} ...
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3answers
49 views

Correctness of proof that every positive rational with square $>2$ is an upper bound for those with square $<2$

I would like to know whether my proof makes sense or not, and if not where should it be corrected. Let $E=\{x \text{ is rational }: x>0 \text{ and } x^2<2\}.$ Claim: Every member of $F=\{x ...
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2answers
57 views

Correctness of the proof that the set $\{x \in \mathbb{Q} : x>0 \text{ and } x^2>2\}$ does not have a smallest element

Let $F=\{x \in \mathbb{Q} : x>0 \text{ and } x^2>2\}$. I am asked to show that $F$ does not have a smallest element. The hint is to simply prove the claim: 'If $p$ is a rational number in ...
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0answers
119 views

The topology generated by open intervals of rational numbers

Let $B = \{ \mathbb{R} \} \cup \{ (a,b) \cap\mathbb {Q} \ ,\ a\lt b \ ,\ a,b \in\mathbb{Q}\}$ Thus, a set $V \in B$ if it is either equal to $\mathbb{R}$ or if it is in the intersection of ...
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1answer
55 views

Rational values of $\sin(\log(x))$

Apart from the trivial solution $\sin(\log(1))=0$, is $$\sin(\log(x))$$ ever rational if $x$ is rational?
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1answer
46 views

Length of a rationals period in base $b$

Okay working in base $b$ we are given a fraction of form $\frac{p}{q}$ with $p$ and $q$ coprime. We also assume that $b$ and $q$ are coprime so $\frac{p}{q}$ is purely periodic in base $b$. The ...
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2answers
463 views

Prove that $x\in\mathbb Q$ [closed]

Let $a\in\mathbb Q$ and $a>\dfrac43$. Let $x\in\mathbb R$ and $x^2-ax,x^3-ax\in\mathbb Q$. Prove that $x\in\mathbb Q$.
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4answers
117 views

Is there a bijection from a bounded open interval of $\mathbb{Q}$ onto $\mathbb{Q}$?

It is easy to create a bijection between two bounded open intervals of $\mathbb{R}$, such as: $$ \begin{align} f : (a,b) &\to (\alpha,\beta) \\ x &\mapsto \alpha+(x-a)(\beta-\alpha). ...
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1answer
35 views

Comparing Fractional Numbers

Does a formula exist for comparing two fractional numbers, without resolving to using anything other than integers and fractions? (Thus not real numbers). In other words: given $\dfrac{a}{b}$ and ...
1
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0answers
61 views

Rational numbers imply reals?

I was solving an inequality today and I proved it for rational numbers (it was easier because I was able to "strengthen" by doing things like "$\frac{p}{q}>\frac{r}{s}\implies ps\ge qr+1$ since ...
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1answer
95 views

Show that $\sqrt{2}$ is irrational using integer root theorem

Show that $\sqrt{2}$ is irrational using integer root theorem. Let $P(x)=x^2-2$. Since $\sqrt{2}$ is a root of this polynomial, had it been a rational (suppose $\sqrt{2}=\frac{p}{q}$) no, by ...
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2answers
139 views

Inequality with $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}$

Inspired by this recent question, I suggest this. Let $n=2,3,4, \ldots .$ Then $$ \frac{7}{12} < \cfrac 1 {1 + \cfrac {1^2} {1 + \cfrac {2^2} {\ddots + \cfrac \vdots { 1 + \, {n^2} \,}}}} \leq ...
60
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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 ...