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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$(-3)^{3/2} \neq (-3)^{6/4}$

$(-3)^{\frac{3}{2}}=-3\sqrt{3}i$ $(-3)^{\frac{6}{4}}=\sqrt{27}$ (not the same thing). What's the deal? It's interesting because people work with fractional exponents all the time and I've never ...
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3answers
92 views

Finding the simplest rational in a closed interval

Given a closed interval [a,b], how would you find the "simplest rational", p/q, contained in that interval. By simplest, I mean the rational with the smallest denominator q. You may, if you wish ...
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2answers
46 views

Is the sum of rational exponentials a rational exponential?.

Prove or disprove that $\forall a,b \in \mathbb{Q}^+$ and $ \forall p,q \in \mathbb{Q}$ there exists $c \in \mathbb{Q}^+$ and $r \in \mathbb{Q}$ such that: $$ a^p+b^q=c^r $$
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3answers
74 views

Path connectedness of the set of points $(x,y)$ where $x$ is rational or $y$ is rational [duplicate]

Prove that $X=\{(x,y) :x\text{ is rational or }y\text{ is rational}\}$ is path connected. So for every $(x,y)$ in $X$, I need to find a continuous function $f$ on $[a,b]$ such that $f(a)=x$ and ...
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0answers
41 views

How many are there triangles with different rational sides, rational area, bisectrixes and 1 rational median?

I've been searching triangles with all elements being rational numbers. However, I've found somewhere on Internet proof that it's not possible. Then, I was searching triangles with maximal possible ...
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0answers
52 views

Does the equation $\tan(x)=y$ have any non-zero rational solution?

Trivially $\tan(0)=0$ but it seems this is the "unique" solution of the equation $\tan(x)=y$ on rational numbers. In fact if we try to make $y$ rational we usually use irrational (transcendental) ...
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3answers
97 views

Quadratic Equations - One rational solution?

I have a question that I am working on: Which of the following will give one rational solution? 4x^2 = 9 4x^2 - 12x = -9 x^2 = 5 x^2 - 2x + 14 = 0 2x^2 = x I am ...
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3answers
71 views

How to prove that the function $f(x) = 2 \left \lfloor x \right \rfloor - x$ is one to one for rational $x$?

How to prove that the function $f(x) = 2 \left \lfloor x \right \rfloor - x$ is one to one for rational $x$? I believe that I will have to somehow use the fact that the $\left \lfloor x \right ...
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2answers
60 views

$a_1^k+a_2^k+\ldots+a_n^k$ integer implies all integers?

Let $n$ be a positive integer, and let $a_1,\ldots,a_n$ be rational numbers. Suppose that $a_1^k+a_2^k+\ldots+a_n^k$ is an integer for all positive integers $k$. Is it true that $a_1,a_2,\ldots,a_n$ ...
3
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1answer
84 views

Prove that $\mathbb{Q}[\sqrt{p}]$ is a field between $\mathbb{Q}$ and $\mathbb{R}$

I really have trouble understanding a task. We've got $p\in$ P, while P are all prime numbers. Now we construct a field $$\mathbb{Q}[\sqrt{p}]:=\{x+y\sqrt{p}:x,y \in \mathbb{Q}\}$$ The Task is to ...
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0answers
13 views

Given conditions on the fraction, can we find a 'best rational approximation'

Just something I thought of and I'm curious about. Say I tell you I want to approximate $\pi$ using a rational number. However, I am going to tell you that the numerator is to be at most $m$ ...
0
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1answer
39 views

Is the set $ \{(p_1,p_2,\dots, p_n):p_i\in \mathbb Q\}$ connected?

Let $X=\{(p_1,p_2,\dots, p_n):p_i\in \mathbb Q\}$. Is $X $ connected or disconnected? My attempt:$X$ is connected iff any two points of $X$ are contained in a connected subset of $X$. This ...
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0answers
29 views

Properties of digit functions for numbers in $[0,1]$

Consider a function $g(n): \mathbb N \to \{0,1,2,3,4,5,6,7,8,9\}$, ie. $g$ maps the natural numbers to natural numbers between $0$ and $9$. Then, no matter what $g(n), \ n\in \mathbb N$ is, the sum ...
4
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1answer
60 views

$\sqrt{x^2+y^3}$ and $\sqrt{x^3+y^2}$ are rational

Are there infinitely many pairs of different positive rational numbers $x,y$ such that $\sqrt{x^2+y^3}$ and $\sqrt{x^3+y^2}$ are rational? Consider such a pair. Then we have $x^2+y^3=a^2$ and ...
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2answers
132 views

Can the b-adic representation of rational numbers (by quote notation) be extended to non-terminating expansions?

The wikipedia article on p-adic numbers warns about $b$-adic expansions where $b$ is not a prime: Although for p-adic numbers p should be a prime, base 10 was chosen to highlight the analogy with ...
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3answers
238 views

Subtraction of two repeating decimals

When I was looking at the proof that every repeating decimal is rational, I came across this example: $x=5.33333333\ldots$ ($3$ repeat indefinitely) $10x=53.3333333\ldots$ ($3$ repeat indefinitely) ...
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2answers
44 views

Prove the result is always a rational number

I am trying to prove the following: If $a$ and $b$ are non-zero rational numbers, then $a^{b}$ is rational. Any ideas or hints how to prove this?
3
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1answer
73 views

Multiplying and adding fractions

Why multiplying fractions is equal to multiply the tops, multiply the bottoms? $$\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b \times d},$$ And why $$\frac{a}{b}\times \frac{c}{c}=\frac{a}{b},$$ ...
3
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1answer
73 views

Identify irrational basis of $\mathbb{Q}$-vector space

A real sequence $\mathbf{x}=(x_k)_{k\in\mathbb{N}_0}$ is of the form $$ x_k=\alpha r_k,\quad \alpha\in\mathbb{R}\backslash\mathbb{Q},\quad r_k\in\mathbb{Q},\tag{*} $$ if and only if all of its terms ...
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2answers
75 views
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7answers
603 views

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 ...
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1answer
34 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 ...
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3answers
87 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
70 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?
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3answers
28 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$\ \lim_{x\to 0} \frac{f_1(x)-f_2(x)}{f_3(x)}$ ...
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3answers
109 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 ...
2
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2answers
47 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 ...
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2answers
154 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
189 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
35 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
113 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 ...
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0answers
57 views

Digamma equation identification

I was messing around with the digamma function the other day, and I discovered this identity: $$\psi\left(\frac ...
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4answers
71 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?
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1answer
111 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
143 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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3answers
186 views

Prove that if $x$ and $y$ are rational numbers and $y\ne 0$, then $x/y$ is a rational number [duplicate]

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
67 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
58 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 ...
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3answers
257 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
983 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
74 views

How find this minimum of the $q$, if such $\frac{95}{36}>\frac{p}{q}>\frac{96}{37}$

let $p,q$ is postive integer,and such $$\dfrac{95}{36}>\dfrac{p}{q}>\dfrac{96}{37}$$ Find the minimum of the $q$ maybe can use $$95q>36p$$ and $$37p>96q$$ and then find this minimum of ...
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2answers
109 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 ...
3
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3answers
108 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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1answer
79 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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1answer
34 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
32 views

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
262 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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3answers
66 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. ...
1
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2answers
59 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} ...