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

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I need to find a rational numbers series that converging to irrational number [duplicate]

I found a series that is $a_{n+1}=\frac{a_n^2 + 2}{2a_n}$ yet I'm not sure. can someone give me a more umm solid example? thanks.
53
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3answers
729 views

A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...
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1answer
102 views

Does there exist a unique closest natural number to each rational number?

So here's the question: Prove or disprove: For every $x \in \mathbb{Q}$, there is a unique $n \in \mathbb{N}$ which is the closest natural number to $x$. I know we can define a rational number ...
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2answers
51 views

If I raise a simplest-terms rational to an integer power, is the result automatically in simplest terms?

Given a rational number $a/b$ expressed in simplest terms (so $GCD(a,b)=1$), I want to raise it to an integer power $n$. I think the result will always automatically be in simplest terms, but it's a ...
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0answers
84 views

Related to $\pi$ and $\tau$ - are these numbers transcendental, irrational, or rational?

Here are three numbers whose decimal expansions are listed in OEIS. Are they transcendental, irrational, or rational numbers? Note: $\tau = 2*\pi$ and the last two values are in radians. A233700. ...
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2answers
266 views

Field containing all square roots of rational numbers

What is the smallest field which contains all square roots of positive rational numbers? I guess I mean “smallest” in terms of set inclusion, i.e. the minimal one with regard to the “$\subseteq$” ...
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4answers
899 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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6answers
864 views

Alternate proof for “$\log_{10}{2}$ is irrational”

I need to prove that $\log_{10}{2}$ is irrational. I understand the way this proof was done using contradiction to show that the even LHS does not equal the odd RHS, but I did it a different way and ...
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22answers
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 ...
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7answers
554 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
153 views

Equality of positive rational numbers, Part-2

I am reading this answer. I have some doubts which I want to clarify. Question 1. The author defines a rational number $\dfrac ab$ as, $$b\times\left(\dfrac{a}{b}\right) = a$$ He presumes that ...
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1answer
20 views

Integrating the normal distribution over rational numbers?

Is it possible to integrate the normal distribution over rational numbers? What is the value of such integral? Is it $\pi$ minus the integral over irrational numbers?
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1answer
31 views

Sets of irrationals whose square contains a rational

Let $S$ be a subset of the irrationals. Also, lets assume that $S$ has infinitely many elements. My very general question is, under what non-trivial conditions does there exist an element $x\in S$ ...
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0answers
52 views

The cube of at least one irrational number is rational

I am supposed to prove the statement above. Here is what I have so far Suppose that the cube of at least one irrational number $n$, is rational. By definition of rational, there exists ...
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1answer
39 views

Real number system

Is the set of rationals a subset of the irrationals? I always assumed it was, but given that irrationals are defined to be numbers that have an infinite, non-repeating decimal expansion, there cannot ...
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1answer
56 views

About the continuity of $f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k}$

Let $q: \mathbb{N} \to \mathbb{Q}$ be a bijection and denote the image of $k \in \mathbb{N}$ by $q_k$. Let $f: \mathbb{R} \to (0,1)$, $$ f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k} ...
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8answers
5k views

Proof that every repeating decimal is rational

Wikipedia claims that every repeating decimal represents a rational number. According to the following definition, how can we prove that fact? Definition: A number is rational if it can be written ...
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3answers
307 views

Double limit of $\cos^{2n}(m! \pi x)$ at rationals and irrationals

I stumbled upon this "relation" (is the name correct?): $$ \lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases} 1,&x\text{ is rational}\\ 0,&x\text{ is ...
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0answers
42 views

Why is this function surjective?

$f: \mathbb Z \times \mathbb N \rightarrow \mathbb Q$ is surjective? When by definition $\frac pq,\; p, q \in \mathbb Z$ and they can be both positive and negative and $q \neq 0$. Is it possible to ...
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1answer
109 views

Multiplication of positive fractional numbers.

I am reading this answer. We know that multiplication of two positive rational number $\dfrac{a}{b}$ and $\dfrac{r}{s}$ respectivly is defined as follows: $\dfrac{a}{b}\times\dfrac{r}{s} = ...
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5answers
207 views

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

How could we prove that for every positive integer $n$, the number $$({\sqrt{2}+1})^{1/n} + ({\sqrt{2}-1})^{1/n}$$ is irrational? I think it could be done inductively from a more general ...
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3answers
100 views

Proving Floor and Ceiling of a Rational Number

Suppose x,y $ \in \mathbb{Z}^+ $ Prove $\lceil x/y \rceil = \lfloor (x-1)/y \rfloor + 1$ I was considering using the definition of floor and ceiling to prove this. But this does not seem like a ...
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2answers
69 views

An isomorphic map from natural numbers to positive rational numbers that preserves addition, multiplication and order

Since $\mathbb{Q}^{+}$ is countable, there is a bijection between $\mathbb{Q}^{+}$ and $\mathbb{N}$ (0 included). Then the question now is, can we go further by constructing an isomorphic map between ...
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1answer
45 views

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

$(-3)^{\frac{3}{2}}=-3\sqrt{3}i$ $(-3)^{\frac{6}{4}}=\sqrt{27}$ (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
46 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
40 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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2answers
114 views

Show that $X = \{ (x,y) \in\mathbb{R}^2\mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\}$ is path connected. [duplicate]

How do I show that $X = \left\{ (x,y) \in \mathbb{R}^2 \mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\right\}$ is path connected? Note that $X$ is a topological space with subspace topology ...
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3answers
42 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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3answers
49 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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4answers
6k views

Is a Whole Number A Rational Number

Is a Whole Number part of A Rational Number or a whole number??
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0answers
32 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
44 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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4answers
3k views

What is a fraction in which the greatest common factor of the numerator and the denominator is 1?

What is this fraction: A fraction in which the greatest common factor of the numerator and the denominator is 1?
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3answers
64 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
59 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 $a_1,a_2,\ldots,a_n$ ...
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3answers
123 views

Irrationality of $n$-th root of positive rationals other than $1$

If $a,b \in \mathbb Z^+ , a \ne b$ then is it true that $\sqrt[a+b]{\dfrac ab}$ is irrational ? This question actually popped up from seeing whether there exists a non-trivial homomorphism from ...
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0answers
9 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 $m$ ...
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3answers
51 views

How to prove that the intersection of $\{p+q \sqrt{2} \mid p,q \in \mathbb{Q}\}$ and $\{r+s \sqrt{3} \mid r,s \in \mathbb{Q}\}$ is $\mathbb Q$? [closed]

Let $S=\{p+q \sqrt{2} \mid p,q \in \mathbb{Q}\}$ and $T=\{r+s \sqrt{3} \mid r,s \in \mathbb{Q}\}$. Prove that $S \cap T = \mathbb{Q}$. We are currently studying proofs by contradiction, but I don't ...
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1answer
43 views

Can any root, such as a square root or a cube root, be rational?

I've heard of this and most roots are irrational such as $\sqrt{8}$ and $\sqrt[3]{25}$. So, can any of these roots be rational? I think so as I'm typing this. I think these are rational: ...
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1answer
37 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
22 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 ...
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2answers
46 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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1answer
57 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
37 views

Approximation of numbers [closed]

How could we approximate an irrational number by rationals?? Could you give me some hints?? I don`t have any idea how we could approximate them by rationals...
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1answer
49 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
42 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?
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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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1answer
33 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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1answer
61 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?