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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12
votes
0answers
128 views

Do all rational numbers repeat in Fibonacci coding?

In a regular, positional number system (like our decimal numbers), every rational number ends in a repeating sequence (even if it's just 0). For example, 1/3 in base 10 is 0.33333..., in base 5 it's ...
0
votes
2answers
79 views

Let a,b be rationals and x irrational. Show that if $\frac{x+a}{x+b}$ is rational, then $a=b$.

I'm trying to solve the following problems: Let $a$,$b$ be rationals and $x$ irrational. Show that if $\frac{x+a}{x+b}$ is rational, then $a=b$ Let $x$,$y$ be rationals such that $\frac{x^2+x+\sqrt{...
4
votes
1answer
110 views

A rational orbit that's provably dense in the reals?

Iterating the map $\ \ x\ \mapsto\ x-\frac{1}{x},\ \ $ the orbit of initial point $2$ is "probably" dense in $\mathbb{R}$. Is there an explicit rational mapping together with an initial rational ...
0
votes
1answer
51 views

How to prove that $\bar{\mathbb{Q}}=\mathbb{R}$?

How to proof that $\bar{\mathbb{Q}}=\mathbb{R}$, where $\bar{\mathbb{Q}}=\mathbb{Q}\cup\mathbb{Q}^{\prime}$ and $\mathbb{Q}^{\prime}$ are the limit points of $\mathbb{Q}$?
6
votes
1answer
404 views

What's an example of a number that is neither rational nor irrational?

Of course in regular logic, the answer is there aren't any. But in intuitionistic logic, there might be, as seen by this answer: http://math.stackexchange.com/a/1437130/49592. My question is, as per ...
1
vote
1answer
52 views

Determine all positive rational numbers $r \neq 1$ such that $r^{\frac{1}{r-1}}$ is rational?

Here's what I've got so far: Let $r = \frac{a}{b}$, where $a$ and $b$ are integers. We then have $$r^{\frac{1}{r-1}} = \frac{a^{\frac{b}{a-b}}}{b^{\frac{b}{a-b}}}$$ Clearly, $a-b=1$ and $a-b=-1$ ...
18
votes
7answers
1k views

How many sequences of rational numbers converging to 1 are there?

I have a problem with this exercise: How many sequences of rational numbers converging to 1 are there? I know that the number of all sequences of rational numbers is $\mathfrak{c}$. But here ...
3
votes
1answer
98 views

Find the maximum number of rational points on the circle with center $(0,\sqrt3)$

Find the maximum number of rational points on the circle with center $(0,\sqrt3)$ Let the equation of the circle be $x^2+(y-\sqrt3)^2=r^2$ Let $(a,b)$ be any rational point on the circle $x^2+(y-\...
2
votes
2answers
35 views

Proving $\mathbb{Q}[\sqrt{2}] = \{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\} = \{x+y\sqrt{2}:x,y\in\mathbb{Q}\}$

I need to prove that: $$\mathbb{Q}[\sqrt{2}] = \{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\} = \{x+y\sqrt{2}:x,y\in\mathbb{Q}\}$$ Well, $ \{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\} $ is the set of $a_nx^n+\...
1
vote
0answers
38 views

How do I know if this will be rational or irrational? ($a^b$)

Usually, when I have $a^b$ when $a$ and $b$ are both irrational, I assume that it will be irrational. But that is not always true, I assume, so when is the result irrational? How will I know? Take ...
5
votes
3answers
492 views

Why is epsilon not a rational number?

I was wondering why epsilon, the smallest positive number, isn't a rational number. I was watching a video a few days ago about surreal numbers, and I've learned that, in the field of surreal numbers, ...
0
votes
4answers
411 views

How can I explain $0.999\ldots=1$? [duplicate]

Possible Duplicate: Does .99999… = 1? I have to explain $0.999\ldots=1$ to people who don't know limit. How can I explain $0.999\ldots=1$? The common procedure is as follows \begin{align} x&...
1
vote
1answer
24 views

What is known about rational points on the ideal of relations / syzygy ideal?

What is known about rational points on the ideal of relations / syzygy ideal? Let $G$ be a finite group, with $|G|=n$. Then $G$ acts on $\mathbb{Q}[x_1,\cdots,x_n]$ through the regular representation (...
2
votes
2answers
61 views

Why is $\{p \in \mathbb{Q},:p<1\} | \{p \in \mathbb{Q}: p \geq 1\}$ a real number?

A basic example of a Dedekind cut is: $A|B$ = $\{p \in \mathbb{Q}:p<1\} | \{p \in \mathbb{Q}: p \geq 1\}$ This is very confusing because what we have here is a pair of subsets of $\mathbb{Q}$ ...
1
vote
1answer
82 views

Given any two real numbers $x<y$, there is a rational $q$ with $x<q<y$

I am trying to prove the following statement: Given any two real numbers $x,y$ with $x<y$, there exists a rational number $q$ that satisfies $x<q<y$. I got stuck at one point of the proof, ...
13
votes
5answers
408 views

Irreducibility of $f(x)=x^4+3x^3-9x^2+7x+27$

Question at hand is: Is $x^4+3x^3-9x^2+7x+27$ irreducible in $\Bbb Q$ and/or $\Bbb Z$. This is for an exam, reasoning is trivial, but no calculators in hand. Clearly, if there is a rational root,...
2
votes
2answers
96 views

show the existence of a real number

Let $a_n > 0$ be any sequence. I want to show that there is a real number $r \in \mathbb R$ such that $ 0 < | r - m/n | < a_n $ for infinitely many points $(m,n ) \in \mathbb N ^2$. I don'...
0
votes
2answers
82 views

$({\mathbb{Q}},+)$ is not finitely generated

I'm trying to prove that $G = ({\mathbb{Q}},+)$ is not finitely generated. I have come up with the following, and would like to check it is correct: $G$ is generated by $\{1/n | n \in \mathbb{N}\}$...
4
votes
3answers
149 views

Can a $\mathbf{Q}$-basis of $\mathbf{R}$ be explicitly defined?

Could someone give me an explicit basis of $\mathbf{R}$ as a vector space over $\mathbf{Q}$? I know some linearly independent subset, namely $1,e,e^2,\dots$ but this seems to be a deep result ...
-1
votes
1answer
63 views

How to proof that a field is complete order field?

I know what an ordered field is but how to actually proof that a field, for example $Q$ (rational numbers), is an ordered field?
1
vote
1answer
61 views

Prove $x \in \mathbb{Q}$ and $y \notin \mathbb{Q} \implies (x + y) \notin \mathbb{Q}$

Looking for tips to prove the homework $\forall x,y \in \mathbb{R}, x \in \mathbb{Q} \land y \notin \mathbb{Q} \implies (x + y) \notin \mathbb{Q}$ Can I assume the hypothesis and to yield a ...
6
votes
2answers
104 views

Solve for Rationals $p,q,r$ Satisfying $\frac{2p}{1+2p-p^2}+\frac{2q}{1+2q-q^2}+\frac{2r}{1+2r-r^2}=1$.

Find all rational solutions $(p,q,r)$ to the Diophantine equation $$\frac{2p}{1+2p-p^2}+\frac{2q}{1+2q-q^2}+\frac{2r}{1+2r-r^2}=1\,.$$ At least, determine an infinite family of $(p,q,r)\in\mathbb{Q}^3$...
0
votes
6answers
158 views

Describe the rational points on $3x^2 + y^2 = 4$

Apart from $(x, y) = (0, 2)$ and $(1, 1)$, are there any nonzero rational points on the curve $3x^2 + y^2 = 4$ ?
1
vote
0answers
40 views

Proof that $M \;$is a field [duplicate]

Let $\; p,d \,$ be prime numbers I wanna proof that $\; M:=\{a_o+a_1\sqrt p +a_2\sqrt d +a_3\sqrt {pd}:a_o,a_1,a_2,a_3 \in \mathbb Q \}\;$ is a field. The only thing I am having trouble with is ...
-1
votes
1answer
23 views

Infinite Sets (real rational integers) [duplicate]

How can the real, integer, and rational number sets be infinite, yet, they aren't all the same size?
32
votes
1answer
693 views

Is $\{\tan(x) : x\in \mathbb{Q}\}$ a group under addition?

A student asked me the following today : Is $S:= \{\tan(x) : x\in \mathbb{Q}\}$ a group under addition? I am quite perplexed by it. Clearly, the only non-trivial part is to check For any $x, ...
8
votes
1answer
180 views

Prove that $\sqrt{2} + \sqrt[3]{3}$ is irrational [duplicate]

$\sqrt{2} + \sqrt[3]{3}$ is irrational ? These are my steps - $\sqrt{2} + \sqrt[3]{3} = a$ $3 = (a-\sqrt{2})^{3}$ $3 = a^{3} -3a^{2}\sqrt{2} + 6a -2\sqrt{2}$ $3a^{2}\sqrt{2}+2\sqrt{2} = a^{3}+6a-...
4
votes
3answers
1k views

Length of period of decimal expansion of a fraction

Each rational number (fraction) can be written as decimal periodic number. Does exists a method or hint that show how long will be the period of arbitrary fraction. For example $1/3=0.3333...=0.(3)$ ...
0
votes
1answer
122 views

Show that the set of polynomials with rational coefficients is countable.

Problem: Show that the set of polynomials with rational coefficients is countable. Idea: We know that the set of rational numbers is denumerable. This implies that the set of rational numbers is ...
6
votes
4answers
922 views

Proof: Is there a line in the xy plane that goes through only rational coordinates?

Question: Is there a line in the XY plane that has all rational coordinates. Prove your answer. Idea: There is most certainly not. I believe it can be shown that between any 2 rational points that ...
2
votes
3answers
109 views

Proving f(x)=0 for all x in [a,b] when we only know that f is continuous and f(x)=0 when x is rational. [duplicate]

The question is as follows a.) Let $f(x)$ be continuous function on an interval [a,b] and suppose that $f(x)=0$ for each rational value $x$ in [a,b]. Prove that $f(x) = 0$ for all $x \in [a,b]$. b.) ...
0
votes
1answer
58 views

Every rational number between $0$ and $1$ is between $1/(n+1)$ and $1/n$ for some $n$

Let $a/b$ be a fraction in lowest terms with $0<a/b<1$. Prove that there exists $n∈\mathbb N$ such that $$\frac{1}{n+1}\le\frac{a}{b}<\frac{1}{n}$$ My proof: If $n∈\mathbb N$ then $n<n+1$...
4
votes
3answers
77 views

Rational solutions $(a,b)$ to the equation $a\sqrt{2}+b\sqrt{3} = 2\sqrt{a} + 3\sqrt{b}$

Find all rational solutions $(a,b)$ to the equation $$a\sqrt{2}+b\sqrt{3} = 2\sqrt{a} + 3\sqrt{b}.$$ I can see that we have the solutions $(0,0), (2,0), (0,3), (3,2), (2,3)$, and I suspect that ...
2
votes
0answers
52 views

Is it possible to reduce Theory of Rationals to Theory of Natural Numbers?

Is the following possible ? $$ Th( \mathbb{Q}, +, \leq ) \leq^{\log}_m Th( \mathbb{N}, +, \leq )$$ I believe it is not possible since Natural Numbers are not dense. It is also not possible $$ Th( \...
4
votes
5answers
2k views

Prove that - for every positive $x \in \mathbb{Q}$, there exists positive $y \in \mathbb{Q}$ for which $y \lt x$

First my apologies if this question has been asked before. Exposition I'm new at learning how to prove theorems and among the given exercises from my reference material it is asked to prove the ...
3
votes
2answers
117 views

Write $0.2154154\overline{154}$ as a fraction

Let $x = 0.2154154\overline{154}$ , I have to prove that it is a rational number just by writing it as a fraction with the proper steps. I note that the repeating part, $154$, is composed by 3 digits....
1
vote
1answer
58 views

Finding multiplicative factors of $p\in[\sqrt{2},2)\cap\mathbb{Q}$

Given $p\in[\sqrt{2},2)\cap\mathbb{Q}$, how to find $q,r\in(0,\sqrt{2})\cap\mathbb{Q}$ such that $p=qr$ ? Context: When constructing $\mathbb{R}$ with Dedekind cuts, this question arises when trying ...
-2
votes
1answer
59 views

Prove that the equation $x^2=x$ has the same solutions in rational numbers as in integers

I was wondering if you could help me start in my discrete math homework. I'm asked to prove that A = B: $A =\{x \in \mathbb{Z}\mid x^2 = x\}$ and $B = \{x \in \mathbb{Q}\mid x^2 = x\}$ I'm having ...
7
votes
1answer
114 views

Is there a function, continuous on the irrationals, with rational values, nowhere locally constant?

Question. Let $\mathbb A=\mathbb R\!\smallsetminus\!\mathbb Q$ be the irrational numbers. Is there a continuous function $\,f:\mathbb A\to\mathbb Q$, which is nowhere locally constant? – i.e., for ...
6
votes
2answers
70 views

Uses of vector spaces over $\mathbb Q$

I know of two applications of vector spaces over $\mathbb Q$ to problems posed by people not specifically interested in vector spaces over $\mathbb Q$: Hilbert's third problem; and The Buckingham pi ...
0
votes
0answers
52 views

When is a finite sum of rational numbers an integer?

Asking in "Which radical equations transformable into a polynomial equation by exponentiating the equation?" and in "When is the number of radicals in a power of a sum of radicals less than or equal ...
1
vote
1answer
47 views

Continuity of a function defined on rationals

So I have a function $$f: \mathbb{Q} \rightarrow \mathbb{R}, f(x)=x$$ and need to state with justification whether or not it is continuous. I seem to be having trouble actually interpreting the domain ...
1
vote
0answers
46 views

Is there a stable probability distribution on the rational numbers?

Does there exist a (non-trivial) probability distribution on the rational numbers $$\sum_{r\in\mathbb{Q}}p_r=1$$ with $0\leq p_r$, which is stable, meaning that the sum of two i.i.d. random variables ...
-1
votes
2answers
70 views

Is the set $\Bbb Q$ a quotient set of $\Bbb Q^*$?

Let $\Bbb Q^*=\{\frac a b: a\in \Bbb Z, b\in \Bbb N\}$. From this definition we can see $c=\frac 2 3$ and $d=\frac 4 6$ are elements of $\Bbb Q^*$. Claim: $$\frac 2 3\neq \frac 4 6$$ Proof: ...
-3
votes
1answer
1k views

(22/7) is a rational number and (π) is irrational number [closed]

Why (22/7) is a rational number and (π) is irrational number. please explain. Edit: How can you say that $22/7=\pi$, when one number if rational and the other is irrational?
1
vote
0answers
27 views

Proof that $\zeta_P$ never does the following…

Let's assume that $\zeta_P(s)$ is the prime-zeta function or: $$\zeta_P(s)=\sum_{n\in P} \frac{1}{n^s}$$ I noted that if $\forall s\in \Bbb{Q},s\not =0, \zeta_P (s)\not\in \Bbb{Q}$ I cant really ...
7
votes
7answers
690 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 ...
1
vote
2answers
69 views
6
votes
1answer
711 views

Show that $\{1, \sqrt{2}, \sqrt{3}\}$ is linearly independent over $\mathbb{Q}$.

My apologies if this question has been asked before, but a quick search gave no results. This is not homework, but I would just like a hint please. The question asks Show that $\{1, \sqrt{2}, \...
1
vote
4answers
110 views

Can all irrational numbers be written in the form $u + v\sqrt{2}$, with $u$ and $v$ rational? [closed]

I am curious to know whether all irrational numbers can be written in the form $u + v\sqrt{2}$, with $u$ and $v$ rational. (Almost similar to how all complex numbers can be written as $x + iy$, ...