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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5
votes
3answers
484 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
407 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 ...
1
vote
1answer
21 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
59 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
77 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
381 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 ...
2
votes
2answers
94 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 ...
0
votes
2answers
76 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 ...
3
votes
3answers
146 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
49 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
60 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
96 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 ...
0
votes
6answers
151 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
38 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
22 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?
30
votes
1answer
663 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
177 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} = ...
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
92 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
908 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
95 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 ...
4
votes
3answers
74 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
51 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
112 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 ...
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
108 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
64 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
51 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
39 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
43 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
381 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
666 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
68 views
6
votes
1answer
654 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}, ...
2
votes
4answers
107 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 $x + iy$, ...
2
votes
1answer
55 views

Examining the nature of mapping of $f(x) = \frac{x}{x^2 - 2}$.

Let f: $\mathbb{Q} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{x}{x^2 - 2}$, $x \in \mathbb{Q}$. Examine the nature of mapping Attempt: I think f(x) is not ...
1
vote
0answers
127 views

Interesting facts/ proofs about rational and irrational numbers

We got set some work to find some interesting facts or proofs regarding rational and irrational numbers. I wonder if anyone could offer some insight or recommend a good book/ website to look at.
3
votes
3answers
147 views

What Will Happen Without Decimal Expansion?

After a discussion on the complexity of decimal expansion (such as $0.\bar{9}=1$), some of my students (middle school) decided to throw away the decimal expansion of some numbers! Namely, the numbers ...
3
votes
1answer
308 views

Show that if m/n is a good approximation of $\sqrt{2}$ than $(m+2n)/(m+n)$ is better

Claim: If $m/n$ is a good approximation of $\sqrt{2}$ then $(m+2n)/(m+n)$ is better. My attempt at the proof: Let d be the distance between $\sqrt{2}$ and some estimate, s. So we have ...
1
vote
2answers
85 views

How many distinct equivalence classes does this equivalence on rationals have?

Let $$A = \{ r\in \mathbb Q \mid \exists p\in \mathbb Z,\text{ and $q\in \mathbb Z$, with $p$ even and $q$ odd, and $r = p/q$} \}$$ For example, $A$ contains such $2/9, 16/(-34)$, and $4$. $A$ does ...
3
votes
2answers
67 views

Intersection of union of crazy intervals in $\mathbb{R}$

I am looking at two sets $X:=[0,1]$ and $V:= X \cap \mathbb{Q}= \{v_1,v_2,...\}$. For each $n,k \in \mathbb{N}_{\ge1}$ I define an interval $I_{n,k}:= X \cap (v_n-2^{-(n+k)},v_n+2^{-(n+k)}) $. Now I ...
3
votes
1answer
69 views

Tautological line bundle over rational projective space

Is the tautological line bundle over $\mathbb{Q}P^{n}$ a non trivial bundle? Here, $\mathbb{Q}P^{n}$ has the natural topology induced from the standard topology of $\mathbb{Q}$ as a subset of ...
5
votes
2answers
217 views

Proving that $x$ is irrational if $x-\lfloor x \rfloor + \frac1x - \left\lfloor \frac1x \right\rfloor = 1$

Prove : $$ \text{If } \; x-\lfloor x \rfloor + \frac{1}{x} - \left\lfloor \frac{1}{x} \right\rfloor = 1 \text{, then } x \text{ is irrational.}$$ I think the way to go here is to falsely assume that ...
2
votes
2answers
55 views

Continuity question: Show that $f(x)=0, \forall x\in\mathbb{R}$. [duplicate]

Assume $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous on $\mathbb{R}$ and such that $f(r)=0$ for every rational number $r$. Show that $f(x)=0, \forall x\in\mathbb{R}$ using the $\varepsilon-\delta$ ...
1
vote
2answers
26 views

$\mathbb{Q}$: Unique operation s.t. $1\star q=q$ and right distributivity hold?

Given $\mathbb{Q}$ and the usual addition $+$ on it, do we have unicity of a binary operation $\star$ such that \begin{align*} \tag{1}1\star q&=q\\ \tag{2}(q+r)\star s&=q\star s+r\star s ...