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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37 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
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1answer
21 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
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1answer
637 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
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2answers
129 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
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3answers
906 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
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1answer
43 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
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4answers
891 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
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3answers
74 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
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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
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3answers
73 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
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0answers
48 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
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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
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2answers
111 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
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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 ...
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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
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1answer
97 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
62 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 ...
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0answers
47 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 ...
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1answer
27 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 ...
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0answers
40 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 ...
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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: ...
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1answer
117 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?
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0answers
26 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
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7answers
639 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
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2answers
66 views
6
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1answer
567 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
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4answers
99 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
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1answer
51 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 ...
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0answers
64 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.
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3answers
132 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
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1answer
255 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 ...
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2answers
84 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
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2answers
62 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
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1answer
66 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
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2answers
191 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
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2answers
48 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 $\varepsilon-\delta$ ...
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2answers
25 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 ...
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5answers
42 views

Log laws proof using only rational exponents [closed]

For all real $a>0$ and rational $b>0$, Show that $\ln(a^b)=b\ln(a)$
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3answers
731 views

Are there infinitely many natural numbers whose square root is rational?

True or false? There are infinitely many natural numbers $n$ for which $\sqrt{n}$ is rational. Is this statement true/false? And how can I construct a proof of its statement or negation?
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0answers
25 views

Smallest Rational Number Proof [duplicate]

Can anybody help me out with solving this mathematical proof? Prove the statement “There is no smallest rational number greater than 2” by contradiction. Contradiction: There is a smallest ...
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1answer
24 views

What is the value of $x$ in $(7+\sqrt{x}+\frac{1}{5-\sqrt{x}})$?

The value of $(7+\sqrt{x}+\frac{1}{5-\sqrt{x}})$ is rational for only one positive integer $x$ that is not a perfect square. What is the value of $x$? I tried $x=1$ and i get ...
2
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2answers
41 views

A group automorphism of real numbers that is not the identity

Is it possible to have a group automorphism of the additive group of the real numbers that fixes a subset of real numbers but is not the identity? The subset might be infinite. I'm thinking of using ...
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1answer
165 views

There is no smallest rational number greater than 2

I have a problem that I am seriously stuck on. I'm not sure what to do I've seen similar proofs online with the least positive rational number but this is apparently different and I'm not sure why. ...
1
vote
1answer
31 views

How to find the closest bounded rational approximation to a rational number?

Say I have a rational number $a/b$ and I want to find its closest rational approximation $x/y$ where $$x_- \leq x \leq x_+$$ $$y_- \leq y \leq y_+$$ for some constants $x_\pm$, $y_\pm$. How can I ...
2
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1answer
32 views

Number of ordered positive rationals (x,y,z) satisfying following conditions.

How many ordered triples $(x,y,z)$ of positive rational numbers satisfy the conditions: $x+y+z$, $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$, and $xyz$ are all integers.
2
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3answers
105 views

proving $ \sqrt 2 + \sqrt 3 $ is irrational [duplicate]

I need to proof that $\sqrt{3} + \sqrt{2}$ is irrational, without using the fact that an irrational number plus a rational number equals irrational. also, i can't use the rational root theorem. that's ...
10
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4answers
2k views

Show that the curve $x^2+y^2-3=0$ has no rational points

Show that the curve $x^2+y^2-3=0$ has no rational points, that is, no points $(x,y)$ with $x,y\in \mathbb{Q}$. Update: Thanks for all the input! I've done my best to incorporate your suggestions ...
1
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1answer
43 views

Exchangeability of union and intersection of open balls around all rational numbers in $[0,1]$

Let $X:=[0,1]$ and $V:= X \cap \mathbb{Q}= \{v_1,v_2,...\}$. For $n,k \ge1$ set $I_{n,k}:= X \cap (v_n-2^{-(n+k)},v_n+2^{-(n+k)}) $. Is it true that $$ \bigcup_{n\ge1} \bigcap_{k\ge1} I_{n,k} = ...
3
votes
2answers
61 views

Is there a mathematical definition for the “divisibility” of rational numbers?

The term divisibility usually refers to integer numbers only. I want to define the divisibility of a rational number $q$ by an integer number $z$ as follows: $q$ is divisible by $z$ if and only if ...
3
votes
2answers
67 views

Numerical polynomials by means of prime powers

Let $\mathbb{E}$ be the set of prime powers (except $1$). Let $f \in \mathbb{Q}[x]$ be a rational polynomial with $f(\mathbb{E}) \subseteq \mathbb{Z}$. Does it follow that $f$ is numerical, i.e. ...