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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0
votes
6answers
156 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?
31
votes
1answer
679 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
179 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
115 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
918 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
106 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
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
116 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
113 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
67 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
42 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
44 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
813 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
681 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
702 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
108 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
56 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
168 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
149 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
320 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 $d=s-\sqrt{...
1
vote
2answers
87 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
224 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
57 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 \...
1
vote
5answers
46 views

Log laws proof using only rational exponents [closed]

For all real $a>0$ and rational $b>0$, Show that $\ln(a^b)=b\ln(a)$
2
votes
3answers
741 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?
1
vote
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 ...
0
votes
1answer
25 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 $$7+1+\frac{1}{4}=\...
2
votes
2answers
57 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 ...
-1
votes
1answer
254 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
33 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
votes
1answer
33 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
votes
3answers
117 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 ...
11
votes
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 and ...
1
vote
1answer
45 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
79 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 $...