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.

learn more… | top users | synonyms

0
votes
1answer
256 views

Bijection of positive rational numbers with the natural numbers

In what position does the number $\frac{14}{15}$ appear in the bijection of the positive rational numbers with the natural numbers? The first few terms of the bijection are: $\frac 11$, $\frac12$, $\...
1
vote
4answers
78 views

$\pi \not\in \mathbb{Q}$?

I've taken this fact for granted; some thinking tells me that indeed, I cannot express it with fractions. So it's not rational. But well, if $p,q \in \mathbb{Q}$ then $p+q \in \mathbb{Q}$ since it is ...
3
votes
0answers
45 views

Rational numbers as angles - where do irrationals fit in?

If we make a rectangular grid with integer coordinates, it's possible to assign a unique angle to any rational number, using the definition $\tan \phi=y/x$ for $\phi \in (-\pi/2, \pi/2)$. For ...
0
votes
1answer
29 views

Square Root of Rational Number $\frac{A}{B}$

Here's the question: Let $x=\frac{A}{B}$ be a positive rational number in lowers terms (i.e., $A, B\in\mathbb{N}$ and $hcf(A,B)=1$). Prove that $\sqrt{x}$ is rational if and only if $A$ and $B$ are ...
2
votes
1answer
38 views

Position or rank of an arbitrary rational number

Rational numbers are countable as shown by the usual table here: https://aminsaied.wordpress.com/2012/05/21/diagonal-arguments/ So, counting in the zig-zag manner as shown in the table, $1/1$ is the ...
2
votes
3answers
209 views

Rational or Irrational number [closed]

we know that "$a$" is a Irrational number .But "$a^2+a$" is Rational. Can You find "$a$"? (more than one answer is available)
0
votes
1answer
44 views

Is it possible to calculate $ \sin(\alpha) $ (and other trigonometric functions) as a rational number? [duplicate]

I am creating a computer library for arbitrary-precision calculations, by expressing numbers as rationals (with an arbitrary-precision numerator and denominator). Now, I am exploring the possibility ...
0
votes
2answers
43 views

Irrational Numbers and their squares

If $s$ is irrational is $s^2$ irrational? Looking at example (a) $s= \sqrt 2$ then $s^2= 2$, which is rational but looking at example (b) $s= 5^{1/3}$, then $s^2= 5^{2/3}$ which is irrational or $\...
10
votes
2answers
1k views

Reversing the digits of an infinite decimal

Let $x$ be a real number in $[0,1)$, with decimal expansion $$ x = 0.d_1 d_2 d_3 \cdots d_i \cdots \;. $$ If the decimal expansion is finite, ending at $d_i$, then extend with zeros: $d_k = 0$ for all ...
1
vote
0answers
62 views

What subsets of rationals have been defined where each element equals the sum of a sequence of numbers?

In particular, I'm interested in the rationals that result from adding a finite sequence of consecutive integer powers of two. Has it been studied somewhere? Update This formalized example might ...
5
votes
1answer
69 views

Have humans proved Schinzel's conjecture for one specific rational number?

I asked the Tooth Fairy about Schinzel's conjecture that if $x$ is a positive rational number, then it can be represented as $$\frac{p + 1}{q + 1}$$ where $p$ and $q$ are primes, for infinitely many ...
1
vote
1answer
42 views

Rationality of $a^2+b^2$

I have looked into this topic lately and have not found an answer to the following question. Is the following true: If $a,b\in\mathbb{R}$ and $a + b$ is rational, then $a^2 + b^2$ is rational
1
vote
1answer
20 views

$\{a x^{z}: a\in \Bbb{Q}, z \in \Bbb{Z}\} \approx \Bbb{Q}^{\times} \otimes_{\Bbb{Z}} \Bbb{Z}^+ \implies$? what about $\Bbb{Q}$-linear sums?

Consider all functions $f: \Bbb{Q} \to \Bbb{Q}$ of the form $f(x) = a x^z$ where $a \in \Bbb{Q}, z \in \Bbb{Z}$, call it $G$. It forms an abelian group under usual multiplication. I think it's ...
0
votes
0answers
38 views

When is $X^n-a$ irreducible over $\mathbb{Q}$?

Is there a general criterion, when $X^n-a$, $a\in\mathbb{Q}$ is irreducible? Clearly, this is the case if there is a prime $p$ such that $\nu_p(a)=\pm 1$: For $\nu_p(a)=1$, this follows from ...
2
votes
1answer
38 views

Countable dense subset claim in Arzela Ascoli proof

In every Arzela Ascoli proof you see the following: Let $S = \mathbb{Q} \cap [a,b]$, where $[a,b]$ is an interval in $\mathbb{R}$, then $S$ is a countable dense subset and there exists a ...
-1
votes
3answers
63 views

If $x$ is rational and $xy$ is irrational, then $y$ is irrational. [closed]

This is a statement that I need to prove. Let $x$ and $y$ be real numbers. If $x$ is rational and $x\times y$ is irrational, then $y$ is irrational. I believe you have to prove this using ...
0
votes
1answer
29 views

Proving that rational equivalence is an equivalence relation on any set.

I seek to prove that the rational equivalence relation is an equivalence relation, in that it is reflexive, symmetric, and transitive. The rational equivalence relation is as follows "Two numbers in ...
2
votes
4answers
75 views

Prove that $\frac{p}{q}$ is a rational number with a finite decimal expression if $p$ is an integer and $q=(2^n)(5^m)$

Let $p,q$ be two integers and $q=(2^n)(5^m)$. Then $\frac pq$ is a rational number with a finite decimal expression. Any ideas how to do this? I've been thinking about it all day but I have no idea ...
1
vote
1answer
24 views

Basel problem over $\mathbb{Q}_{\geq 1}$

Let $\mathbb{Q}_{\geq 1}=\{r\in\mathbb{Q}\,|\,r\geq 1\}$. Once $\mathbb{Q}$ is enumerable, $\mathbb{Q}_{\geq 1}$ is also enumerable. Let $\{r_1,r_2,\ldots\}$ be such an enumeration. What can we say ...
0
votes
2answers
52 views

What is a good way to show that $[0,1]$ is not complete in $\mathbb{Q}$

To show a set is not complete, the best way is always produce a Cauchy sequence that does not converge in the set. I wish to show $[0,1]$ is not complete in $\mathbb{Q}$ I am a little stucking ...
0
votes
0answers
39 views

How many rational values of x are not integers and satisfy the following equation?

How many rational values of x are not integers and satisfy the following equation: $$x^7 - 6x^6 + 5x^5 - 4x^4 + 3x^3 - 2x^2 + 1 = 0 ?$$ Well, I got this question from one of the Mathcounts ...
1
vote
3answers
60 views

Conditions under which $\frac{ax+b}{cx+d}$ will be rational.

Suppose $x$ is an irrational number and $a,b,c,d$ are rational numbers. If we know that $$ \frac{(ax+b)}{(cx+d)} $$ is rational, then it follows that: a.) $a=c=0$, b.) $a=c$ and $b=d$, c.) $a+b = ...
1
vote
3answers
152 views

Why is the set of Rational numbers countably infinite? [duplicate]

Why is the set of Rational numbers ,$\mathbb Q$, a countably finite set? I think that - if we assign $n$ to a rational number, and $n+1$ to another rational number, Then I can surely find a rational ...
2
votes
2answers
109 views

T/F: any number that can be written as a fraction is rational.

Any number that can be written as a fraction is rational. I am being asked this question, and I believe it is true but for some reason,I feel that there is a trick. However, the definition of ...
1
vote
1answer
88 views

Which set is more dense: set of irrational numbers or set of rational numbers? [duplicate]

Is the infinity of irrational numbers equal to the infinity of rational numbers? Or is one is greater than other? And what is the proof? I could not find out a rigorous proof about this. P.S. I am ...
1
vote
1answer
70 views

Set of Rational numbers a countable set?

How can we say that rational numbers is a countable set? I can divide a rational number by infinite different number of natural numbers so shouldn't there be infinite rational numbers. http://www....
3
votes
0answers
37 views

Nonlinear regular bijection from $\mathbb Q$ to itself

Is there a bijection $\phi: \mathbb Q \to \mathbb Q$ such that $\phi$ is nonlinear (i.e. different from $ax+b$), $\phi$ is regular: the extension $\hat{\phi}$ of $\phi$ over $\mathbb R$ is $\mathcal ...
0
votes
2answers
53 views

Onto function with domain of rational numbers and co-domain of natural numbers

I'm trying to find an onto function $f: \mathbb{Q} \to\mathbb{N}$ I'm somewhere along the lines of $f(q) = |(1 - q)| + q$ for non integers, but I'm not sure where to go from there.
1
vote
2answers
85 views

(H.W question) In Mathemaical Analysis of Rudin example 1.1 Pg 2

The author went on and proved that 1) There exists no rational $p$ such that $p^2=2$ 2) He defined two sets $A$ and $B$ such that if $p\in A$ then $p^2 <2$ and if $p\in B$ then $p^2>2$ and ...
1
vote
1answer
44 views

Does the inner pentagon inside a Robbins pentagon $also$ have a rational area?

The Heron triangle has integer sides and area. The Robbins pentagon is just the generalization: it also has integer sides and area. The example below has sides $78, 126, 66, 50, 32$ and area $A_R = \...
0
votes
2answers
66 views

Given a rational number and an irrational number, both greater than 0, prove that the product between them is irrational.

Does this proof I made make sense? Proof// $\mathbf a$ is the rational number, $\mathbf b$ is the irrational number. Assume that $\mathbf {a * b}$ is rational due to proof by contradiction. ...
1
vote
2answers
56 views

Sets which are order-isomorphic to the (extended) rationals

Let $(S,<)$ be a totaly ordered set under the strict order relation $<$. Suppose that, for any $a,b\in S$, if $a<b$, then there exists $c\in S$ such that $a<c<b$. We also assume that $...
2
votes
1answer
53 views

Proof of part of properties of exponentiation Tao proposition $4.3.12$

If you let $x,y$ be non-zero rational numbers, and let $n, m$ be integers, I need to prove that if $x \geq y>0$, then $x^{n} \geq y^{n}>0$ if n is positive, and $0< x^{n} \leq y^{n}$ if $n$ ...
1
vote
1answer
59 views

The set of rational numbers has continuum many subsets that are order-wise inequivalent

How to show that there are continuum many subsets of $\mathbb Q$, no two of which are similar? Two sets are called similar if there is an order-preserving bijection between them.
1
vote
1answer
51 views

Is this proof correct? Show $\mathbb{Q}$ is dense in $\mathbb{R}$

I like proof by contradictions in showing that $\mathbb{Q}$ is dense in $\mathbb{R}$. But I can't understand this one> https://math.dartmouth.edu/archive/m54x12/public_html/m54densitynote.pdf ...
1
vote
1answer
48 views

When $\frac{\pi ^{x}}{\zeta (x)}$ is rational?

When $n$ is a positive integer, we know $$\zeta (2n)=\frac{(-1)^{n+1}B_{2n}(2\pi )^{2n}}{2(2n)!}$$ Now let's say $x>1$ is a real number. Can we say if $\frac{\pi ^{x}}{\zeta (x)}$ is a rational ...
1
vote
1answer
28 views

Property of a sequence being an enumeration of the rationals.

Let $(r_n)$ be an enumeration of the rationals and $x\in\mathbb{R}$. Is it possible to find out whether the set $\left\{n\in\mathbb{N}:\left|x-r_n\right|<\frac{1}{2^n}\right\}$ is finite or ...
5
votes
2answers
91 views

Is it true that $\mathbb{Q}(\sqrt{2}) \cap \mathbb{Q}(i) = \mathbb{Q}$?

Is it true that $\mathbb{Q}(\sqrt{2}) \cap \mathbb{Q}(i) = \mathbb{Q}$? I know that \begin{align*} \mathbb{Q}(\sqrt{2}) &= \{a+b\sqrt{2} \mid a,b \in \mathbb{Q}\}, \\ \mathbb{Q}(i) &= \{a+...
4
votes
1answer
91 views

Proving that the Calkin-Wilf tree enumerates the rationals.

The Calkin-Wilf tree is an infinite undirected graph (tree) which is constructed as follows: starting from the root at $\frac{1}{1}$, each node $\frac{a}{b}$ has two children: a left child $\frac{a}{...
4
votes
1answer
66 views

if $r,s$ are rational numbers, then $r+s\sqrt2$ is irrational unless $s=0$?

if $r,s$ are rational numbers, Prove $r+s\sqrt2$ is irrational unless $s=0$? I need to prove this simple question, but not sure if my method is acceptable I'm trying to prove it by contradiction, ...
0
votes
1answer
27 views

Why is it that for any rational numbers $a < b$, the interval $[a, b]$ in $\mathbb{Q}$ is not compact with respect to this metric?

Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$. We can ...
0
votes
1answer
30 views

Rational solutions for $\sin(n)$ in radians

This is completely for my own curiosity. Does $y = \sin(n)$ have rational solutions for $n$, an integer number of radians. I know that this is strange because usually integers are only used in ...
2
votes
1answer
47 views

$\sup$ and $\inf$ of $E=\{p/q\in\mathbb{Q}:p^2<5q^2 \text{ and } p, q > 0\}$

I'd appreciate if you could please check to see if my proof is valid. Find $\sup$ and $\inf$ of $E=\{p/q\in\mathbb{Q}:p^2<5q^2 \text{ and } p, q > 0\}$. Solution: $q^2 > p^2/5 \iff q > ...
5
votes
2answers
98 views

Is there a first order formula $\varphi[x]$ in $(\mathbb Q, +, \cdot, 0)$ such that $x≥0$ iff $\varphi[x]$?

In the first-order language $\mathscr L$ having $(+, \cdot, 0)$ as signature, it is easy to define a formula $\phi[x]$, namely $\exists y \; x = y^2$, satisfying : $$\text{for all } x \in \Bbb R, \...
2
votes
1answer
60 views

How can all of them be irrational ??

Assume that $\{x,y,x^2,y^2,xy\}$ are all irrational. Can it be true that all of $\{x-y,x+y,x^2-y^2,x^2+y^2\}$ are irrational? Details: $|x|\ne|y|$ and $x,y\in\mathbb R$. In the question ...
1
vote
0answers
38 views

Is a value that tends to infinity considered rational?

This really confused me. I know a rational number is any number that can be written in the form of ${a\over b} \space \space \forall a,b\in Z$. We also know all Integers are clearly Rational. But ...
0
votes
2answers
43 views

Rational mean of irrational numbers?

My teacher tells me that in the vicinity of any rational number, an irrational exists. To elucidate, I presume, he further went on to say, if a function, if defined to give 1 for every rational number ...
3
votes
2answers
50 views

For a.e. $x \in [0, 1]$, there are finitely many $p/q$ such that $\left| x - p/q \right| < 1 / \left( q \log q \right)^2$

I am stuck on a qualifying exam problem and was hoping to get some help. Show for a.e. $x \in [0, 1]$ that there are finitely many $p/q \in \mathbf{Q}$ in reduced form such that $q \geq 2$ and $\...
1
vote
3answers
69 views

Proving that there is no continuous function $f:\Bbb R\to\Bbb R$ satisfying $f(\Bbb Q)\subset\Bbb R-\Bbb Q$ and $f(\Bbb R-\Bbb Q) \subset\Bbb Q$. [duplicate]

How can I prove that there is no continuous function $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(\mathbb{Q}) \subset \mathbb{R}\backslash \mathbb{Q}$ and $f(\mathbb{R}\backslash \mathbb{Q} ) \subset \...
2
votes
1answer
37 views

Rational points on a line

This question is quite unique. Does there exist some point in the coordinate system such that any line passing through it has at most 2 rational points lying on it?