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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3
votes
7answers
145 views

Rational Expression equivalent form

EDIT: I know how to find the answer, but does anyone know why plugging in numbers for x does not work? The Question: If the rational expression $\frac {3x^2}{3x-1}$ is rewritten in the equivalent ...
1
vote
2answers
69 views

If a series grows more slowly than any geometric series, can it ever converge to a rational?

I was reading a proof of $e$'s irrationality which, in some sense, uses the fact that the series $\sum \frac{1}{n!} = e$ grows slowly. This got me thinking: can we generalize this and say "oh, $\sum \...
-8
votes
3answers
111 views

The dilemma of Pi [closed]

Is Pi rational or irrational ? Pi can be represented as 22/7 which is a rational number. Whereas 3.14 is a non terminating and non recurring number which is a irrational number
2
votes
1answer
31 views

Does an analytic $f$ need be polynomial to close $\mathbb{Q}$

If an analytic function $f : \mathbb{R}\to\mathbb{R}$ satisfies $f(\mathbb{Q}) \subseteq \mathbb{Q}$, can we conclude that $f$ is a polynomial?
0
votes
1answer
24 views

Creating a periodic sequence from a given subsequence

You are given the odd elements of an infinite binary sequence: $$ a_1, a_3, a_5, \dots $$ You have to add even elements $a_2,a_4,a_6,\dots$ such that the resulting sequence is periodic (i.e, a ...
17
votes
9answers
1k views

Function that maps the “pureness” of a rational number?

By pureness I mean a number that shows how much the numerator and denominator are small. E.g. $\frac{1}{1}$ is purest, $\frac{1}{2}$ is less pure (but the same as $\frac{2}{1}$), $\frac{2}{3}$ is ...
0
votes
1answer
22 views

Proof that 1/x + 1/y is distinct for distinct unordered pairs of (x,y), xy = k.

Take xy = k, for nonzero k. There are many (x,y) that can satisfy this. However, how do I prove that the sums of the members of any two distinct, unordered pairs, is distinct? (This is an equivalent ...
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 = ...
8
votes
3answers
2k views

How can we find and categorize the subgroups of R?

$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}\newcommand{\Z}{\Bbb Z}$ What are all the subgroups of R = $(\R, +)$ and how can we categorize them? I started thinking about this question last night ...
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
1answer
34 views

What is the name of the set obtained by multiplying a given number by any rational?

Given a number, is there a name for the set where each element results of multiplying this number by a rational? For a given $ n \in \mathbb N $: $$ \{ r \cdot n \mid r \in \mathbb Q \} $$
1
vote
2answers
60 views

show that this statement is false (counterexample) if $a,b \in \mathbb R \backslash \mathbb Q $ then $a \cdot b \in \mathbb R \backslash \mathbb Q $

if $a,b \in \mathbb R \backslash \mathbb Q $ then $a \cdot b \in \mathbb R \backslash \mathbb Q $ Okay so the question asks to show, with a counter example, that the above statement is false. Here ...
0
votes
5answers
369 views

Why can't the reals be constructed from the infinitesimal?

If the infinitesimal gives an unlimited precision as 1/∞ --> 0 Which can be thought of as the decimal 0.000000.....00000... then Why can't the reals, which demands, simply, unlimited precision (this ...
6
votes
2answers
134 views

Show that $x=y+z$ for all $x \in S$

We are given a set $S$ as a subset of the rational numbers defined by: $0 \notin S$ If $s_1 , s_2 \in S$, then $\frac {s_1}{s_2} \in S$ There exists a nonzero rational number $q \notin S$ such ...
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 ...
5
votes
3answers
3k views

Proving the rationals are dense in R

I know this is a common proof. I'm following Rudin's proof and I'm following everything except for one step. Suppose $x, y \in \Bbb R$ and $x < y$. Then there exists an $n \in \Bbb N$ such that $n(...
2
votes
1answer
39 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
210 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 $\...
9
votes
2answers
534 views

Can someone clarify Example I.I.2 from Hardy's Course of Pure Mathematics?

"If $\lambda, m,$ and $n$ are positive rational numbers, and $m > n$, then $\lambda(m^2 − n^2), 2\lambda mn$, and $\lambda(m^2 + n^2)$ are positive rational numbers. Hence show how to determine any ...
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 ...
0
votes
0answers
17 views

Continuity - approximating an irrational number via rationals [duplicate]

If $x=p/q$, where $(p,q)=1$ are integers, then $f(x)=1/q$. If x is irrational then f(x)=0. Prove that: a) f is continuous for all irrationals b) f is not continuous for all rationals. I think ...
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 ...
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 ...
0
votes
0answers
95 views

Limits as a representation of the Dirichlet function

I read that the Dirichlet function ($1$ if Rational, $0$ else) can be written as: $$ D(x)=\lim_{m\to\infty}\lim_{n\to\infty} \cos^{2n}(m!\pi x)$$ What is the proof of that? Are those limits ...
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
25 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
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 ...
1
vote
1answer
113 views

Image of ring homomorphism $\phi : \mathbb{Z}[t] \to \mathbb{Q}$?

Here is a problem I face practicing the theory of rings: Define $\phi : \mathbb{Z}[t] \to \mathbb{Q}$, a ring homomorphism (it does map $1$ to $1$). I'm trying to show that if $\phi(t)=\frac{u}{v}$...
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 ...
8
votes
1answer
277 views

For each irrational number $b$, does there exist an irrational number $a$ such that $a^b$ is rational?

It is well known that there exist two irrational numbers $a$ and $b$ such that $a^b$ is rational. By the way, I've been interested in the following two propositions. Proposition 1 : For each ...
1
vote
4answers
9k views

What is a fraction in which the greatest common factor of the numerator and the denominator is 1?

What is this fraction: A fraction in which the greatest common factor of the numerator and the denominator is 1?
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....
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 ...
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
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 $...
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
67 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. ...