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
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0answers
43 views

Non-constructive proofs for the rationality of a number

One of the key ideas in transcedental number theory is proving that a number is transcedental (i.e. not the root of any polynomial with integer coefficients) by showing a sequence of rational numbers ...
1
vote
2answers
62 views

Show that a subset $E$ $\subset \Bbb Q$ is not compact in $(\Bbb Q, d)$ and decide whether it is open or not

Assume $(\Bbb Q, d),$ $d(p, q):= |p -q|$ is a metric space and $E := \{p \in \Bbb Q : 2 < p^2 < 3\} = \{p \in \Bbb Q : \sqrt2 < p < \sqrt3\} \subset \Bbb Q.$ I have to show ...
0
votes
1answer
41 views

Show that $E \subset \Bbb Q$ is closed in $(\Bbb Q, d)$

Assume $(\Bbb Q, d),$ $d(p, q):= |p -q|$ is a metric space and $E := \{p \in \Bbb Q : 2 < p^2 < 3\} \subset \Bbb Q.$ I have to show that $E$ is closed. I see two ways of proving ...
0
votes
1answer
53 views

Assume $r,s \in\mathbb{Q}$. Prove $\frac{r}{s},r-s \in\mathbb{Q}$ [closed]

I have attempted this proof by contradiction. Beginning with assuming to the contrary that each a and b are irrational but was not sure if I did it correctly. Any help would be greatly appreciated. ...
0
votes
2answers
32 views

Countable set of number rational, prove with $\mathbb{Z}$.

Good morning, I need to prove $ \mathbb{Q} $ is a countable set, but I prove $ \mathbb{Z} $ is a countable set, now, can I use this for proving $ \mathbb{Q} $ is countable set? I was thinking about a ...
2
votes
1answer
55 views

What's the numerator and the denominator of a fraction called?

Just a quick question, is it right to call the numerator and the denominator of a fraction by "terms"? I don't think that "terms" is the right word here, but i don't know any alternatives. Can any ...
12
votes
6answers
2k views

Visual representation of the fact that there are more irrational than rational numbers.

Would anybody know of a visual or even (preferably) geometric representation of this? To make it more specific: Text, symbols and written numbers are predominantly used as labels, and and less to ...
0
votes
0answers
40 views

Is LCM of rationals used in higher math? [duplicate]

I read in this school book, an algorithm to find the LCM of rationals. It goes in the following manner. $[\frac{a}{b}, \frac{c}{d}]=\frac{[a,c]}{(b,d)}$. If you inspect as to why the formula is given ...
1
vote
2answers
59 views

How to prove the power set of the rationals is uncountable?

Recently a professor of mine remarked that the rational numbers make an "incomplete" field, because not every subsequence of rational numbers tends to another rational number - the easiest example ...
7
votes
3answers
173 views

The equation $\{x^2\} + \{x\}=1$ has no solution over positive rationals

Prove there is no positive rational $x$ so that $$\{x^2\} + \{x\}=1 \tag1 $$ Let $x=\frac p q$ and $p=qc+r, p, q, c, r \in \mathbb{N}, 0 \le r \lt q$ From (1) $\{ 2c \frac r q + (\frac r q)^2\} +...
-1
votes
2answers
27 views

Proving by contradiction (6/9)

I have been given a statement that I need to prove using the contradiction method and I am just a little unsure of how to go about setting this up and executing. Here is the statement: If x is any ...
1
vote
2answers
43 views

How do you deduce the integer whose multiplicative inverse decimal has a digit sequence or repetend length of 3 digits?

A positive integer's, n, reciprocal, $\frac{1}{n}$, in which the decimal's repetend has a length of three digits which starts at the decimal mark. e.g. 0.037037... of the integer, 27 ,reciprocal $\...
0
votes
1answer
25 views

A sum of irrational numbers ending rational

Let $x$ be a positive irrational number I know that there exists $y$ such that: $$\begin{cases} y>0 \\ x+y\in \mathbb Q.\end{cases}$$ How would you construct explicitly such $y$ ? For instance ...
4
votes
3answers
45 views

is there any convergent sub-sequence of a sequence of all rational numbers?

Let $(a_n)$ be a sequence of rational numbers, where all rational numbers are terms. (i.e. enumeration of rational numbers) Then, is there any convergent sub-sequence of $(a_n)$?
0
votes
1answer
51 views

General Conic and its Rational Solutions

Suppose you have a rational conic $ax^2+bxy+cy^2+dx+ey+f=0$. There is a theorem that states if a conic has 1 rational solution it has infinitely many rational solutions. How can you prove this ...
1
vote
2answers
42 views

Irreducible fraction of a given rational

Given a rational $ r \in \mathbb Q $, how to find the irreducible fraction $ \frac a b = r $? Any direct formula based on the digits of $ r $, instead of successive approximations by increasing ...
1
vote
2answers
36 views

Show that there are at most two rational points on $(x - a)^2 + (y - b)^2 = r^2$ for $a, b$ irrational.

For any given irrational numbers $a, b$ and real number $r \gt 0$, show that there are at most two rational points (points whose coordinates are both rational numbers) on the circle $(x - a)^2 + (y - ...
0
votes
0answers
13 views

Should we expect any anomalies when dealing with rational differential equations?

Some time ago, I found myself reading a short article that proposed that the rational numbers where the 'appropiate' number system that we should use for most of mathematics (where we use the real ...
0
votes
3answers
33 views

How does one find a rational number in fraction form, knowing the repeating decimal?

For example, I have 0.786786786... How do I find the fraction equivalent?
3
votes
2answers
80 views

Is the value of $\log_27$ a rational number?

Is $\log_27$ a rational number?
1
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2answers
85 views

Minimizing rational solutions of $ x^3+y^3=9$

I´m trying to solve this problem: An old alchemist had two sphercial flasks, one with a circunference of 12 inches and the other with a circunference of 24 inches. He desired to transfer their ...
-2
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1answer
25 views

If $\cos\pi\theta$ is algebraic and $\theta$ is irrational, what is the set of possible $\theta$?

I know that $a= \cos \pi \theta$ is an algebraic number ($\theta$ is rational). I want to prove that if $\cos\pi\theta$ is rational, then the possible only possible values of $\theta$ are $0,±1/2,±1$ ...
4
votes
1answer
42 views

Why do metric spaces that produce the same topology have different number theoretical difficulties?

Consider finding a a point with rational distance to the corners of unit square. Under the Euclidean metric this is very hard. (unsolved) Under the "city block" or taxicab metric this is very easy ...
1
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1answer
94 views

For which $a,b\in \mathbb{N},$ is $\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$ is a rational number.

I found the following problem on a Olympiad question paper: For which $a,b\in \mathbb{N},$ is $$\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$$ a rational number. I am unable to solve it. Any help ...
2
votes
2answers
55 views

Rational Distance Problem triple — irrational point

Many points with rational coordinates are known with rational distances to three vertices of a unit square. For example, the following points are rational distances from $a=(0,0)$, $b=(1,0)$, and $c=(...
0
votes
0answers
21 views

Proof that the union of rational and irrational numbers sets is a set of real numbers [duplicate]

I see it all the time but is there a nice way to show that this is true? Or is this just a definition? I know that $\mathbb{Q} \subset \mathbb{R}$ and $\mathbb{I} \subset \mathbb{R}$, but how do we ...
2
votes
1answer
48 views

If $a$ and $b$ are positive rational numbers with $a < b$, show that $\frac{1}{a} >\frac {1}{b}$

Since both $a,b\in \mathbb{Q}^+$ and $a<b$, then of course $\frac{1}{a}$ is greater than $\frac{1}{b}$. However, I don't know how to prove that. I suppose I could do the greater than property in an ...
1
vote
4answers
143 views

Approximation of $\sqrt{2}$

I got the following problem in a chapter of approximations: If $\frac{m}{n}$ is an approximation to $\sqrt{2}$ then prove that $\frac{m}{2n}+\frac{n}{m}$ is a better approximation to $\sqrt{2}.$(...
2
votes
1answer
36 views

Prove that the group of the rational points on the conic $u^2-Av^2=1$ is not finitely generated.

This is an exercise from Rational Points on Elliptic Curves by Silverman. Let $H$ be the conic $u^2-Av^2=1$ where $\sqrt{A}\notin \mathbb{Q}$. If $(u_1,v_1), (u_2,v_2)$ are two points in $H(\...
6
votes
4answers
251 views

'Almost rational' integrals with no known closed form?

I recently stumbled upon an 'almost rational' integral, namely: $$\int_0^{\pi/2} x \frac{\sqrt{\sin x}-\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}} dx=0.231231222\dots \approx 0.231231231\dots= \frac{...
1
vote
1answer
13 views

Let $H = \{2^m : m \in \mathbb{Z}\}$ & define a relation $R$ on the set $\mathbb{Q^{+}}$ of positive rationals by $a\mathbin{R}b$ iff $a/b \in H$.

Let $H = \{2^m : m \in \mathbb{Z}\}$ and define a relation $R$ on the set $\mathbb{Q^{+}}$ of positive rational numbers by $a\mathbin{R}b$ if and only if $a/b \in H$. Prove that $R$ is an equivalence ...
0
votes
2answers
34 views

Convert this into fractional number step by step?

3.41287548754875... Convert the above number to a rational number? I was reviewing some pre calculus on my own but couldn't figure this out.
1
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2answers
23 views

Let $S = { r \in \mathbb{Q} : r \lt 2}$. Prove that $S$ does not have a largest element.

Let $S$ = $[{ r \in \mathbb{Q} : r \lt 2}]$. Prove that $S$ does not have a largest element. My method: Assume to the contrary that $S$ does have a largest element, where $S$ = $[{r \in \mathbb{Q} :...
4
votes
5answers
267 views

Complement of rationals has empty interior

This question refers to How to prove closure of $\mathbb{Q}$ is $\mathbb{R}$ I want to prove that the closure of $\mathbb{Q}$ is $\mathbb{R}$. I am trying to understand the accepted answer, but when ...
5
votes
2answers
51 views

Rational Question for $a + b$ and Irrationality of $a^2 + b^2$

I have looked into the question and need help. Find some $a,b$ ${\in}$ $\mathbb{R}$ such that $a + b$ ${\in}$ $\mathbb{Q}$, $a^2 + b^2 \not\in \mathbb{Q}$, and $\frac{a}{2} < b < a$. Or prove ...
21
votes
2answers
372 views

Is $\frac{1}{11}+\frac{1}{111}+\frac{1}{1111}+\cdots$ an irrational number?

Obviously: $$\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\cdots=0.1111\dots=\frac{1}{9}$$ is a rational number. Now, if we make terms with demoninators in the form: $$q_n=\sum_{k=0}^{n} 10^k$$ Then ...
0
votes
2answers
26 views

Validity of certain arguments about the countability of infinite sets

I am trying to get an understanding, in layman's terms / on an intuitive level, why some arguments about the countability of infinite sets are valid, and some arguments which seem almost identical on ...
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 ...
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 ...
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 ...
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 ...
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 ...
0
votes
1answer
254 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 ...