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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4answers
411 views

Integral of rationals

Define $f(x)$ as $$f(x)=\begin{cases}0,&\text{if }x\in \mathbb{Q}\\ 1,&\text{if }x\notin \mathbb{Q}\;. \end{cases}$$ Considering the fact that there is a countable infinity of rationals yet an ...
4
votes
3answers
168 views

Prove that the product of an irrational number and a rational number is irrational.

If $x$ is an irrational number and $r$ is a rational number then $xr$ is an irrational number. Proof. Suppose that $xr$ is a rational number. By defintion of a rational number $xr= m/n$ where ...
5
votes
5answers
727 views

Odd divided by even is a fraction

How can we prove that an odd number divided by an even number is a fraction? I started with odd $=2m+1$ and even $=2n$ and get left with with $(m+2)/n$.
0
votes
2answers
21 views

Proof about rational neighbors

Two rational numbers $\frac{a}{b}$ < $\frac{c}{d}$ will be called neighbors if $\frac{c}{d}$ - $\frac{a}{b}$ = $\frac{bc-ad}{bd}$ = $\frac{1}{bd}$. Suppose $\frac{a}{b}$ and $\frac{c}{d}$ are ...
0
votes
0answers
35 views

In which way to prove that the set has the measure zero in R3?

I can understand this task in the way that we should prove that there are less numbers in the rational set compared to the numbers in the real set ? TO PROVE: The set A is described as follows: ...
0
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2answers
47 views

prove or disprove if a number is irrational

Prove or disprove : I'm pretty sure this isn't true yet i can't find a counter example. Thanks in advance !
1
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2answers
19 views

If you apply the Distributive Property to a Rational and an Irrational number, which will your solution be?

Say that "A" and "B" are Rational, and C is irrational, would the solution to "A(B+C)" be Rational or Irrational? An example for clarification would be wonderful.
1
vote
3answers
72 views

Why rational numbers are dense?

So the books says that rational numbers are dense, meaning that for every two rational numbers there is another rational number in between them. Is it actually true? Why? It feels to me that there ...
8
votes
3answers
2k views

subtraction of two irrational numbers to get a rational [duplicate]

Say you have a number like $\pi$ or e. Is it possible to subtract another number from it and end up with a rational number? I mean I guess you could write an equation like $\pi-x=3$ But could there ...
3
votes
1answer
78 views

Is every homeomorphism of $\mathbb{Q}$ monotone?

It is well known that every continuous injective map $\mathbb{R}\rightarrow\mathbb{R}$ is monotone. This statement is false for maps $\mathbb{Q}\rightarrow\mathbb{Q}$. (That is becaus $\mathbb{Q}$ is ...
0
votes
2answers
20 views

equivalence class of function, picking proper x

Defining R to be the relationship on real numbers given by xRy iff x-y is rational, I've been asked to find the equivalence class of $\sqrt2$. My instincts say that the equivalence class of $\sqrt2$ ...
4
votes
1answer
124 views

Prove that $x$ is rational iff $a=b=0$

So the question goes let $x = a\sqrt3 + b\sqrt 5$ where $a,b$ rational. Prove that $x$ is rational iff $a=b=0$. I think I can prove this but I'm not sure if my proof is correct or rigorous. Well ...
8
votes
2answers
921 views

Is there any basis transformation under which all irrational numbers are rationals and vice-versa?

For example, if you change the length of your "unit scale" or basis for numbers to $\sqrt{2}$, then you may represent all fractional multiples of $\sqrt{2}$ as "rational numbers" in the new basis ...
0
votes
1answer
57 views

determine the frontier

determine the frontier of the set R\Q (where R is the real numbers and Q is the rational numbers). I figured R\Q is the same as saying the real line minus all the rational numbers which would just ...
1
vote
1answer
17 views

A question of rationality of integral powers

Given $a,b\in\mathbb{Z}$, is $x$ from $a^x=b$ ever rational? More specifically, is $x$ in $2^x=3$ irrational?
1
vote
4answers
60 views

Prove rational nums

For all real number x : R(x) -> there exist two integers k, l such that x = k/l. (i.e. x is a rational number) Prove/Disprove: For all real number x : R(x) -> R(x+1) My answer: Let x be a real ...
3
votes
2answers
91 views

Does $\sin n$ have a maximum value for natural number $n$?

In formal, does there exist $k\in\mathbb{N}$ such that $\sin n\leq\sin k$ for all $n\in\mathbb{N}$?
2
votes
3answers
48 views

Why proof about a rational on open interval (a,b) works…

For the case $0 < a < b$ which is what I'm interested in, there is a proof that there exists a rational number on the open interval which I've seen many times but I don't really understand it. ...
0
votes
1answer
38 views

Intuition for a proof that the rationals are incomplete. [duplicate]

Let A be a set of positive rationals $p$ such that $p^2<2$. Now this set contains no upper bound. To prove this, for every rational $p$, a number $p- \frac{p^2-2}{p+2}$ is associated. This ...
0
votes
1answer
39 views

Looking for a simple proof of why you can't mathematically tune a piano

https://www.youtube.com/watch?v=1Hqm0dYKUx4 Video states that a corollary of the Rational Root Theorem is that $\left(\frac{a}{b}\right)^n != 2$ for integers $a,b,n$, where $n \gt 1$. I'm simply ...
0
votes
2answers
60 views

How to prove that $y = 0,273273273…,$ is a rational number?

How to prove that $$y = 0.273273273...$$ is a rational number? I don't have any experience with proofs... Can I get your help and your advice?
1
vote
1answer
79 views

Neighbors of Irrational Numbers on Real Number Line

I was looking at a post on MathOverflow about "What is your favorite 'strange' function?" One of the answers mentioned Thomae's function claiming that the function was "continuous at all irrationals ...
1
vote
1answer
30 views

Characterizing the roots of rational numbers

I am trying to prove the statement: if $n \in \mathbb{Q}$ and $\sqrt[m]{n} \in \mathbb{Q}$ for all positive integers, then $n = 1$. In my work, I have done all the work given by the top answer to ...
1
vote
2answers
36 views

definition of rational powers of real numbers

Suppose that $b\gt1$ and x is a real number. Rudin defines $B(x)$ to be the set of all numbers $b^{t}$, where $t$ is a rational number and $t\le x$. I want to prove that if $r$ is a rational number ...
20
votes
5answers
13k views

Is there a rational number between any two irrationals?

Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 < i_2$. Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$? How would you construct such ...
3
votes
2answers
214 views

Is the infinite table argument for the countability of Q unsound?

The first "proof" I learned for why the rationals are countably infinite relied on arranging the rational numbers in a two-dimensional array and using the well-known traversal shown below to construct ...
2
votes
4answers
105 views

Find an increasing sequence of rationals that converges to $\pi$

I am not sure how to construct a sequence that would convey convergence to $\pi$. Except maybe $a_n=\{\pi + 1/n\}$ but the terms would not be rational. Looking for an adequate way to show to satisfy ...
0
votes
1answer
37 views

Finding the next rational number

A rational number is one that can be written as $a/b$ where $a$ and $b$ are integers, $b\gt0$ ($a$ can take care of negative rationals), and I suppose $\gcd(a,b) = 1$. Given some $n\in\mathbb{Q}$ ...
0
votes
0answers
44 views

for what x, is $\frac{1}{\pi} \cdot cos^{-1}(x) \in \mathbb{Q}$

While solving a question, I met the next problem, for what x, is: $$ \frac{1}{\pi} \cdot cos^{-1}(x) \in \mathbb{Q} $$ I found in this paper that for $ 0 \leq r \leq 1, r \in \mathbb{Q} $, $$ ...
8
votes
2answers
128 views

How to show this cover of $\mathbb{Q}$ doesn't cover $\mathbb{R}$?

Let $\{q_n : n \in \mathbb{N}\}$ be an enumeration of $\mathbb{Q}$ and define $\mathcal{O} = \{I_n : n \in \mathbb{N}\}$ being $$I_n = \left(q_n - \frac{1}{2^n}, q_n + \frac{1}{2^n}\right).$$ It is ...
0
votes
5answers
79 views

Prove rational sum and product of two irrational numbers

I need to prove that $$\exists a,b \in \mathbb{R} \setminus \mathbb{Q} : a + b, ab \in \mathbb{Q}$$ Any ideas? I, unfortunately, don't have one yet. The most obvious way with equations in integers ...
1
vote
4answers
23 views

Convert rational number in $\frac {p}{q}$ form

Convert rational number in $\frac {p}{q}$ form $0.40\bar 7$ (here bar is over $7$). solution: By solving I got the answer $367/900$ by multiplying by $10$ My friends are getting answer $4037/9900$ ...
2
votes
1answer
55 views

Difference between $\mathbb{Q}$ and $\mathbb{R}$ - countability proof

We know $\mathbb{Q}$, the rational numbers, is countable; the real numbers is not. My professor in the course of real analysis proved the title by showing $\{0,1\}^{\mathbb{N}}$ is not countable, ...
0
votes
1answer
28 views

Simplifying Rational Expressions

Simplify the following rational expression: $5/(x+3) - 7x/(x-1)$ I came across this question in my homework and because it is a fraction, I decided that I needed to establish a common denominator ...
0
votes
1answer
36 views

Is it possible to construct a maximal set with irrational distance between elements?

As part of my algebra homework a few weeks ago, I was asked to prove some things about the relation $R$, defined by $(x,y) \in R$ if $x - y \in \mathbb{Q}$. The homework problem itself wasn't ...
1
vote
1answer
2k views

Prove that the difference between two rational numbers is rational

This is a terribly simple question I'm sure, but I can't find a work-around in my proof. I must prove that the difference between two rational numbers is thus rational. Here is my attempt: Let $a$ ...
1
vote
1answer
127 views

How do I write the opposite of a rational number? [closed]

Write the opposite of each rational number A)$ 9$ B)$-17.6$ C) $6.12 $ D) $-7 \frac{5}{7 }$ Some one please help! I am not doing very good in Math I'm in grade 9 and I'm struggling I would ...
-1
votes
3answers
118 views

What decimal is between 0.5 and 0.625 [closed]

I would really appreciate some help with this. I have been literally stumped with it for an hour. So if you know the answer please comment below! Thank you for your time:)
-2
votes
1answer
615 views

The difference between two rational numbers always is a rational number [duplicate]

Claim: The difference between two rational numbers always is a rational number Proof: You have a/b - c/d with a,b,c,d being integers and b,d not equal to 0. Then: a/b - c/d ----> ad/bd - bc/bd ...
2
votes
2answers
103 views

Length of digits before the period in decimal expansion for rational numbers

I'm a newbie with number theory and I've been reading this page and trying to figure out how to calculate the length of the digits before the period and digits of the period of a rational number of ...
2
votes
2answers
536 views

Prove that $x\in\mathbb Q$

Let $a\in\mathbb Q$ and $a>\dfrac43$. Let $x\in\mathbb R$ and $x^2-ax,x^3-ax\in\mathbb Q$. Prove that $x\in\mathbb Q$. EDIT: Thsi is my attempt: Let $x^2-ax=b$ and $x^3-ax=q$ for some ...
0
votes
1answer
30 views

Can reflection across a line segment be done using the rational field?

Assume that I have a point and a line segment, all specified using rational coordinates. Can I compute the reflection of the point across the line segment using only rational numbers? This previous ...
4
votes
1answer
203 views

Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number.

Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number. I thought this question might have been asked by someone else, but I couldn't find any. Currently I'm studying ...
0
votes
3answers
64 views

How Can I calculate this expression?

I have this repeating expression $5+\dfrac {6} {5+\dfrac {6} {5+..}}$ I saw a solution on a book. which is: $5+\dfrac {6} {5+\dfrac {6} {5+..}}=x$ $5+\dfrac {6} {x}=x$ $x^2-5x-6=0$ $x=6 $ or ...
3
votes
3answers
100 views

Find all cluster points for the sequence $x_{n}$ = The $n$-th rational number

Find all cluster points for the sequence $x_{n}$ = The $n$-th rational number Note: In this problem a labeling of rational numbers by positive integers is used. Such labellings do exist because ...
2
votes
1answer
45 views

How to express $15.3\dot{9}$ in fractional form

In the number $15.3\dot{9}$, $9$ is repeated forever. If the number is rational then it can be expressed as a fraction (i suppose it is rational since it's an exercise for me to find it's rational ...
6
votes
1answer
115 views

What are some of the implications of $\pi + e$ being rational?

Whether or not $\pi + e$ is rational is an open question. If it were rational, what would some of the implications be?
1
vote
1answer
140 views

Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational? [duplicate]

I know the famous proof that uses $x={\sqrt{2}}^{\sqrt{2}}$ to prove that there must exist an irrational to an irrational power that evaluates to a rational. But I don't know if $x$ itself is known to ...
1
vote
5answers
187 views

Why is $[0, 1] \cap \mathbb{Q}$ not compact in $\mathbb{Q}$?

Statement: $[a, b] \cap \mathbb{Q}$ in $\mathbb{Q}$ is not compact. Thus the interior of all compact subsets of $\mathbb{Q}$ is $\emptyset$. I am trying to understand the first sentence. I read ...
4
votes
1answer
35 views

Rationals $(\mathbb{Q},<)$ are isomorphic to a part of a finite partition

I believe the following statement is true but I can't find or figure out a proof: For any partition of the set $\mathbb{Q}$ of rationals into a finite number of parts, there is a part containing an ...