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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2answers
49 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 !
5
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2answers
220 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 ...
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2answers
20 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.
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3answers
83 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 ...
-2
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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 ...
0
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2answers
23 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 $\sqrt2$ ...
8
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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 ...
4
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1answer
147 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
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2answers
962 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
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1answer
62 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 ...
3
votes
2answers
113 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 ...
1
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1answer
18 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?
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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
97 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 $n\in\mathbb{N}$?
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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
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1answer
48 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
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1answer
42 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
63 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
100 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
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1answer
35 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
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2answers
41 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 ...
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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: ...
2
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4answers
248 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
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1answer
52 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 $n\in\mathbb{Q}$ ...
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0answers
47 views

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

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} $, $$ ...
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2answers
136 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
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5answers
91 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
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4answers
26 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 $4037/9900$ ...
0
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1answer
29 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 ...
2
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1answer
58 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
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1answer
37 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
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1answer
199 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 ...
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3answers
201 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:)
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1answer
1k 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 ...
0
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1answer
34 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 ...
3
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2answers
130 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 ...
0
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3answers
65 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
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3answers
177 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
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1answer
47 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
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1answer
144 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?
6
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1answer
347 views

What's an example of a number that is neither rational nor irrational?

Of course in regular logic, the answer is there aren't any. But in intuitionistic logic, there might be, as seen by this answer: http://math.stackexchange.com/a/1437130/49592. My question is, as per ...
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5answers
304 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 ...
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1answer
39 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 ...
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1answer
35 views

The minimum cardinal of a geometrical set

Let $S$ be a set of points in a plane $P$, having the following property: for any point $X \in P$ there is at least one point $M \in S$ so that the distance $|XM|$ is rational. Find the minimum ...
1
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1answer
172 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 ...
3
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1answer
67 views

Coloring rational numbers

Here is my problem. Fix a color for the number $1$, for example yellow. Choose another color, for example green. Now, for a positive rational denoted $x$, there are two rules : $x$ and $1/x$ have ...
2
votes
3answers
338 views

Find the supremum and infimum of {x $\in$ [0,1]: x $\notin$ $\mathbb Q$}. Prove why your assertions are correct

Ok I am lost from this question. Does that mean $x$ can only be $0$ or $1$? And it can't be any rational?
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1answer
28 views

a dense set in (0,1)

Define for $\epsilon > 0 $ $$V_\epsilon = \left( \bigcup_{j \in \mathbb{N}} (x_n - \frac{\epsilon}{2^{n+1}} , x_n + \frac{\epsilon}{2^{n+1}}) \right) \ \bigcap \ (0,1)$$ where $x_n$ stems from ...
0
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2answers
39 views

Equivalence classes and rational numbers

We defined $\mathbb{Q}$ as the set of equivalence classes for the relation $\sim$. Tentatively define operations $+,\cdot:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{Q}$ by $[(a,b)]+[(c,d)] = ...
0
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1answer
34 views

Given a rational number produce a larger rational number

Given Dedekind cuts $A|B$ and $C|D$ in $\mathbb{Q}$, let $E=\{a+c:a\in A,c\in C \}$ Prove that E has no largest element. If I understand the first statement $A|B$ and $C|D$ are real numbers, but ...