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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Log laws proof using only rational exponents [closed]

For all real $a>0$ and rational $b>0$, Show that $\ln(a^b)=b\ln(a)$
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0answers
25 views

Smallest Rational Number Proof [duplicate]

Can anybody help me out with solving this mathematical proof? Prove the statement “There is no smallest rational number greater than 2” by contradiction. Contradiction: There is a smallest ...
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1answer
25 views

What is the value of $x$ in $(7+\sqrt{x}+\frac{1}{5-\sqrt{x}})$?

The value of $(7+\sqrt{x}+\frac{1}{5-\sqrt{x}})$ is rational for only one positive integer $x$ that is not a perfect square. What is the value of $x$? I tried $x=1$ and i get $$7+1+\frac{1}{4}=\...
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2answers
57 views

A group automorphism of real numbers that is not the identity

Is it possible to have a group automorphism of the additive group of the real numbers that fixes a subset of real numbers but is not the identity? The subset might be infinite. I'm thinking of using ...
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1answer
249 views

There is no smallest rational number greater than 2

I have a problem that I am seriously stuck on. I'm not sure what to do I've seen similar proofs online with the least positive rational number but this is apparently different and I'm not sure why. ...
1
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1answer
33 views

How to find the closest bounded rational approximation to a rational number?

Say I have a rational number $a/b$ and I want to find its closest rational approximation $x/y$ where $$x_- \leq x \leq x_+$$ $$y_- \leq y \leq y_+$$ for some constants $x_\pm$, $y_\pm$. How can I ...
2
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1answer
33 views

Number of ordered positive rationals (x,y,z) satisfying following conditions.

How many ordered triples $(x,y,z)$ of positive rational numbers satisfy the conditions: $x+y+z$, $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$, and $xyz$ are all integers.
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3answers
117 views

proving $ \sqrt 2 + \sqrt 3 $ is irrational [duplicate]

I need to proof that $\sqrt{3} + \sqrt{2}$ is irrational, without using the fact that an irrational number plus a rational number equals irrational. also, i can't use the rational root theorem. that's ...
1
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1answer
45 views

Exchangeability of union and intersection of open balls around all rational numbers in $[0,1]$

Let $X:=[0,1]$ and $V:= X \cap \mathbb{Q}= \{v_1,v_2,...\}$. For $n,k \ge1$ set $I_{n,k}:= X \cap (v_n-2^{-(n+k)},v_n+2^{-(n+k)}) $. Is it true that $$ \bigcup_{n\ge1} \bigcap_{k\ge1} I_{n,k} = \...
3
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2answers
76 views

Is there a mathematical definition for the “divisibility” of rational numbers?

The term divisibility usually refers to integer numbers only. I want to define the divisibility of a rational number $q$ by an integer number $z$ as follows: $q$ is divisible by $z$ if and only if $...
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4answers
429 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 ...
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3answers
199 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 $m,n$ ...
3
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2answers
67 views

Numerical polynomials by means of prime powers

Let $\mathbb{E}$ be the set of prime powers (except $1$). Let $f \in \mathbb{Q}[x]$ be a rational polynomial with $f(\mathbb{E}) \subseteq \mathbb{Z}$. Does it follow that $f$ is numerical, i.e. ...
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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 ...
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2answers
86 views

How many distinct equivalence classes does this equivalence on rationals have?

Let $$A = \{ r\in \mathbb Q \mid \exists p\in \mathbb Z,\text{ and $q\in \mathbb Z$, with $p$ even and $q$ odd, and $r = p/q$} \}$$ For example, $A$ contains such $2/9, 16/(-34)$, and $4$. $A$ does ...
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0answers
43 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: $A=\...
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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 !
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2answers
223 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
21 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
85 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 ...
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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 ...
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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 $\sqrt2$ ...
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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 ...
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1answer
151 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
973 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 ...
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1answer
66 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
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2answers
115 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 digits....
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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
63 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 ...
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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. ...
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1answer
49 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 number (...
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1answer
43 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 ...
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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?
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1answer
105 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 ...
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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 ...
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2answers
42 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: ...
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4answers
264 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
54 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 ...
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5answers
95 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 (...
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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 $4037/9900$ ...
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1answer
30 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 of:...
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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, ...
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1answer
38 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 ...
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1answer
214 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
222 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 ---...