Questions tagged [rational-numbers]

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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28 views

How do I prove by contradiction?

I'm stuck on this question and I don't know how where to start. Let $a$ and $b$ be rational numbers with $a$ is not equal to $b$ Prove that $a+\frac{b-a}{\sqrt2}$ is irrational. (You may ...
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2answers
41 views

Rationalising fractions proof

Suppose that a, b, c and d are positive integers and c is not a square. Given that $$\frac a{b+\sqrt c}+\frac d{\sqrt c}\in \mathbb Q$$ prove that $b^2d = c(a + d)$ What I did was try and ...
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1answer
36 views

Dedekind Cuts and showing that they satisfy the additive axioms

Let α,β be cuts and let α+β={r+s|r∈αands∈β}. How can I show that for all cuts in R with the addition defined here, we can satisfy the additive axioms of commutativity, closure, identity, inverse, and ...
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5answers
137 views

How to find the limit $\lim_{x\to0}\frac{x\tan x-x\sin x}{x\sin^2x/\cos x}$ [closed]

Here is the limit I'm struggling with: $$\lim_{x\to0}\cfrac{x\tan x-x\sin x}{x\sin^2x/\cos x}.$$ Worked so hard to find it, but couldn't.
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2answers
133 views

Prove $a < (p+1)/n < b$ if $a,b$ are irrationals with $0<a<b$ and $p$ is greatest integer with $p/n < a$

Suppose that $a$ and $b$ are positive irrational numbers, where $a < b$. Choose any positive integer $n$ such that $1/n < b - a$, and let $p$ be the greatest integer such that $p/n < a$. ...
1
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1answer
49 views

In $\mathbf{Sets}^\mathbf{Q}$, prove the subobject classifier $\Omega$ is given by $\Omega(q)=\{r\mid r\in\mathbf{R^+},r\ge q\}.$

This is Exercise I.9 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". The Details: A definition of a subobject classifier is given on page 32, ibid. Definition: In a category $...
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4answers
44 views

Prove that $3^x,3^y$ and $3^z$ are successive terms of a G.P.

Prove that $3^x,3^y$ and $3^z$ are successive terms of a geometric progression , if $x,y$ and $z$ are successive terms of an arithmetic progression. How should this be proven? I have no clue
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0answers
47 views

Question About A Problem About Rational Numbers [duplicate]

I'm a bit confused about what I am supposed to prove in this step of the problem. Am I supposed to prove that for any positive integer $T$, no matter it's size, I can always find positive integers $b&...
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1answer
41 views

Obtain an expression for the nth term of the Geometric Progression.

The 2nd, 6th and 8th terms of an Arithmetic progression are three successive terms of a Geometric progression. Find the common ratio of the Geometric progression and obtain an expression for the $n$...
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2answers
49 views

$P(x)\in \mathbb{R}[x]$, If $P(x)=n$ has at least one rational zero for $\forall n \in \mathbb{N}$, $P(x)=ax+b$

Question: Let $P(x)\in \mathbb{R}[x]$. If $P(x)=n$ has at least one rational zero for $\forall n \in \mathbb{N}$, show that $P(x)=ax+b$ where $a$ and $b$ are rational. I totally do not know how I ...
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1answer
1k views

Rational vs Irrational distribution

Imagine I draw a number line, and I took two points. What's the distribution of rational and irrational numbers between them? If I put it in a diagram where I color rational with a color and ...
2
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1answer
34 views

What is the standard notation for the set of rationals with finite fractional part?

As the title says. What is the standard notation for the set of rationals with finite fractional part, when written in base $n$ with a radix point? I expected $ℚ_n$, but that's taken for n-adic ...
2
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1answer
47 views

Determine the value of $\frac{\sqrt[3]{4\sqrt[3]{4\sqrt[3]{4…}}}}{\sqrt{2\sqrt {2\sqrt{2…}}}}$

Determine the value for $$\frac{\sqrt[3]{4\sqrt[3]{4\sqrt[3]{4...}}}}{\sqrt{2\sqrt {2\sqrt{2...}}}}$$ I think the formula $S_\infty =\frac {a}{1-r}$ should be used for this question but I don’t ...
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0answers
41 views

What periodic functions have to do with rational numbers (2)?

Two harmonic oscillators satisfying $$ \begin {cases} x (0) = y (0) = 1 \\ x '(0) = y' (0) = 1 \\ \end{cases} $$ ,have movements governed by the equations $$ \begin {cases} x '' = - x \\ y '' = - Ky ...
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1answer
25 views

What periodic functions have to do with rational numbers?

I would like to understand why is the answer of this question letter d. Question: The solution φ(t) = (x (t) , y(t), z(t), w(t)) of the system of equations: x' =y, y' = - x, z' =w, w' =-(k^2)*z, ...
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3answers
89 views

Show that an explicit formula for $u_r$ is given by $u_r = 1+ \frac {10}{3} [4^{r-1} -1]$

A sequence $u_1, u_2, u_3$,... is such that $u_1=1$ and $u_{n+1}=4u_n +7$ for $n \geqslant 1$. Write down the first four terms of the sequence. I have solved the first half of the question. $T_1 =1$...
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3answers
30 views

If ${\sqrt 3} - {\sqrt 2}, 4- {\sqrt 6}, p {\sqrt 3} - q {\sqrt 2}$ form a geometric progression, find the values of p and q.

If ${\sqrt 3} - {\sqrt 2}, 4- {\sqrt 6}, p{\sqrt 3} - q {\sqrt 2}$ form a geometric progression, find the values of p and q. So I take the second term $4-{\sqrt 6} =( {\sqrt 3} - {\sqrt 2}) (r)$ , ...
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2answers
61 views

Factor $x^2+2351x+125$ in $\mathbb{Q}[x]$.

Factor $x^2+2351x+125$ in $\mathbb{Q}[x]$ I applied the quadratic formula and received a bizarre answer that I don't believe is held in $\mathbb Q[x]$. Is there a specific method to factor in $\...
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2answers
86 views

The following relation $\frac{a}{b} =\frac{b}{a+b}$ is satisfied for $a = 1.6b$ approximately. How to reach this result? [closed]

The following relation $\frac{a}{b}=\frac{b}{a+b}$ is satisfied for $a = 1.6 b$ approximately. My question is how to reach this result, I need a detailed explanation of each step of how $\frac{a}{b}=\...
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0answers
56 views

Rational numbers and coordinates

Due to my poor English, I make the following question from this paper: What does it mean are identical off of their N Tails in the snippet below: I.e. what does it technically mean ?
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1answer
83 views

Proving if $x$ is rational then $\sqrt x$ is irrational

I'm trying to prove the statement: Let $x\in\mathbb R$. If $x$ is rational then $\sqrt x$ is irrational. I know that a number $x$ is rational if we can write it as $x=\frac{p}{q}$ for some integers $p,...
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0answers
38 views

Representation of a rational number

the euclidean algorithm of division can represent an element of $\mathbb{Q} $ by decimal expansion, I know this euclidean algorithm can't generate a period of 9. Well, what is an example of an ...
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3answers
49 views

Does set of rational numbers with odd denominators form a group with binary operation of $+$?

Does set of rational numbers with odd denominators form a group with binary operation of $+$? I think no, because it doesn't have an identity element since $0$ is not in the set because it doesn't ...
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1answer
38 views

Inclusion of $Z$ in $\mathbb{Q}$ [duplicate]

When constructing $\mathbb{Q}$ as equivalence classes containing pairs of integers, a natural inclusion of $\mathbb{Z}$ in $\mathbb{Q}$ arises: $$f: \mathbb{Z} \to \mathbb{Q}, \; n \to \frac{n}{1}. $$...
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7answers
39k views

Produce an explicit bijection between rationals and naturals?

I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but ...
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2answers
70 views

Rational approximations for $\pi$ using Fibonacci numbers?

It is (well?) known that $$\frac{\pi} 4 = \sum_{k=1}^\infty \arctan \left ( \frac1{F_{2k+1}} \right )$$ Where $F_k$ denotes the $k$-th Fibonacci number. However, any truncation of this sum is ...
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1answer
77 views

Is $\tan^{-1}\tan^{-1}1$ irrational?

Here, it is proven that $\arctan(2)$ is irrational. Here, it is proven that $\arctan(x)$ is irrational for natural $x$. By a proof similar to that from the last linked post, it can easily be shown ...
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1answer
43 views

Find the values of a, b, c

Given $$\sum_{k=0}^{\infty}\frac{k!}{\Pi_{j=0}^{k}(2j+3)} = a + b\pi^c$$ where $a$, $c$ - integers, $b$ - is a rational number. Find a, b, c. Please help or give me a tip on how to approach this ...
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2answers
38 views

Are binary, octal, decimal and hexadecimal numbers natural, whole, integer, real numbers?

I think that the title is clear in itself. But still i wanna ask whether can we consider these binary, octal, decimal and hexadecimal number formats at least as rational numbers or is it illogical to ...
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0answers
188 views

left riemann sum of dirichlet function

Note: Let $n\in Z>0$. Let $a, b\in R$ with a < b. Let y = f(x) be a continuous real-valued function on [a, b]. Let $P=\{{x_i}\}_{i=0}^n$ be a Riemann partition of [a, b], i.e., define $\...
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0answers
263 views

Is $z=e^{\frac{1}{\log(x)}}$ a solution to a polynomial?

Is $z=e^{\frac{1}{\log(x)}}$ a solution to a polynomial? $x\in\Bbb Q~\cap(0,1).$ Wolfram is telling me $z$ is algebraic but I'm not sure that I believe this. I believe it would follow from Schanuel'...
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0answers
64 views

Best possible rational approximation of $\pi$

$\frac{355}{113}$ is a surprisingly good rational approximation of $\pi$. If you define the quality of a rational approximation $\frac{a}{b}$ as minimizing $\log(ab)+\log(|\frac{a}{b}-\pi|)$, is it ...
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7answers
4k views

Axiomatic characterization of the rational numbers

We have the well-known Peano axioms for the natural numbers and the real numbers can be characterized by demanding them to be a Dedekind-complete, totally ordered field (or some variation of this). ...
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2answers
52 views

$0 < \frac{a}{b}, \frac{c}{d} < 1$, when do we have $\frac{a+c}{b+d} = \frac{a}{b} + \frac{c}{d}$? [duplicate]

Given natural numbers $a,b,c,d$, let $a,b$ be coprime with $b>a$ and let $c,d$ be coprime with $d>c$. Define a function $f:\mathbb{Q}^2 \to \mathbb{Q}$ as $$f\left(\frac xy, \frac zw\right)= \...
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1answer
32 views

Creating a sequance with rationals [closed]

I am wondering whether there exists a sequance which consists of all rational numbers.
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7answers
30k views

Why is $\frac{987654321}{123456789} = 8.0000000729?!$

Many years ago, I noticed that $987654321/123456789 = 8.0000000729\ldots$. I sent it in to Martin Gardner at Scientific American and he published it in his column!!! My life has gone downhill since ...
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1answer
838 views

Dimension of the rationals over the integers

What is the dimension of $\mathbb Q$ when it is seen as a module over the integers $\mathbb Z$ (with the usual definitions of addition and multiplication)? Initially I thought that the dimension ...
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1answer
61 views

Question about 12-tone musical scale and rational approximations

On a modern tuned instrument, an octave has twelve notes with a common frequency ratio of $2^{\frac{1}{12}}$ Of course, twelve is a very good choice for the number of notes, as $2^\frac{12}{12}=1$ ...
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1answer
45 views

Subsets $A$ of rational numbers with $| \Bbb{Q} \setminus (A-A)|=2m$

Fix a positive integer $m$ and let $\mathbb{Q} \cap (m, +\infty) = \{ r_k \}_{k \ge m}$. If $$A:= \left\{ m+r_m+\sum_{k=m}^n r_k : n \ge m-1\right\},$$ then we know that $A-A= \Bbb{Q} \setminus ([-m,m]...
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2answers
94 views

Number theory prime number conjecture

I came up with a theorem that a number $n$is prime if it is not divisible by any prime number $a \le \approx \sqrt{n}$ My proof is that past that set limit any prime number divisible would share a ...
3
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6answers
184 views

Show that $a^2+b^2+c^2$ is a square when $\frac{1}{a}+\frac{1}{b} = \frac{1}{c}$ and $a,b,c\in\mathbb{Q}$ [closed]

Knowing that $$\dfrac1a+ \dfrac1b=\dfrac1c$$ Prove that $a^2+b^2+c^2$ is a square, where $a,b,c\not=0$ are rational numbers. It can probably be solved by a quick factoring trick, but I really can’t ...
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0answers
48 views

Is this harmonic number in lowest terms?

Take the harmonic number $$\sum_{n=1}^m \frac{1}{n}=\frac{a}{b}$$ Define $a$ and $b$ as the actual integers produced by the summation - i.e., treat the numerator and denominator separately and do no ...
2
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1answer
31 views

open balls and rational centers and radii

Every open ball in $\mathbb{R}^n$ is a union of balls with rational centers and rational radii. My proof: Let $B(x,r)$ be an open ball in $\mathbb{R}^n$. Let $y\in \mathbb{Q}^n$ be such that $|x_i-...
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4answers
136 views

Is every integer a rational number? [closed]

Why is this the case? $0$ is an integer and it can't be divided by $0$... It's on my textbook, as it says We conclude that every integer is a rational number, and so the rational numbers form an ...
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4answers
195 views

Intuition behind Diophantine approximation: why do we express the bound as function of denominators?

The field $\mathbb Q$ is dense in $\mathbb R$, so we can approximate a real number $\alpha$ by an arbitrarly close rational number $r=\frac{a}{b}$. The purpose of Diophantine approximation is to find ...
39
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10answers
7k views

Can sum of a rational number and its reciprocal be an integer?

Can sum of a rational number and its reciprocal be an integer? My brother asked me this question and I was unable to answer it. The only trivial solutions which I could think of are $1$ and $-1$. ...
4
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3answers
84 views

The set of differences of square rationals

At first, we observe that $A:=\{ p^2-m^2 : p,m\in \Bbb{Z}\}=\mathbb{Z}\setminus (4\mathbb{Z}+2)$ (because an integer $a$ can be written as the form $a=p^2-m^2$ if and only if $a\neq 4k+2$, for every ...
12
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1answer
306 views

For $A \in \mathcal{M}_3(\mathbb{C})$, does $\mathrm{tr}(A^2) = \mathrm{tr}(A^3) \in \mathbb{Q}$ imply $\mathrm{tr}(A^4) \in \mathbb{Q}$?

Let $A \in \mathcal{M}_3(\mathbb{C})$ such that $\mathrm{tr}(A^2) = \mathrm{tr}(A^3) \in \mathbb{Q},$ where $\mathrm{tr}(A)$ is the trace of $A.$ It is possible to prove that $\mathrm{tr}(A^4) \in \...
2
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1answer
54 views

Proving that $\mathbb{R}$ is a field

Assuming that we have the definition of the rational numbers $\mathbb{Q}$ we define $\mathbb{R}$ as the completion of $\mathbb{Q}$ with respect to the usual norm. I want to understand why this gives a ...
1
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1answer
77 views

Why do we see the sets in this order? [closed]

There is the following question: Why do we see in maths first the set of integers $\mathbb{Z}$ and then the set of rational numbers $\mathbb{Q}$ but in school we see first the rationals and the ...

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