# 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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### 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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### 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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### 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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### 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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### 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 $p/n < a$. ...
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### 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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### $P(x)\in \mathbb{R}[x]$, 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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### 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 ...
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### 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 ...
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### 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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### Creating a sequance with rationals [closed]

I am wondering whether there exists a sequance which consists of all rational numbers.
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### 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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### 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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### 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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### 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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### 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 ...
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### 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 $-1$. ...
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 ...