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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21 views

Rational number sets which grow in cardinality and a related sequence of numbers.

Define the collection of sets $\left\{Q_n\right\}$ as follows: $$Q_1 = \{0\};\quad Q_2 = \left\{\frac{1}{2}\right\} \cup Q_1; \quad Q_3 = \left\{\frac{1}{3}, \frac{2}{3}\right\} \cup Q_2;$$ $$\left\{\...
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1answer
103 views

Five different positive non-integer rational numbers such that increased by 1 the product of any two numbers is a square of some rational number

Find at least one example of five rational numbers $x_1, \; x_2, ..., \; x_5$ such that i) $x_k > 0$ for all $k=1,2,...,5$; ii) $x_k$ is not an integer for all $k=1,2,...,5$; iii) if $k \ne m$ ...
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20 views

Image of rational points of a morphism (defined over $\mathbb{Q}$) between affine spaces

Let $\phi: \mathbb{A}^n \rightarrow \mathbb{A}^m$ be a polynomial map, where each coordinate is a polynomial defined over $\mathbb{Q}$. Suppose $\phi$ is dominant so the Zarishi closure of $\phi(\...
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1answer
58 views

Are the authors sloppy when forgetting the property “with unity” of $\mathbb Z$ in Remark (b)?

I am reading Section 9. The Rational Numbers from textbook Analysis I by Amann/Escher. They present two theorems: and then say It is stated in Theorem 9.1 that $\mathbb Z$ is a smallest ...
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37 views

There is, up to isomorphism, a unique smallest field $\mathbb{Q}$, which contains $\mathbb{Z}$ as a subring

I am reading Section 9. The Rational Numbers from textbook Analysis I by Amann/Escher, where there is a theorem: I would like to confirm if my understanding about the proof (which leaves ...
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1answer
42 views

How do the authors justify that $\mathbb{Z}$, by construction, is clearly minimal?

I am reading Section 9. The Rational Numbers from textbook Analysis I by Amann/Escher, where there is a theorem: My thought: smallest means that If there is another domain with unity, $\...
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46 views

Help with this proof; Numbers arbitrarily close to square root of 2

I am having a little trouble understanding this proof. For every rational number $\varepsilon > 0$, there exists a non-negative rational number $x$ such that $x^2 < 2 < (x + \varepsilon)^2$. ...
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26 views

Optimal Fixed-Digit Rational Approximations of $\pi$

Is there a systematic way of finding optimal rational approximations to $\pi$ whose numerator and denominator have at most $n$ digits? More precisely: Let $D_n$ be the set of all positive integers ...
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1answer
87 views

Niven’s theorem proof.

I couldn’t find any proof on the internet. I would appreciate an elementary proof , but any help would be appreciated! Thank you.
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42 views

Is there an ordering on $\mathrm Z$ to make it dense?

It is well-known that the set of rational numbers $\mathrm Q$ is dense; that is, given any two rationals, say $r\ne s,$ then there exist infinitely many rationals $r_i$ such that $r<r_i<s$ for ...
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25 views

Simplify the denominator

having difficulties getting this..How to simplify by rationalizing the denominator? $(\frac{\sqrt2+1}{\sqrt2-1})^2-(\frac{\sqrt2-1}{\sqrt2+1})^2$
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$\sqrt{11}$ as rational number within $10^{-4}$?

Use continued fractions to find a rational number which approximates $\sqrt{11}$ to within $10^{−4}$. I know how to solve for continued fractions like this: $$\sqrt{11}=3+x$$ $$11=9+6x+x^2$$ $$11=9+(...
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39 views

Sec. 6 in G.F. Simmons' INTRO TO TOPOLOGY & MODERN ANALYSIS: A formula for the general terms of these two sequences please?

The set $\mathbb{Q}^+$ of all the positive rational numbers is given by $$ \mathbf{Q}^+ = \left\{ \ \frac{p}{q} \ \colon \ p \in \mathbf{N}, q \in \mathbf{N} \ \right\}.$$ This set is countably ...
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299 views

LCM of irrational numbers [closed]

So i read in a book that irrational and rational numbers do not have a common multiple and it said that lcm of irrational numbers is also only possible when both the irrational numbers have the same ...
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1answer
30 views

Dedekind cut corresponding to the limit of a Cauchy sequence

Let $a : \mathbb{N} \rightarrow \mathbb{Q}$ be a Cauchy sequence of rationals. Then is it correct to say that $$\lim_{n \rightarrow \infty} a_n = \{x \in \mathbb{Q} : \exists n \in \mathbb{N} : \...
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26 views

Means and Set of Rational Numbers

Let $S$ be the set $\{0, 1\}$. Given any subset of $S$ we may add its arithmetic mean to $S$ (provided it is not already included - $S$ never includes duplicates). Show that by repeating this process ...
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98 views

Rudin's proof of theorem 1.20b (Archimedean Principle)

The theorem states: (a) If $x\in \mathbb{R}, y\in \mathbb{R}$, and $x>0$, then there is a positive integer $n$ such that $$nx>y.$$ (b) If $x\in \mathbb{R}, y\in \mathbb{R}$, and $x<y$, ...
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1answer
32 views

Proof of Density of Rational Numbers as given in Howie

I'm working out of John M. Howie's Real Analysis (2001). In section 1.4 (Exercise 1.5), we're asked the following: Let $x$ and $y$ be real numbers, with $ x < y $. Show that, if $x$ and $y$ are ...
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1answer
53 views

Alternation of Rationals and Irrationals?

I'm in a lunch group at work of recreational math geeks and we came up with a question which we need help to resolve. I apologize in advance, if my explanation is not perfectly rigorous. Given these ...
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What numbers can be approximated by ratios of numbers containing only specified digits?

Let $D$ be a subset of the decimal digits $ \{ 0, 1, 2, \ldots, 9\}$, with $D \neq \{0\}$ or $\emptyset$. Let $N$ be the set of positive integers whose decimal representations (without leading $0$'s) ...
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90 views

Are Rationals constructed from infinite Naturals valid? [closed]

I can construct a Rational like $3/4$. And then I can construct anther one like $31/41$, and then another like $311/411$. I can envisage a Rational whose numerator is $31111111...$ and denominator ...
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1answer
37 views

Numerator and denominator of rational number

I have to prove that $\sqrt5$ is irrational. I prove it by contradiction. I assume that $p$ is an integer and $q$ is a positive integer such that $gcd(p,q)=1$ and $(\frac{p}{q})^2 = 5$. And then ...
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2answers
36 views

Prove that $a$ and $b$ are rational numbers [closed]

If $a+b$, $a^2+b$ and $b^2+a$ are rational numbers and $a+b\neq 1$ then $a$ and $b$ are rational. I try, sum the expresions but I only got that $a^2+b^2$ and $ab$ are rational. Any suggestion?
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Can we fill the plane with a certain operation?

Background: Paint the origin $(0,0)$ black in $\mathbb{R}^2$. Let $S$ be a set $\{ (x,y) \in \mathbb{R}^2 ~|~ x^2 + y^2 =1 \}$. Paint $S$ black. Paint $(u,v) +S$ black for all $(u,v) \in S$. (...
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Recurring Decimal in base 3

I have this decimal $y=0.012012012012....$ I was wondering how I could put this into the form of a rational number in base 3. So far I have $y= \frac{0}{3}+\frac{1}{3^2}+\frac{2}{3^3}+\frac{0}{3^4}......
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3answers
87 views

Famous fractions: Can any “special” numbers be approximated by simple ratios like $3.14\ldots$ as $22/7$?

The ratio $22/7$ dates back to antiquity as an approximation of $3.14\ldots$. I'm wondering whether there are any other "famous" numbers with a similar situation. That is, something like $e$ or $\phi$ ...
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1answer
71 views

Are there $2^{\aleph_{0} }$ sets of natural numbers such that each two have finite intersection [duplicate]

Question: Are there $2^{\aleph_{0} }$ sets of natural numbers such that each two have finite intersection. From what I've read about infinite families, I need to ignore those who have the properpty $...
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127 views

Prob. 1, Sec. 29, in Munkres' TOPOLOGY, 2nd ed: The subspace $\mathbb{Q}$ of $\mathbb{R}$ is not locally compact

Here is Prob. 1, Sec. 29, in the book Topology by James R. Munkres, 2nd edition: Show that the rationals $\mathbb{Q}$ are not locally compact. My Attempt: Here the topology on the set $\mathbb{...
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179 views

What is $\mathbb{Q}$?

When we say set of rationals $\mathbb{Q}$, which of the following does it refer to? $$\left\{\frac{p}{q}~|~p,q\in\mathbb{Z},q\neq 0\right\}$$ or $$\left\{\left[\frac{p}{q}\right]~|~p,q\in\mathbb{Z},...
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3answers
264 views

Suppose that $x$ and $y$ are irrational, but $x + y$ is rational. Prove that $x -y$ is irrational.

i was wondering if someone could check my proof $Q= \{a/b , c,d : a,c ∈ \mathbb Z , b,d ∈ N>0\}$ $a/b =x+y$ $a/b -y=x$ proof by contradiction. Let $x-y$ is rational $c/d = x-y$ sub $a/b -y = ...
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38 views

Proving the negation of a conditional using proof by contradiction

CONTEXT: Question made up by uni maths lecturer Prove the following statement using a proof by contradiction: For all nonzero rational numbers $x$, if $y$ is irrational then $\frac{x}{y}-3$ is ...
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1answer
85 views

Define the set $E(\mathbb{Q})$ of $\mathbb{Q}$-rational points on an elliptic curve

I'm a bit struggling with defining the set of $\mathbb{Q}$-rational points on an elliptic curve $E:\;y^2=x^3+ax^2+bx+c$ with $a,b,c\in\mathbb{Q}$. I'm actually guessing that If we let $K$ and $L$ be ...
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74 views

Dedekind cut/additional property

Consider the following lemma and its proof. My question follows. Let $(P,<)$ be a dense unbounded linearly ordered set. Then there is a complete unbounded linearly ordered set $(C,\prec)$ such ...
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75 views

Does $x^x$ exists for certain rational numbers at $x<0$?

There have been debates in mathematics that $x^x$ cannot exist for any rational numbers for $x<0$ since $${x}^{x}=e^{x\ln(x)} \ \text{for all}\ x$$ and for $x=-1/1=-2/2$ $$(-1)^{-1/1}\neq(-1)^{-...
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Analytic functions on $\mathbb{Q}$

$\mathbb{Q}$ has the topology induced from $\mathbb{R}$, therefore it is in principle possible to talk about power series and define analytic functions on $\mathbb{Q}$ to be power series (with ...
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138 views

Does there exist a bijection $f$ from $\mathbb{N}$ to $\mathbb{Q}^+$ such that $\lim_{n \to \infty} \frac{f(n+1)}{f(n)}$ exists?

Does there exist a bijection $f$ from $\mathbb{N}$ to $\mathbb{Q}^+$ such that $$\lim_{n \to \infty} \frac{f(n+1)}{f(n)}$$ exists? My guess that no such $f$ exists.
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Numbers between decimal number

So there is a question that says I can choose a number between $0.00001$ and $0.1$, but the problem is what exactly comes after $0.00001$? Would it be $0.0001$, continued by $0.001$, $0.01$, $0.1$? ...
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Rainbow numbers: Can mapping digits to different bases produce different varieties of irrationality?

This is a follow-up to the question, "Irrationality of 0.123456789101112 … and similar numbers." There I took some decimal number, in one case Champernowne's constant, $$ n_{10} = 0....
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If $n \in \mathbb{Z}^+$ and $x \in \mathbb{R}$, show that $x \in \mathbb{Q}$ if $n x \in \mathbb{Q}$

For this question would I be correct in stating the following If $n$ is a positive integer and $n x$ is a rational number then $n x = n (a / b)$. Simplifying this = $x = (a / b)$ so $x$ must also ...
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1answer
61 views

De-rationalisation of a surd expression $\sqrt p - \sqrt {pq} + q$

Consider two dissimilar surds $\sqrt p$ and $\sqrt q$. Then the problem asks to find rational numbers $a,b,c$ and $d$ such that for $x=\sqrt p + \sqrt q$ we can write, $$ \sqrt p - \sqrt {pq} + q = \...
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44 views

Cauchy sequences of rationals with limit irrational: natural, or geometric examples

As we know, real numbers are constructed by filling up gaps between rationals by the limits of all Cauchy sequences of rationals. Q. What are examples of sequence of rationals such that its easy ...
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17 views

How to express that for $\beta \in (0,1)$, $\bigg(\frac{|x_i|}{i} \bigg)^2 \leq \beta^2 |x_i|^2$?

How to express that for $\beta \in (0,1)$, $\bigg(\frac{|x_i|}{i} \bigg)^2 \leq \beta^2 |x_i|^2$? What I want to say is that in R.H.S. the $|x_i|^2$ is divided by something smaller than $i$. and thus ...
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1answer
53 views

Abelian groups about rationals [duplicate]

Is the set $\mathbb{Q}$ under $×$ an abelian group? It is sure for $\mathbb{Q} - {0}$, but i think the whole set of rationals is not an abelian group as $0 × a = a × 0 = 0$, but the identity element ...
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1answer
30 views

“The order of a torus link can be understood as a rational number”

The order of a torus link consists of a pair of integers $(m,n)$, with at least one of them nonzero, and it is such that if the two integers are not coprime, i.e. of the form $(km, kn)$, the link ...
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2answers
54 views

Integer solutions of $ X+Y+Z=X\cdot Y\cdot Z $ [closed]

An integer solution of above equation is $(X,Y,Z)=(1,2,3)$. But I am wondering: are there other natural solutions? And what about rational or irrational solutions, where $X,Y,Z$ are different ...
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268 views

Why do we run in diagonals when proving that $\mathbb{Q}$ is countable?

Why do we index the elements like this but not finishing the 1/x elements and then going through 2/x then 3/x...
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49 views

Construction of Rational Numbers without quotients

The context is Intensional Type Theory, where quotients are unavailable. I managed to construct Integers in this way: $\mathbb{Z}:=(\mathbb{N}^+\times\{{+,-\}})+\{{0\}}$, but I can't see a way to ...
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1answer
81 views

If $x^2$ and $x^3$ are rational, does it imply that $x$ is rational? [closed]

It is given that $x^2$ is rational and $x^3$ is rational. Is $x$ rational for all cases satisfying these conditions or is there are case where $x$ won't be rational? If so, then what other condition(...
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1answer
98 views

Questions about 0.999… equals 1 [closed]

Being 0.999... = 1, I expect that they have the same behaviour when applying the same algorithm/operation, but: If we define >, <, =, as checking digit by digit two number, we have that 0 < 1 ...
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1answer
71 views

Does $x^3 - \frac{m}{n}\sqrt{5}x - 1$ has rational root?

I am trying to show whether $p(x) = x^3 - \frac{m}{n}\sqrt{5}x - 1$ has a rational root or not, where $\frac{m}{n}$ is rational. My attempt so far is to turn $p(x)$ into another polynomial $q(x) = ( - ...