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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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Alternative solution to a problem involving an enumeration of rationals

I'm working on the same problem as in this post. I understand the solutions provided in the answers. The question basically asks us to find an enumeration of the rationals $\{r_n\}_{n≥1}$ such that ...
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3answers
49 views

Can the product of two rational numbers be an irrational number? (Kindly see the example in description)

I checked in many sources and I saw "Multiplication is closed under Rational Numbers Q". But consider $$ a = \frac{1}{7} ; \;\;\; b = \frac{22}{1} ;$$ both a, b are individually rational (either ...
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How to show that $\mathbb{Q}$ and $\mathbb{Q}^2$ are elementarily equivalent?

I try to prove that $(\mathbb{Q},P)$ and $(\mathbb{Q}^2,P)$ are elementarily equivalent using Ehrenfeucht–Fraïssé games $P(a,b,c)=True$ when $a+b=c$ $+$ on $\mathbb{Q}^2$ is defined as $(x,x')+(y,y'...
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2answers
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Prove or disprove that there is a rational number $x$ and an irrational number $y$ such that $x^y$ is irrational.

In solving the following problem: Prove or disprove that there is a rational number $x$ and an irrational number $y$ such that $x^y$ is irrational. I let $x=2$ and $y = \sqrt 2$, so that $x^y = 2^\...
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4answers
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Cardinality of set of sequences from $\mathbb{Q}$ that converge to $0$

I'm looking for cardinality of a set of sequences from $\mathbb{Q}$ which are convergent to $0$. I think the answer is continuum, but I don't know how to prove it.
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Can nth root of 2 be a rational number for any natural number n > 1?

We know square and cube root of 2 are irrational numbers, but is it possible for any a rational number to be multiplied by itself finite times and it results in 2.
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1answer
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How can an convergent series of rational numbers result in a irrational number?

In mathematics, nearly all significant irrational numbers can be expressed as a sum of an infinite convergent series, but according to law of addition of rational numbers, adding any to rational ...
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2answers
222 views

Sixth grade math (number related) problem

We have this statement (about rational numbers, btw): If $m-n+p = p$ and $ m \neq n \neq 0$ then $ m = -n$ Is this true? a) always b) never c) sometimes The given answer is b) but: ...
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2answers
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Can the square root of a non square be rational? [duplicate]

It seems pretty obvious that the square root of a non-square is irrational, but is it always true? If so, what is the proof? It would help if there was any intuitive geometric proof if the math is too ...
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Roots of polynomial irreducible over the rationals

If a polynomial is "irreducible over the rationals", does it mean that it has no rational roots? I would say yes because otherwise I could divide out the linear factors (i.e. rational roots) but ...
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Proving that if $x_1,\dots,x_n$ are rational numbers and $\sqrt{x_1}+\dots\sqrt{x_n}$ is rational, then each $\sqrt{x_i}$ is rational as well

I'm having a hard time with the following problem: Let $x_1,x_2...x_n$ be rational numbers. Prove that if the sum $\sqrt{x_1}+\sqrt{x_2}+...+\sqrt{x_n}$ is rational, then all $\sqrt{x_i}$ are ...
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1answer
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Check my proof of 0 < |r - q| < epsilon. (Real # - Rational #) [duplicate]

I am working on this exercise: $$ \forall \varepsilon > 0, \ \exists q \in Q \text{ where } 0 < |r - q| < \varepsilon $$ To clarify, r is a real number, q is a rational number. This is ...
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5answers
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Find two irrational numbers $x,y$ such that $x+y$ and $xy$ are both rational.

I know how to satisfy one of the statements but never both together. $(a+b)*(a-b)=a^2-b^2$ so taking $a=\sqrt k_1$ and $b=\sqrt k_2$, $k_1,k_2\in\mathbb{Q}$ such that $a,b \notin\mathbb{Q}$ would ...
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1answer
55 views

Determine all triples $(m,n,p)$ of positive rational numbers.

Determine all triples $(m,n,p)$ of positive rational numbers such that the numbers $m+\frac{1}{np}, n+\frac{1}{pm}, p+\frac{1}{mn}$ are integers. I have no idea how to go about. Please help.
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Real Analysis Inequality Proof Involving Reals and Rationals $0 < |r - q| < \varepsilon$

I'm having difficulties making progress in proving: $$\forall \varepsilon > 0, \ \exists q \in Q \text{ where } 0 < |r - q| < \varepsilon $$ To clarify, $r$ is a real number and $q$ is a ...
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1answer
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Definition of rational numbers from real numbers

Usually the set of numbers are introduced starting from integers, from wich the rational numbers are defined using equivalence classes of couples of integer numbers. Than, from these rational numbers, ...
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3answers
102 views

Showing the irreducibility of $x^8 - 60 x^6 + 1160 x^4 - 7800 x^2 + 8836$ in $\mathbb{Q}[x]$

I would like to show the irreducibility of $x^8 - 60 x^6 + 1160 x^4 - 7800 x^2 + 8836$ and $x^8 - 120 x^6 + 4360 x^4 - 45600 x^2 + 15376$ in $\mathbb{Q}[x]$. In both cases Eisenstein criterion fails. ...
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2answers
132 views

$\frac{7x+1}2, \frac{7x+2}3, \frac{7x+3}4, \ldots ,\frac{7x+2016}{2017}$ are reduced fractions for integers $x\in(0,301)$. [closed]

BdMO 2017 junior catagory Question 7. $$\dfrac{7x+1}2, \dfrac{7x+2}3, \dfrac{7x+3}4, \ldots ,\dfrac{7x+2016}{2017}$$ Here $x$ is a positive integer and $x < 301$. For some values of $x$ it is ...
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2answers
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Finding irrational entries such that the determinant will never be zero

Context. The main goal is to find whether or not a subspace of $\mathbb R^5$ of dimension $3$ intersects a rational subspace of dimension $2$. By rational subspace, we mean a subspace of $\mathbb R^5$...
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For which x the following sequences converges

If $q_n$ be an enumeration of rational numbers, for which $x$ the following sequence converges? $$\sum_{n=1}^{\infty}e^{-n^2|x-q_n|}.$$ I guess that for no $x$ the sequence converges. I tried to ...
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1answer
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How is it that there are 'gaps' in rational numbers and yet between any two rational numbers, there exists another rational number?

If there are gaps in rational numbers then lets assume we have a gap between a and b, both being rational. Then we have $\frac{a+b}{2}$ which is inside the gap which essentially makes it a non-gap. ...
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2answers
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Apostol proof for $\mathbb{Q}$ being countable.

I am trying to understand a proof from Mathematical Analysis by Apostol for the following theorem: The set of rationals $\mathbb{Q}$ is countable. Here is the proof (I rewrote a few things): ...
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1answer
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Proving the Set of Rational Numbers is the collection of equivalence classes of ratios of integers with nonzero denominators.

I read in a textbook that the set of Rational Numbers is the collection of equivalence classes of ratios of integers with nonzero denominators. I found it interesting and tried to prove it but ended ...
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1answer
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Recurring Decimal Expansion

For any natural number $n>1$, we write the infinite decimal expansion of $\frac 1n$ (for example, $\frac 14$ is written as $0.24999$... instead of $0.25$). We need to determine the length of the ...
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Question on proving that the rationals are countably infinite

I just a question on a proof for the rationals being countably infinite from a textbook. We consider the following function, a mapping from $\Bbb Q$ to $\Bbb N$ $$f(x) = \begin{cases} 0, & \text{...
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Showing $\mathbb{Q} \cap [a,b]$ is an open set in $\mathbb{Q}$ for irrational $a$, $b$.

I came up with this lemma (although not confident enough about it) while solving Baby Rudin. In the chapter "Basic Topology", I attempted to solve question 16, in which $\mathbb{Q}$ is regarded as the ...
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1answer
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Prove at least one of the length, width, height of the box must be rational.

Assume there is a big box be combined by finite small boxes and that the small boxes are not necessarily be the same. The statement in my note is " If there is at least one of length, width, height ...
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1answer
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Is this a valid definition of the rationals?

$$\mathbb{Q}=\left\{\sum_{n=1}^k f(n)\mid k,n\in\mathbb{N}\land f\text{ is a finite composition of $+$, $-$, $\div$, $\times$}\right\}$$ Reasoning: Any real number can be described by a (sometimes ...
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1answer
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Handling opposites when adding and subtracting rational expressions

I'm following example 8.47 from the OpenStax book Elementary Algebra. When trying to create a common denominator it is sometimes necessary to handle opposites on either side of an equation. In the ...
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Proving fraction is irreducible

Example: The fraction $\frac{4n+7}{3n+5}$ is irreducible for all $n \in \mathbb{N}$, because $3(4n+7) - 4(3n+5) = 1$ and if $d$ is divisor of $4n+7$ and $3n+5$, it divides $1$, so $d=1$. I want to ...
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Sequence of Number System Construction

After constructing the naturals, why construct integers before rationals? Is there a historical explanation? Couldn't ordered pairs of fractions constructed from the naturals be used to represent ...
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Roots of polynomials are Gaussian integers

I have got a question. I want to show the following: Let P be a normalized polynomial with integer coefficients and let w be a root of this polynomial (in $\mathbb{Q}[i]$), then w is a Gaussian ...
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Finding a subspace of dimension $3$ which does not intersect a rational subspace of dimension $2$

Some context. By rational subspace, I mean a subspace of $\mathbb R^5$ which admits a rational basis. In other words, a basis formed with vectors of $\mathbb Q^5$. For instance, the vector $v=(0,\pi,...
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Strange sum that always end up with 9

If we have any number, example 4896, and sum all digits sum = 4+8+9+6 = 27 and than substract this number from the original number, we always get a number that is divisible by 9: 4896-27=4869 -> ...
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Help proving there is a sequence of rational numbers

I'm trying to prove the following: Let $\Bbb Q$ be the countable set of rational numbers and $\{x_n\}_{n=1}^\infty$ be a sequence such that for every q $\in$ $\Bbb Q$ there is a $n \in \Bbb N$ with $...
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1answer
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Find natural number $0 < n < 30,000$ such that $\sqrt[3]{5n}+\sqrt{10n}$ is rational

I was thinking that I could try to make some sort of substitution to convert $\sqrt[3]{5n}+\sqrt{10n}$ into a polynomial with integer coefficients then use the Rational Roots Theorem to find a ...
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Am I right in this proof of a criterion for the nonsingularity of a conic curve?

$\newcommand{\C}{\mathcal{C}}$ This is an exercise in Silverman and Tate's Rational Points on Elliptic Curves: Let $\C$ be the conic given by the equation $$ F(x,y)=ax^2+bxy+cy^2+dx+ey+f=0.$$ ...
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Unique Rational Approximation With “Small” Denominator

Suppose we have some irrational $x > 0$ and some $\epsilon > 0$. I want to show that there is at most one rational approximation $\frac{a}{b}$ such that both $| x - \frac{a}{b}| < \epsilon$ ...
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1answer
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The oscillation of a bounded function at a point

Enumerate the rationals in $[0,1]$ (ie. $\mathbb{Q}\cap[0,1]$) by $q_n$. Define $f:[0,1]\to\mathbb{R}$ by $$ f(x)= \begin{cases} 1/n & \text{if } x=q_n \text{ for some }n\\ 0 & \text{...
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$f(a)-f(b)$ is rational iff $f(a-b) $ is rational

Prove that the continuous function $f:\mathbb{R} \to \mathbb{R}$ satisfying $f\left(x\right)-f\left(y\right) \in\mathbb{Q} \iff f\left(x-y\right) \in \mathbb{Q}$ is of the form $ f\left(x\right)=ax+...
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1answer
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For $0 \le \theta \le \pi/2$, When are both $\theta/\pi$ and $\sqrt2\sin\theta$ rational?

For $0 \le \theta \le \pi/2$, when are both $\theta/\pi$ and $\sqrt2\sin\theta$ rational? I think $\theta=0, \pi/4$ is the only cases. This problem seems to be related to Niven's theorem, but I ...
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1answer
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Definition of rational numbers

We can uniquely define the set of real numbers as a complete ordered field. But can we do something similar with the set of rational numbers ? I think we need to change the completness axiom with some ...
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2answers
79 views

Rational solution to a system of equations

Some context. By rational subspace, I mean a subspace of $\mathbb R^5$ which admits a rational basis. In other words, a basis formed with vectors of $\mathbb Q^5$. For instance, the vector $v=(0,\pi,...
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48 views

Show $(3 + \sqrt{2})^{2/3}$ is irrational using RZT

I am asked to prove that $(3+\sqrt{2})^{2/3}$ is irrational via the rational zeroes theorem. This is what I have so far: $ x = (3+\sqrt{2})^{2/3} $ $ x^3 = (3+\sqrt{2})^{2} $ $ x^3 - 11 - 6\sqrt{2}...
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Prove or disprove non-constructively there exist irrationals $a, b, c$ such that $a^{b^c}$ is rational.

Consider the interesting question: Do there exist irrationals $a$ and $b$ such that $a^b$ is a rational? Alternatively, prove or disprove that there exist irrationals $a$ and $b$ such that $a^b$ is ...
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3answers
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Determine all $q \in \mathbb{Q}$, so that $ \sum_{n=1}^{\infty}{\frac{\sqrt{n+1}-\sqrt{n}}{n^q}} $ converges

I already tried the ratio and root criterion, but it didn't get me anywhere. I'd be thrilled if you had any suggestions. (also applied the third binomial formula, so it would "look nicer", maybe it ...
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1answer
328 views

Does this polynomial have a rational value which is the square of a rational number?

I have the following polynomial: $$P(x,y,z):=9y^2z^2-30x^2z+90xyz+54yz-270x+81\in\mathbb Q[x].$$ It came up in a larger proof, and I would need in order to complete the proof to prove the following ...
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3answers
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Proof by Induction Question including Rational Numbers

I just recently covered 'rational numbers' in class and was assigned the following question to solve using induction for n, so that for all $q \in \mathbb{Q}$ \ {1}:...
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A question construction of rational numbers

In $\mathbb{Z}$ we can do addition, substraction and multiplication, but not division. For example we cannot divide $2$ by $3$ (i.e. the equation $3x=2$ has no solution in $\mathbb{Z}$, because $\...
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3answers
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Silly Question about $π$ [closed]

In our junior classes, we learnt that $π$ is an irrational number. Now, also we know about rational numbers. So, if I say that I have a thread of 44cm long and we may convert it into a circle. Then ...