For questions on rational numbers, numbers that can be expressed as the quotient or fraction $\frac pq$ of two integers.

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0
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1answer
71 views

Dimension of the rationals over the integers

What is the dimension of the $Q$, when they are seen as a vector space over the integers $Z$ (with the usual definitions of addition and multiplication)? Initially I thought that the dimension ought ...
1
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1answer
103 views

An example of a group operation on the rationals, which is not isomorphic to the additive group

I'm looking for an example of a group operation on the rationals, which is not isomorphic to the rational additive group. Can you find such an example?
-4
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2answers
259 views

Do irrational numbers really exist?

Isn't it possible that an irrational number is in reality the quotient of two infinitely long integers that even if there were repeating sections in it, it would take infinite digits to find the first ...
18
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2answers
514 views

Order preserving bijection from $\mathbb Q\times \mathbb Q$ to $\mathbb Q$

Do you have a simple example of order preserving bijection from $\mathbb Q\times \mathbb Q$ to $\mathbb Q$? On $\mathbb Q$, I use the usual order and on $\mathbb Q\times\mathbb Q$, I define the ...
2
votes
1answer
300 views

Is every injective rational function $f:\mathbb Q\to\mathbb Q$ a polynomial?

I thought this might be quite easy to show, and then realized that the tools I know from real analysis aren't going to help here. Suppose we have a rational function: $$ f(X)=\frac{P(X)}{Q(X)} $$ ...
13
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6answers
821 views

Is there a function that gives the same result for a number and its reciprocal?

Is there a (non-piecewise, non-trivial) function where $f(x) = f(\frac{1}{x})$? Why? It would be nice to compare ratios without worrying about the ordering of numerator and denominator. For example, ...
5
votes
5answers
357 views

Odd divided by even is a fraction

How can we prove that an odd number divided by an even number is a fraction? I started with odd $=2m+1$ and even $=2n$ and get left with with $(m+2)/n$.
1
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1answer
51 views

Complex Numbers, Complicated Powers

We know there are two non-real imaginary numbers like $a$, $b$ such that the power $a^{b}$ is a real number. For example we have $i^{i}=\frac{1}{\sqrt{e^{\pi}}}$. Question: Are there two non-real ...
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1answer
571 views
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2answers
89 views

Rational numbers in base 6

Express the fractions , for several small values of , in base 6. Determine which rational numbers have terminating expressions in base 6. I am unsure how to reduce fractions in base 6. For example, ...
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4answers
247 views

How can I explain $0.999\ldots=1$? [duplicate]

Possible Duplicate: Does .99999… = 1? I have to explain $0.999\ldots=1$ to people who don't know limit. How can I explain $0.999\ldots=1$? The common procedure is as follows ...
3
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2answers
73 views

Existence of five real numbers satisfying a given condition.

Let $a_1,\dots,a_5$ be five distinct non-zero real numbers. Suppose that for $i\neq j$ either $a_i+a_j$ or $a_ia_j$ or both are rational numbers, does it implies that $a_i^2$ are rational numbers for ...
0
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1answer
111 views

subring of rational numbers and its ideal

Let $p$ be a prime number. For any $p$ the subring $\mathbb{Q}_p$ of of the field of rational numbers is defined: $\mathbb{Q}_p=\{\frac{a}{b}|a,b\mbox{ are integers, $p$ does not divide $b$}\}$ Let ...
4
votes
2answers
141 views

Rational solutions of $x^3+y^3=2$

I came along the problem of finding three perfect cubes that are consecutive numbers of an arithmetic progression, i.e: $a^3-b^3=b^3-c^3$, where $a>b>c$ (to avoid trivial solutions). Clearly it ...
3
votes
0answers
131 views

Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number.

Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number. I thought this question might have been asked by someone else, but I couldn't find any. Currently I'm studying ...
5
votes
1answer
143 views

If $q>1$ is not an integer, can $q^n$ be made arbitrarily close to integers?

This question arose when I heard about Mill's constant: the number $A$ such that $\lfloor A^{3^n} \rfloor$ is prime for all $n$. It made me wonder whether $A^{3^n}$ could be made arbitrarily close to ...
3
votes
1answer
40 views

Is the fraction of the irrational exponentiations of two coprime integers by a rational an irrational?

Consider two strictly positive integer coprimes $n, m\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. Consider furthermore that the three number statifies the following condition: ...
2
votes
2answers
93 views

Can the exponentiation of an integer by a rational be a non-integer rational?

Consider a strictly positive integer $n\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. My question is the following: what is the nature of $n^r$? My first guess is that $n^r$ is an ...
1
vote
0answers
22 views

Rational exponentiation?

Consider the following operation: $\left(\frac{a}{b}\right)^\frac{n}{m}$ where $a, n\in\mathbb{Z}$ and $b, m\in\mathbb{N^*}$. My question is: when the result is a rational number, how (formula or ...
1
vote
7answers
393 views

Why is a repeating decimal a rational number?

$$\frac{1}{3}=.33\bar{3}$$ is a rational number, but the $3$ keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ...
1
vote
3answers
104 views

Wrong Answer - Rewrite Rational Number as a Fraction.

This number 2.962962 can be rational $$x=2.962962$$ $$10x=29.62962$$ $$100x=296.2962$$ $$1000x=2962.962$$ $$1000x-10x=\frac{990x}{990}=\frac{2933}{990}$$ why is this wrong? That way of getting the ...
5
votes
4answers
2k views

Infinite number of rationals between any two reals.

Let $a$ and $b$ be reals with $a<b$. Show that there are infinitely many rationals $x$ such that $a<x<b$. My plan of action was to assume that $x$ is the smallest such rational and find ...
2
votes
4answers
291 views

Show that the integers have infinite index in the additive group of rational numbers.

Show that the integers have infinite index in the additive group of rational numbers. Anybody in a good enough mood to tell me how this is done?
1
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2answers
161 views

Does the set of rational numbers between 0 and 2 have the least upper bound property?

Let $A = \{ a \in Q : 0 < a < 2\}$ Does A have the least upper bound property? Definition: $A$ has the least upper bound property if $\forall$ nonempty $B \subseteq A$, if $B$ has an upper ...
21
votes
2answers
445 views

Is $\{\tan(x) : x\in \mathbb{Q}\}$ a group under addition?

A student asked me the following today : Is $S:= \{\tan(x) : x\in \mathbb{Q}\}$ a group under addition? I am quite perplexed by it. Clearly, the only non-trivial part is to check For any $x, ...
6
votes
4answers
305 views

Is there a bijection $f: \mathbb{Q} \to \mathbb{Q}_{>0}$?

For $\mathbb{R}$, we have the exponential function. Is there also a bijection $f: \mathbb{Q} \to \mathbb{Q}_{>0}$ or to $\mathbb{Q}_{\geq 0}$?
4
votes
1answer
146 views

Is $\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$ a rational number for $m,n\ge 2\in\mathbb N$?

Question : Is $$\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$$ a rational number for $m,n\ge 2\in\mathbb N$ where $\zeta (s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$? Motivation : We know that $$\zeta ...
1
vote
1answer
112 views

Does this equation have a rational point? (Elliptic curve?)

Can someone check pls if, $$852 + 3017 x - 1104 x^2 + 2009 x^3 - 3362 x^4=y^2$$ has a rational point? (This arose in an equal sums of like powers problem.) P.S. I've checked $x=p/q$ for ...
4
votes
3answers
1k views

Prove that $x^3 + x^2 = 1$ has no rational solutions?

Is this enough for a proof?: $$x^3+x^2 = 1$$ I would factor and get: $x^2(x+1) = 1$ I would show that $x = \sqrt1$, which is irrational but then do I have to show more? $x+1=1$ which gives me $x=0$ ...
5
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6answers
2k views

what's the difference between a rational number and an irrational number?

I tried to understand the difference between rational numbers and irrational numbers. I understand what is a rational number (a number that can be expressed as the ratio of two numbers p/q). what ...
0
votes
1answer
69 views

For any $a \in \mathbb R$ and any $n \in \mathbb N^+$ there exists $q \in \mathbb Q$ such that $|a-q|< \frac{1}{n}$.

For any $a \in \mathbb R$ and any $n \in \mathbb N^+$ there exists $q \in \mathbb Q$ such that $|a-q|< \frac{1}{n}$. I think i can prove this is false, let $a=2,n=2,q=1/2$ so $|2-\frac{1}{2}|< ...
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2answers
110 views

supremum of rational numbers true or false

I'm trying to figure out if this is false or true. $\sup \{a \in \mathbb{Q} : 0 \le a <1\} = 1$ I'd say it's false because we can $a=1/2$ $\sup\{1/2\}= 1/2$
1
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1answer
66 views

Question on a subset $S$ of $[0,1]\times[0,1]$ where for each $(x,y)\in S$ at least one of $x$ and $y$ is irrational

If $S$ is a subset of $[0,1]\times[0,1]$ such that one point of the ordered pair is rational and the other is irrational or both are irrationals. Then which of the following is true? a) $S$ is closed ...
1
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1answer
37 views

Linear operators over rational coefficients

This is related to a question I answered earlier which raised a question in my mind. My question is the following, Suppose we have a vector space $\mathbb{V}$ with real coefficients. Let ...
11
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3answers
209 views

Prove that x is rational

Let $x$ be a real number with the properties that $x^3+x$ and $x^5+x$ are rational. Prove that $x$ is rational. Denote $a=x^3+x$; $b=x^5+x$. We can multiply and add them together until we get the ...
1
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3answers
276 views

Proving supremum for non-empty, bounded subsets of Q iff supremum in R is rational

Let E be a nonempty bounded subset of ℚ. Prove that E has a supremum in ℚ if and only if its supremum in ℝ is rational and that in this case, the two are equal. This seems intuitive enough, and I ...
0
votes
2answers
1k views

Can we ever get an irrational number by dividing two rational numbers?

If we try to divide any two random arbitrarily long rational numbers like 103850.2387209375029375092730958297836958623986868349693868398659825528365... and ...
0
votes
1answer
35 views

How to solve this Ratio problem

I am preparing for an olympiad and came across the following question in a workbook. There is answer but no explanation: The ratio between the number of passengers travelling by Ist and IInd class ...
2
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0answers
42 views

Experimental calculation and $\mathbb{Q}$

I have been reading this article and have a question about the first line of the second paragraph on the first page. It says: The basis for this suggestion is the simple fact that all experimental ...
4
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2answers
578 views

Proving the rationals are dense in R

I know this is a common proof. I'm following Rudin's proof and I'm following everything except for one step. Suppose $x, y \in \Bbb R$ and $x < y$. Then there exists an $n \in \Bbb N$ such that ...
3
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1answer
298 views

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem.

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using the Gel'fond-Schneider's theorem. I'm interested in this problem because I knew that ${\sqrt2}^{\sqrt2}$ is a transcendental ...
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1answer
201 views

Is this a rational or irrational number?

It is given that $$z=\sqrt\frac{\sqrt{3x+1}}{\sqrt{3x-1}}$$ How does one find whether $z$ is a rational or irrational number?
3
votes
1answer
90 views

Is the infinite table argument for the countability of Q unsound?

The first "proof" I learned for why the rationals are countably infinite relied on arranging the rational numbers in a two-dimensional array and using the well-known traversal shown below to construct ...
1
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2answers
38 views

A necessary and sufficient on the co-efficients of a quadratic to give an integer

Le $f(n) := an^2 + bn + c$ for all integers $n$, where $a$, $b$, $c$ are rational. What are the necessary and sufficient conditions on $a$, $b$, and $c$ such that $f(n)$ be an integer for all $n$?
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3answers
139 views

Finding Rational numbers

Please help with the following question: Find rational numbers a and b such that: $$\left(7 + 5\sqrt2\right)^{\frac13} = a + b \sqrt2$$ Thank you
2
votes
1answer
58 views

Analysis, Density of Rational Numbers

Suppose p/q and k/l are rational numbers with abs(p/q - k/l) < 1/ql. Prove p/q = k/l. Similarly, let p/q be a fixed rational number and suppose k/l is a rational number with 0 < abs(p/q - k/l) ...
0
votes
3answers
109 views

If a triangle has rational coordinates, does it have rational area?

Basically, the topic says it all. If a triangle has rational coordinates (say, in $\mathbb Q^2$), must it have rational area? I realize the side-lengths are usually irrational; that's fine. Heron's ...
2
votes
3answers
317 views

What condition that if imposed on $\alpha$ imply that $\cos^{-1} \alpha$ is a rational multiple of $\pi$?

It is well known that if $x$ is a rational multiple of $\pi$ then $\cos x$, $\sin x$, etc, are algebraic numbers. What is known about the inverse problem? That is, is there a set of conditions that ...
0
votes
1answer
49 views

Linear order, Löwenheim-Skolem

LO is the theory of linear ordering. Suppose T a theory, which contains at least the symbol {<} in her language and T $\vDash$ LO. Suppose T has an infinite model. Prove that there's a model M for ...
1
vote
1answer
258 views

Hausdorff dimension of the set of rational numbers within a certain interval?

Intro: The Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative real number associated with any metric space. In general the Hausdorff dimension ...