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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-3
votes
1answer
16 views

Find whether a given rational number has a terminating decimal expansion

Without actual division find whether the rational number $\dfrac{1323}{264600}$ is terminating or non terminating. I know that to solve this, we have to convert the denominator into the formula ...
7
votes
3answers
445 views

“Length” of rationals in an interval

For $x \in \mathbb{R}$, define $r(x)$ as follows: $$ r(x)= \begin{cases} 1 &\text{if $x$ is rational},\\ 0 &\text{if $x$ is irrational}. \end{cases} $$ Q. What is $\int_0^1 r(x) dx$ ? I ...
0
votes
1answer
23 views

Bounded Set for Rational Number

$$A = \{x \in \Bbb Q: x^2 < 11\}$$ I am looking for the upper and lower bounds of set $A$. I know they should be the greatest and the smallest rational number within $(- \sqrt{11}, \sqrt{11})$ ...
3
votes
2answers
59 views

Closeness of $n! \ x$ to integers for irrational $x$

This question came up in the comments to another question. Is there an irrational number $x$ such that, for sufficiently large $n$, the product $$ n! \ x $$ is arbitrarily close to an integer? More ...
1
vote
1answer
35 views

Why is the $\mathbb Z$-module $\mathbb Q$ projective in category $\sigma[\mathbb Q]$?

We know that $\mathbb Q$ is not a projective $\mathbb Z$-module, but, I've read this paper and it says in Example 2.11 that it's easy to check that "$\mathbb Q$ as $\mathbb Z$-module is projective ...
0
votes
0answers
31 views

Determining if a rational number has a terminating decimal expansion (proof)

Theorem: $x=\frac pq$ is any given rational number, $n$ and $m$ are any whole numbers (including zero) which you can choose. a) If $q=2^n5^m$ is possible, $x$ has a terminating decimal expansion. ...
1
vote
1answer
53 views

About subspaces of $\mathbb{R}$ as vector space over $\mathbb{Q}$.

In many texts is noted the analogy between the transcendence degree of a field extension and the dimension of a vector space, so I'm tempting to use such analogy to better understand the structure of ...
0
votes
3answers
109 views

All sets of rational numbers are bigger than the set containing infinite integers - or are they?

Intro This started with me learning the different types of infinity. I like to call them types instead of sizes due to the fact, that infinite is defined by being endless or "not-finite" - meaning ...
1
vote
0answers
51 views

Natural bijection between $\mathbb{N}$ and algebraic numbers?

Q. Is there a canonical, explicit bijection between the natural numbers $\mathbb{N}$ and the algebraic numbers? The earlier MSE question, "Bijection for algebraic numbers," does not quite ...
2
votes
4answers
58 views

Characterization of the $x$ such that $\sin(x)$ is rational?

For $x \in [0,\pi/2]$, $\sin(x)$ ranges over $[0,1]$. So every rational number in $[0,1]$ is the sine of some $x \in [0,\pi/2]$. Q. Is there any characterization of the $x$ for which $\sin(x)$ is ...
5
votes
1answer
115 views

For each irrational number $b$, does there exist an irrational number $a$ such that $a^b$ is rational?

It is well known that there exist two irrational numbers $a$ and $b$ such that $a^b$ is rational. By the way, I've been interested in the following two propositions. Proposition 1 : For each ...
3
votes
2answers
79 views

Dedekind Cut Roots

I was studying Dedekind Cuts in my history of math class and was looking at the following formulas for the multiplication of Dedekind Cuts: $L_\sqrt2 = \{ r \in \mathbb{Q} : r^2 < 2 $ and $ r>0 ...
1
vote
1answer
31 views

Rational Zeros Theorem, get the possible solutions

In my book there is a section about rational numbers. The first example show that:6^(1/3) cannot represent a rational number. In the proof it says that the ...
1
vote
0answers
31 views

How many elements are in the following set?

The set is $$\{ x \in Q:x^2 =64/25 \} $$ I thought the answer was $\{ \frac{8}{5}, -\frac{8}{5} \}$ but I am told there are in fact 4 distinct elements: $$\{ \frac{8}{5}, \frac{8}{-5}, \frac{-8}{5}, ...
1
vote
1answer
56 views

Set in $\mathbb{R}^2$ that contains no (non-null) measurable rectangles [closed]

From Torchinsky's Real Variables text, presented in a chapter on abstract Fubini's Theorem. Define the following set: $$E := \mathbb{R}^2 \setminus \{(x,y) \in \mathbb{R}^2: x-y \in \mathbb{Q}\}.$$ ...
2
votes
1answer
45 views

Dedekind cuts in Rudin's PMA

I'm working on Appendix to chapter I of Rudin's Principles of mathematical analysis and I have the following problem: Given a positive cut $\alpha$ and a rational $x>1,$ how can I prove that there ...
3
votes
1answer
38 views

Limit of sequences and integers

If $a$ is a non zero real number , $x \ge 1$ is a rational number and $(r_n)$ is a sequence of positive integers such that $\lim _{n \to \infty}ax^n-r_n=0$ , then is it true that $x$ is an integer ?
3
votes
3answers
94 views

Find rational points on $x^2 + y^2 = 3$ and on $x^2 + y^2 = 17$

$(a)$ Find all rational points on the circle $x^2 + y^2 = 3$, if there are any. If there is none, prove so. $(b)$ Find all rational points on the circle $x^2 + y^2 = 17$, if there are any. If there ...
0
votes
0answers
50 views

Irrational numbers to irrational powers being rational?

So some of you may be familiar with the proof that some irrational numbers to irrational powers are rational, that is: if $A = \sqrt2^\sqrt{2}$ then it follows that $A^\sqrt{2} = 2$. So, I've found a ...
1
vote
0answers
31 views

Rational number in form $(a^x+b^y)/(c^z+d^w)$

Find all positive integers $(x,y,z,w)$ such that for any positive rational number $r=p/q$, there exist positive integers $(a,b,c,d)$ for which $$r=\frac{a^x+b^y}{c^z+d^w}.$$ For instance, for ...
3
votes
1answer
58 views

Is a subset of $\mathbb{Q}\times\mathbb{Q}$ that all variants of an exponentiation equation have answers in it, infinite?

Note that we have: $$A=\{(a,b)\in\mathbb{Q}\times\mathbb{Q}~|~\text{Both equations}~a+x=b, b+y=a~\text{have answers in }~\mathbb{Q}\}=\mathbb{Q}\times\mathbb{Q}$$ ...
2
votes
2answers
46 views

Seeing the plane as a four (or more) dimensional vector space on $\mathbb Q$

As I was trying to answer a question about the enumeration of circuits one can build with a set of miniature train track elements, I realized that all plane positions that could be reached had ...
41
votes
1answer
680 views

Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

About a month ago, I got the following : For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that ...
3
votes
2answers
53 views

Determine whether the decimal expansion of a rational number is infinite

This may be a naive question but I would like to know whether we can determine if a fraction (say $1/3$) will produce a rational number with an infinite number of digits after the decimal when ...
0
votes
0answers
38 views

Not including rational number in inequality

If you have a linear inequality like $x < 7$ where $x$ belongs to rational numbers. Then on graphing it on a number line, a unfilled circle is used to denote that $7$ is not included. But that ...
0
votes
1answer
36 views

There does not exist rational numbers $x$ and $y$ such that $x^y$ is a positive integer and $y^x$ is a negative integer

I want to prove or disprove: There does not exist rational numbers $x$ and $y$ such that $x^y$ is a positive integer and $y^x$ is a negative integer. For the integers $-3$ and $4$, $(-3)^4 = 81$ ...
0
votes
1answer
37 views

Prove that the numbers of the form $a+b\sqrt{2}$, where $a$ and $b$ are rational numbers, form a subfield of $\mathbb{C}$.

I'm having trouble proving that a multiplicative inverse exists in the following problem: Prove that the numbers of the form $a+b\sqrt{2}$, where $a$ and $b$ are rational numbers, form a subfield of ...
0
votes
0answers
13 views

How do you calculate certain variables of two or more events that occur simultaneously compared to the same events happening subsequently.

Say you have two hoses, A and B, that fill up a pool of equal size at different rates. Hose A fills up a pool in 10 mins, hose B in 20 mins. Thus A = 1p/10m, B = 1p/20m. Lets say that Hose A filling ...
0
votes
1answer
42 views

$a, b, x \in \mathbb{Q}$ with $a \neq 0$. Is the $\frac{b}{a}$ the only possible value for x in $a \cdot x = b$

I have an exercise in my last assignment for calculus which is the following: Let $a, b, x \in \mathbb{Q}$ with $a \neq 0$. Use only the field axioms and the properties which we showed in class ...
3
votes
1answer
44 views

Show using ordering axioms that $x^2 < y^2$ for $x, y \in \mathbb{Q}$, with $0 < x < y$

I have an exercise in my last assignment of calculus: Show using ordering axioms that $$x^2 < y^2$$ for $x, y \in > \mathbb{Q}$, with $0 < x < y$ This is my solution: We have that ...
0
votes
0answers
60 views

Set theory proof question on rational numbers

I was assigned a problem by my Discrete Mathematics professor that goes as follows: Prove that on $\mathbb{Q}$ (the set of all rational numbers), the relation "$<$" satisfies " $< \circ <~ = ...
9
votes
3answers
1k views

If $ f(x \cdot f(y) + f(x)) = y \cdot f(x) + x $, then $f(x)=x$

Let $ f : \mathbb{Q} \rightarrow \mathbb{Q} $ be a function which has the following property: $$ f(x \cdot f(y) + f(x)) = y \cdot f(x) + x \;,\; \forall \; x, y \in \mathbb{Q} $$ Prove that $ f(x) = ...
11
votes
4answers
1k views

Are there any bases which represent all rationals in a finite number of digits?

In base 10, 1/3 cannot be represented in a finite number of digits. Examples exist in many other bases (notably base 2, as it's relevant to computing). I'm wondering: does there exist any base in ...
0
votes
1answer
30 views

for $p$ given, $\zeta_p$ a primitive root of unity, fow which $d\in \mathbb{Z}$ does $\zeta_p \in \mathbb{Q}(\sqrt{d})$?

Here is a question that I am trying to answer: Let $p$ be a prime greater than $2$. For which $d \in \mathbb{Z}$ contains $\mathbb{Q}(\sqrt{d})$ a primitive root of power $p$? What I did If ...
2
votes
3answers
52 views

Finding a sequence of sets whose intersection is a null set

Find a sequence of sets $I_n=\{r:r \in \mathbb{Q}, a_n\le r \le b_n\} $ in $\mathbb{Q}$, where $a_n, b_n \in\mathbb{Q}$ such that $$I_{n+1} \subset I_n\forall n\in\mathbb{N}$$ $\lim_{n \to ...
0
votes
1answer
27 views

A Elementary fact but proof needed

Let $n,q\in\mathbb{N}$, $r\in\mathbb{R}$ and $m,p\in\mathbb{Z}$ such that $\frac{m}{n}<r<\frac{m+1}{n}$ and $|\frac{p}{q}-r|<\min(r-\frac{m}{n};\frac{m+1}{n}-r)$. It does seem obvious that we ...
5
votes
1answer
76 views

Find the functions

Find all the functions $ f : \mathbb{Q} \rightarrow \mathbb{Q} $ with the following property: $$ f(x + 3f(y)) = f(x) + f(y) + 2y, \: \forall x, y \in \mathbb{Q} $$
3
votes
1answer
43 views

Algebraic number with bounded coefficients

How many algebraic numbers $z$ are there satisfying $P(z)=0$ where $P(z)$ is some polynomial with integer coefficients of degree less than or equal to $n$ such that the absolute value of every ...
0
votes
2answers
70 views

Let $S=\{x\in\mathbb Q\mid x>2\}$. Prove $\inf S = 2$.

Okay, so I think I kind of get this one already. Since 2 is the lowest rational number in the set that's less than $x$, then $\inf S = 2$. But is there is any other way to explain this? I feel like ...
2
votes
1answer
37 views

Proof for number of rational ordered pairs on a line

It is given that the function $y=ax+b,\; a \neq 0$ has an ordered pair $(x,y)=( \sqrt{2}, 0)$. Prove that $y=ax+b$ does not have two or more rational ordered pairs. From the above I know that ...
5
votes
1answer
68 views

Is $\Bbb Q$ homeomorphic to $\Bbb Q^2$? [duplicate]

It's an easy excercise in set theory to exhibit a bijection $\Bbb Q \cong \Bbb Q\times \Bbb Q$. However, none of the bijections I'm aware of respect the topologies on $\Bbb Q$ and $\Bbb Q^2$, ...
0
votes
1answer
39 views

Image of ring homomorphism $\phi : \mathbb{Z}[t] \to \mathbb{Q}$?

Here is a problem I face practicing the theory of rings: Define $\phi : \mathbb{Z}[t] \to \mathbb{Q}$, a ring homomorphism (it does map $1$ to $1$). I'm trying to show that if $\phi(t)=\frac{u}{v}$ ...
4
votes
6answers
104 views

Why is $\mathbb Q $ (rational numbers) countable? [duplicate]

By definition, a set $S$ is called countable if there exists an bijective function $f$ from $S$ to the natural numbers $N$. If we take a function $g\colon\mathbb{Z\times N\to Q}$ given by $g(m, n) = ...
1
vote
1answer
24 views

Rational number in $\mathbb{Z}[\omega]$ should be integer.

Let $\omega = \cos \frac{2\pi}{p} + i \sin \frac{2\pi}{p}$ for some prime number $p > 2$. Then how to prove that if $q \in \mathbb{Q} \cap \mathbb{Z}[\omega]$, $q$ must be integer.
2
votes
2answers
75 views

Why must the decimal representation of a rational number in any base always either terminate or repeat?

Wikipedia makes the following statement about rational numbers. The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same ...
0
votes
1answer
38 views

Relationship of basis vectors of the complex plane

I am working on learning more about the connection of complex numbers and rotations in the context of rational geometry. Thanks ahead of time for any corrections on my best assertions. Let $B$ ...
1
vote
0answers
24 views

Rationals in an interval $[a,b] \in \Bbb R$

(i) For which real values $a$ and $b$, ($a < b$), is the set $[a,b] \cap \Bbb Q$ open in $(\Bbb Q, d)$, (where $d(x,y)= \lvert x-y \rvert$)? (ii)For which real values $a,b$ is the set $[a,b] \cap ...
2
votes
1answer
32 views

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational.

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational. From defintion $a=\frac m n$ such that $m,n\in \mathbb Z, n\neq 0$. Take ...
1
vote
0answers
49 views

Why rational numbers in stopping times for continuous time processes

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{\ge 0},P)$ be a filtered probability space. Let $X_t \in \mathbb{R}^n$ be a continuous stochastic process adapted to $\mathcal{F}_t$. Let $A \subset ...
0
votes
0answers
45 views

$\{p\in\mathbb Q:p^2<2\}$ having no large element and beyond

In Baby Rudin, the proof of $\{p\in\mathbb Q:p^2<2\}$ having no largest element, a number $q$ larger than $p$ in this set is defined as: $$ q=p-\frac{p^2-2}{p+2}=\frac{2p+2}{p+2} $$ and $$ ...