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

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0answers
27 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}, ...
2
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1answer
49 views

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

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
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1answer
30 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
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1answer
36 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 ?
2
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3answers
73 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 ...
2
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6answers
90 views

What type of number is this $\frac{\sqrt2}{2}$?

$$\frac{\sqrt{2}}{2}$$ In this polynomial, an irrational number is divided by a rational number. However this is not a general case but can any one tell me that when we divide an irrational number or ...
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0answers
46 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 ...
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0answers
22 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
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1answer
57 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: $$A=\{(a,b)\in\mathbb{Q}\times\mathbb{Q}~|~\text{Both equations}~a+x=b, b+y=a~\text{have answers in }~\mathbb{Q}\}=\mathbb{Q}\times\mathbb{Q}$$ ...
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1answer
61 views

Do irrational numbers exist? [closed]

Assuming the question is asked in the real number system: If an irrational number, such as Pi is infinitely long, then do parts of this number have to repeat? and if so, does it then become a ...
1
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2answers
62 views

Rationalizing the fraction $\frac{1}{1-\sqrt2 -\sqrt3}$

I'm having problem in rationalizing the following root with the fraction $$\frac{1}{1-\sqrt2 -\sqrt3}$$ Eventually after many tries, I found the solution which was : $$\frac{-\sqrt2 (1-\sqrt2 +\sqrt3 ...
2
votes
2answers
44 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 ...
36
votes
0answers
488 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 ...
0
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0answers
33 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 ...
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0answers
67 views

If (a+b)/2 is rational can we say that a,b are rational? Prove

The question is if it's given that $$ {a+b\over 2} \in \Bbb Q $$ prove or disprove $a,b \in \Bbb Q$. Since it is to disprove, i tried the following method by using examples Take $$a = 1 + \sqrt{2} ...
0
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1answer
34 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
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1answer
36 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
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0answers
11 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
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1answer
41 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
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1answer
41 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 ...
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0answers
56 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 <~ = ...
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3answers
1k views

Prove that 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) = ...
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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
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1answer
27 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
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1answer
26 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
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1answer
75 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
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1answer
42 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
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2answers
62 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
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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 ...
3
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1answer
56 views

Is there a number $x\neq0$ whose products with $\pi$ and with $e$ are both rational?

Does there exist a number $x\neq0$, such that $[x\cdot\pi\in\mathbb{Q}]\wedge[x\cdot{e}\in\mathbb{Q}]$? I thought this question would be easy to answer, but it turns out otherwise. Obviously ...
5
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1answer
67 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
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1answer
38 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 $\phi(t)=\frac{u}{v}$ ...
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6answers
97 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) = ...
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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.
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2answers
60 views

Does there exist any positive integer $n$ such that $e^n$ is an integer ( to show $\log 2$ is irrational)?

Does there exist any positive integer $n$ such that $e^n$ is an integer ? I was in particular trying to prove $\log 2$ is irrational ; now if it is rational , then there are relatively prime integers ...
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2answers
300 views

'Rational' solutions of sine

Do there exist rational numbers $q \in (0,1) \cap \mathbb Q$ such that $$\sin\left(\frac{\pi}{2}q\right) \in \mathbb Q$$ Clearly if $q \in \mathbb Z$, yes. But what about the case $0 < q < 1$? ...
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1answer
35 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 $B$ ...
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0answers
23 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
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1answer
30 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 ...
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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 ...
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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 $$ ...
2
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3answers
66 views

Proving that rational numbers are dense

I am trying to show that for any real number a, there exist infinitely many rational numbers m/n with $ |a - m/n| < 1 /n^{2} $. I've tried to attempt the question by assuming there are finite ...
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2answers
31 views

Problem with the rational root theorem

Consider this polynomial: $f(x)=(2x+5)(x-3)(x+8/3)=0$. Then $f(x)=2x^3+...+(-40)$ Here is a list of all factors of $40$ and $2$: $40$: $±1$, $±2$, $±4$, $±5$, $±8$, $±10$, $±20$ $2$: $±2$, $±1$ ...
2
votes
4answers
123 views

Rationality of $e + \pi$

I found just one question similar to this, but it had been edited, so hopefully this isn't asked too often. Given the formulas via infinite sums for expressing $e$ and $\pi$... $$ e = ...
0
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1answer
21 views

Repeated averaging of rational numbers to get zero

I have a set of rational numbers, and the only allowed operation is calculating the mean of a subset and adding it to the set. The goal is to generate zero. I tried brute-forcing this problem with S ...
0
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0answers
19 views

How do I prove that as 2 integers p, s tend to infinity, p/s tends to x?

Forgive me for asking such a broad question, but I really do have very little knowledge on how to do this and it came up in a problem that I have been working on for some time now, so any help would ...
0
votes
2answers
36 views

Is $\mathbb Z$ the only proper sub-domain ( a subring that is an integral domain ) with unity of the ring $\mathbb Q$?

Is $\mathbb Z$ the only proper sub-domain ( a subring that is an integral domain ) with unity of the ring $\mathbb Q$ ? ( I can easily prove that if $D$ is any subring with unity then $\mathbb Z ...
3
votes
1answer
62 views

Dedekind's Cuts Lemma

I'm studying Dedekind's Cuts and his construction of Real numbers from the Rational ones. Here we are allowed to use $\Bbb{Q}$ as an ordered field and all all its properties (Archimedean Property, his ...
2
votes
1answer
47 views

Is it possible to express $\Gamma\!\left(\tfrac{1}{50}\right)$ through values of the $\Gamma$-function at rational points with smaller denominators?

Sometimes it is possible to express a value of the $\Gamma$-function at a rational point through values of the $\Gamma$-function at rational points with smaller denominators, e.g. ...