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
2answers
302 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
40 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
38 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
60 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 ...
3
votes
2answers
45 views

Simplifying $\frac{1/(\frac{1}{z_1}(1-t)+\frac{1}{z_2}t) - z_1}{(z_2 - z_1)}$

This drives me mad! I am not very good in math but thought I could at least do basic things like this one, but can't figure it out and I spent a day on it. I am trying to simplify: ...
0
votes
0answers
23 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
45 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
51 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
66 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
34 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
59 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
28 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
78 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
87 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
39 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
82 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
43 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
128 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
26 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
291 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
51 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
30 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
35 views

Prove that the quotient of a nonzero rational number and an irrational number 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 $m,n\in \mathbb Z, n\neq 0$. ...
1
vote
0answers
50 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
48 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
votes
3answers
85 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 ...
1
vote
2answers
34 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 $2$: $40$: $±1$, $±2$, $±4$, $±5$, $±8$, $±10$, $±20$ $2$: $±2$, $±1$ ...
0
votes
1answer
23 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
votes
0answers
20 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
50 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
70 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
65 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. ...
3
votes
3answers
480 views

Irrationals forming rationals

Can we obtain every rational number from the multiplication of two irrational numbers? If not, which ones can we not obtain?
8
votes
2answers
135 views

Given dividend and divisor, can we know the length of nonrepeating part and repeating part?

$13/92=0.14\overline{1304347826086956521739}$ In this example, the length of nonrepeating part is $3$. The length of repeating part (repeating period) is $21$. I collected some properties related to ...
0
votes
2answers
78 views

Rational and irrational numbers

Consider $x$ a rational number. Let $\epsilon \geq 0$ be the minimal value such that $x + \epsilon$ is irrational, and let also $\gamma > 0$ be the minimal value such that $x+\gamma$ is rational. ...
2
votes
3answers
149 views

There is no largest rational number $p$ such that $p^2 < 2$

In Rudin's analysis example 1.1, he tried to show the following Let $A$ be the set of all positive rationals $p$ such that $p^2<2$ and let $B$ consist of all positive rationals $p$ such that $p^2 ...
0
votes
0answers
34 views

I need to find a rational numbers series that converging to irrational number [duplicate]

I found a series that is $a_{n+1}=\frac{a_n^2 + 2}{2a_n}$ yet I'm not sure. can someone give me a more umm solid example? thanks.
2
votes
2answers
61 views

If I raise a simplest-terms rational to an integer power, is the result automatically in simplest terms?

Given a rational number $a/b$ expressed in simplest terms (so $GCD(a,b)=1$), I want to raise it to an integer power $n$. I think the result will always automatically be in simplest terms, but it's a ...
1
vote
1answer
121 views

Does there exist a unique closest natural number to each rational number?

So here's the question: Prove or disprove: For every $x \in \mathbb{Q}$, there is a unique $n \in \mathbb{N}$ which is the closest natural number to $x$. I know we can define a rational number ...
0
votes
2answers
81 views

Integrating the normal distribution over rational numbers?

Is it possible to integrate the normal distribution over rational numbers? What is the value of such integral? Is it $\pi$ minus the integral over irrational numbers?
0
votes
1answer
41 views

Sets of irrationals whose square contains a rational

Let $S$ be a subset of the irrationals. Also, lets assume that $S$ has infinitely many elements. My very general question is, under what non-trivial conditions does there exist an element $x\in S$ ...
1
vote
1answer
97 views

Real number system

Is the set of rationals a subset of the irrationals? I always assumed it was, but given that irrationals are defined to be numbers that have an infinite, non-repeating decimal expansion, there cannot ...
0
votes
1answer
93 views

About the continuity of $f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k}$

Let $q: \mathbb{N} \to \mathbb{Q}$ be a bijection and denote the image of $k \in \mathbb{N}$ by $q_k$. Let $f: \mathbb{R} \to (0,1)$, $$ f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k} ...
5
votes
1answer
188 views

Equality of positive rational numbers, Part-2

I am reading this answer. I have some doubts which I want to clarify. Question 1. The author defines a rational number $\dfrac ab$ as, $$b\times\left(\dfrac{a}{b}\right) = a$$ He presumes that ...
2
votes
3answers
256 views

Proving Floor and Ceiling of a Rational Number

Suppose x,y $ \in \mathbb{Z}^+ $ Prove $\lceil x/y \rceil = \lfloor (x-1)/y \rfloor + 1$ I was considering using the definition of floor and ceiling to prove this. But this does not seem like a ...
1
vote
1answer
50 views

$(-3)^{3/2} \neq (-3)^{6/4}$

$(-3)^{\frac{3}{2}}=-3\sqrt{3}i$ $(-3)^{\frac{6}{4}}=\sqrt{27}$ (not the same thing). What's the deal? It's interesting because people work with fractional exponents all the time and I've never ...
1
vote
3answers
92 views

Finding the simplest rational in a closed interval

Given a closed interval [a,b], how would you find the "simplest rational", p/q, contained in that interval. By simplest, I mean the rational with the smallest denominator q. You may, if you wish ...