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.

learn more… | top users | synonyms

2
votes
2answers
44 views

rational number plane vector space or not?

Two questions: 1. Is $\mathbb{Q}^2$ a vector space over the field $\mathbb{Q}$? 2. Is $\mathbb{Q}^2$ a vector space over the field $\mathbb{R}$? My answer to the first question is yes. Because the ...
2
votes
2answers
124 views

Is the set of rationals between $\sqrt{2}$ and $\sqrt{3}$ open or closed in $\mathbb Q$?

Consider the set of all rationals, $\mathbb Q$ as a subset of the set of all reals $\mathbb R$. Assign $\mathbb Q$ the subspace topology induced by the standard topology on $\mathbb R$. ...
1
vote
4answers
78 views

question about the proof about the square root of natural numbers [duplicate]

Could someone please help me to prove that for $t \in \mathbb{N}$ , $\sqrt{t} \in \mathbb{Q} $ if only if $\sqrt{t} \in \mathbb{N}$
0
votes
2answers
28 views

Decimals and Rational numbers

How do you prove: Q1) Why is every rational number (say m/n, where m and n are both positive integers) either a terminating or a repeating decimal? Q2) Why is every repeating decimal (or terminating ...
-2
votes
2answers
103 views

lowest denominator to lie between to rational numbers.

What's the lowest $ m\in \mathbb {N} $ such that the exists an $ n $ with $1/3 >\frac {n}{m}>33/100$? note that there used to be a typo in the inequality which gives the opposite sign I'm on my ...
0
votes
1answer
65 views

Irrational power of root

Let $a$ and $b$ be rational numbers, such that $\sqrt{a}$ and $\sqrt{b}$ are irrational. Can $\sqrt{a}^\sqrt{b}$ be rational? I found examples, where the irrational power of an irrational number is ...
2
votes
0answers
44 views

Digamma equation identification

I was messing around with the digamma function the other day, and I discovered this identity: $$\psi\left(\frac ...
1
vote
4answers
69 views

Real Between Rationals

Let $x$ be a real number. Show that, for any $\varepsilon>0$, there exist two rationals $q$ and $q'$ such that $q<x<q'$ and $|q-q'|<\varepsilon$ How should I approach this prove?
0
votes
1answer
100 views

Monotone increasing sequence of rationals with an irrational limit

I am trying to use rationals in order to approximate irrationals. Is it possible to construct a monotonically increasing sequence of rationals the limit of which is an irrational? If so, how?
1
vote
4answers
124 views

Example of a non-trivial function such that $f(2x)=f(x)$

Could you give an example of a non-constant function $f$ such that $$ f(x) = f(2x). $$ The one that I can think of is the trivial one, namely $\chi_{\mathbb{Q}}$, the characteristic function on the ...
1
vote
1answer
101 views

What can be said about $\pi+e$ and $\pi e$? Are these numbers rational or irrational? [duplicate]

"homework" What can be said about $\pi+e$ and $\pi e$? Are these numbers rational or irrational? I know that both $\pi$ and $e$ are irrational. What can be said about $\pi+e$, and $\pi e$?
1
vote
3answers
78 views

Prove that if $x$ and $y$ are rational numbers and $y\ne 0$, then $x/y$ is a rational number

Prove that if $x$ and $y$ are rational numbers and $y\ne 0$, then $\frac{x}{y}$ is a rational number. How do I prove this, and also which proving method would I use? I'm confused between that and ...
0
votes
3answers
65 views

There is at most one way to represent a number as $a+b\sqrt 2$ with rational $a,b$

If $a,b,c,d\in\mathbb Q$ and $a+b\sqrt 2= c + d\sqrt 2$, then prove $a=c$ and $b=d$ ? I don't have any idea to solve this , it's freaking me out.
0
votes
3answers
56 views

Find all $n\in \mathbb N$ such that $\sqrt{n+7}+\sqrt{n}$ is rational.

Find all $n\in \mathbb N$ such that $\sqrt{n+7}+\sqrt{n}$ is rational. By inspection it is pretty easy to see that the only $n$ that will work is $n=9$. Because the distance between perfect squares ...
3
votes
3answers
254 views

Using a given identity to solve for the value of an expression

This problem caught my eye in the book yesterday. Till now I still get stuck. Here it is: If $$\frac{x}{x^2+1}=\frac{1}{3},$$ what is the value of $$\frac{x^3}{x^6+x^5+x^4+x^3+x^2+x+1}?$$ The ...
6
votes
4answers
954 views

Can any two irrational numbers NOT of the form (m+A) and (n-A) be added to produce a rational number?

$m$ and $n$ being rational numbers, A being an irrational number. I was wondering if two irrational numbers when added always yield an irrational number. All the counter-examples I could find were of ...
-5
votes
2answers
101 views

$\mathbb{Q}$ can not be embedded in $\mathbb{Z}$

Show that $\mathbb{Q}$ can not be embedded in $\mathbb{Z}$ (where both has the subspace topology of $\mathbb{R}$) My attempt at a solution Since Z is discrete, {k} is open in $\mathbb{Z}$ with ...
3
votes
3answers
98 views

The supremum of rationals that are less than a given number is equal to that number

I have the following theorem to prove. Given a real number $a$, define the set $S$ such that $S = \{x \in \mathbb Q: x < a\}$. Show that $a = \sup S$. My attempt at a proof is as follows ...
0
votes
1answer
66 views

Usage of decimal expansion

I learned about the rigorous construction of rationals as a set of equivalence classes of ordered integers with operations defined on this set. I understand that the decimal expansion is another way ...
0
votes
1answer
29 views

Given that $a>1$, show that the exponential function $a^x$ is increasing for $x\in\mathbb{Q}$

The assumption one can make here is that it is increasing for $x\in\mathbb{Z}$. I have tried to make a proof but I'm not sure if it is valid. Here it goes. Say $x,y\in\mathbb{Q}$. Then they can be ...
1
vote
0answers
32 views

How many rationals for a given $n \in \Bbb N \;\backslash \{1\}$?

Fix $n \in \Bbb N, n> 1$. Now choose a two digit base-$n$ number $ab $ say. There's $n^2$ choices for this. Consider the number $0.c_1 c_2 c_3 \ldots$ where the $c_i$ are defined recursively: ...
5
votes
9answers
2k views

Can rational numbers have decimals?

I had a question in my exam paper - Which of the following is not a rational number? a) $\sqrt{25}$ b) $\sqrt{45}$ c) $\sqrt\frac{256}{225}$ d) $\frac{3}{4}$ The answer to this is b. Now, ...
3
votes
2answers
262 views

What do rationals represent?

While learning about the construction of number systems, I realized that I had many misunderstandings of crucial concepts which I was learning intuitively. I recently learned about the construction of ...
0
votes
0answers
18 views

What is a class of equations? Which class would require the rational numbers to guarantee a solution?

What is a class of equations? Which class would require the rational numbers to guarantee a solution? The biggest problem I am having is that I am unsure what exactly a class of equations is. The ...
1
vote
3answers
60 views

Is a complex fraction considered part of the rationals?

I have always been taught that $\mathbb{Q}=\{ \frac{a}{b}|\,\,a,b\in \mathbb{Z},\, \,b\neq0\}$. Is this definition of the rationals limited? Could it also be true that a complex fraction, i.e. ...
1
vote
2answers
55 views

Cantor diagonal argument; related number

I was reading another question on mse about cantors proof and I'm curious about a number that could be defined from it. Well there could be a whole heap of them, but one for now. Define $A=\Bbb{Q} ...
0
votes
3answers
51 views

Correctness of proof that every positive rational with square $>2$ is an upper bound for those with square $<2$

I would like to know whether my proof makes sense or not, and if not where should it be corrected. Let $E=\{x \text{ is rational }: x>0 \text{ and } x^2<2\}.$ Claim: Every member of $F=\{x ...
0
votes
2answers
67 views

Correctness of the proof that the set $\{x \in \mathbb{Q} : x>0 \text{ and } x^2>2\}$ does not have a smallest element

Let $F=\{x \in \mathbb{Q} : x>0 \text{ and } x^2>2\}$. I am asked to show that $F$ does not have a smallest element. The hint is to simply prove the claim: 'If $p$ is a rational number in ...
1
vote
0answers
187 views

The topology generated by open intervals of rational numbers

Let $B = \{ \mathbb{R} \} \cup \{ (a,b) \cap\mathbb {Q} \ ,\ a\lt b \ ,\ a,b \in\mathbb{Q}\}$ Thus, a set $V \in B$ if it is either equal to $\mathbb{R}$ or if it is in the intersection of ...
3
votes
1answer
78 views

Rational values of $\sin(\log(x))$

Apart from the trivial solution $\sin(\log(1))=0$, is $$\sin(\log(x))$$ ever rational if $x$ is rational?
2
votes
1answer
51 views

Length of a rationals period in base $b$

Okay working in base $b$ we are given a fraction of form $\frac{p}{q}$ with $p$ and $q$ coprime. We also assume that $b$ and $q$ are coprime so $\frac{p}{q}$ is purely periodic in base $b$. The ...
-2
votes
2answers
501 views

Prove that $x\in\mathbb Q$ [closed]

Let $a\in\mathbb Q$ and $a>\dfrac43$. Let $x\in\mathbb R$ and $x^2-ax,x^3-ax\in\mathbb Q$. Prove that $x\in\mathbb Q$. EDIT: Thsi is my attempt: Let $x^2-ax=b$ and $x^3-ax=q$ for some ...
3
votes
4answers
169 views

Is there a bijection from a bounded open interval of $\mathbb{Q}$ onto $\mathbb{Q}$?

It is easy to create a bijection between two bounded open intervals of $\mathbb{R}$, such as: $$ \begin{align} f : (a,b) &\to (\alpha,\beta) \\ x &\mapsto \alpha+(x-a)(\beta-\alpha). ...
0
votes
1answer
40 views

Comparing Fractional Numbers

Does a formula exist for comparing two fractional numbers, without resolving to using anything other than integers and fractions? (Thus not real numbers). In other words: given $\dfrac{a}{b}$ and ...
1
vote
0answers
77 views

Rational numbers imply reals?

I was solving an inequality today and I proved it for rational numbers (it was easier because I was able to "strengthen" by doing things like "$\frac{p}{q}>\frac{r}{s}\implies ps\ge qr+1$ since ...
0
votes
1answer
129 views

Show that $\sqrt{2}$ is irrational using integer root theorem

Show that $\sqrt{2}$ is irrational using integer root theorem. Let $P(x)=x^2-2$. Since $\sqrt{2}$ is a root of this polynomial, had it been a rational (suppose $\sqrt{2}=\frac{p}{q}$) no, by ...
5
votes
2answers
149 views

Inequality with $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}$

Inspired by this recent question, I suggest this. Let $n=2,3,4, \ldots .$ Then $$ \frac{7}{12} < \cfrac 1 {1 + \cfrac {1^2} {1 + \cfrac {2^2} {\ddots + \cfrac \vdots { 1 + \, {n^2} \,}}}} \leq ...
66
votes
23answers
13k views

Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical. Numerical computations, to my understanding, never deal ...
0
votes
1answer
95 views

Rational number, dense but measure zero

When calculating the measure of Q in real number interval [0, 1], an interval $ (q_n-\epsilon, q_n + \epsilon)$ around each rational number $ q_n $ is defined to show the measure of Q is zero. Is ...
3
votes
0answers
126 views

Rational analysis

I found myself thinking about how much of real analysis that can also be developed within the rational numbers. Of course, $\Bbb Q$ is lacking what is perhaps the most important property of the real ...
7
votes
3answers
961 views

Is it possible to find square root using only rational numbers and elementary arithmetic operators

Suppose I have a number a How can I find it's square root using only +, -, /, ...
1
vote
2answers
91 views

Definition of Rational/ Irrational Numbers reguarding denominators

The definition of a Irrational number is "Irrational numbers don't include integers OR fractions. However, irrational numbers can have a decimal value that continues forever WITHOUT a pattern." So ...
1
vote
1answer
51 views

Finding rational points at rational distance in the plane

Take any point $p$ in the real plane. Does there always exist a rational point at a rational distance from $p$? (A rational point is a point $(q,r)$ where $q$ and $r$ are rational.)
2
votes
0answers
53 views

Approximating real vectors in the unit hypercube by rational vectors in the unit hypercube

I am currently looking for an error bound for an problem where I have to approximate a real vector in the unit hypercube using a rational vector in the unit hypercube. Specifically: given a ...
13
votes
1answer
161 views

Existence of rational sequence such that a polynomial is split over $\Bbb{Q}$

Does there exist a sequence $(a_n)_{n\in \Bbb{N}}$ of rationals such that for all $n\in \Bbb{N}$, $a_n\neq 0$ and the polynomial $a_0+a_1X+\cdots+a_nX^n$ is split over $\Bbb{Q}$? I was asked this ...
0
votes
3answers
50 views

Help solve rational expression

I need help solving this rational expression. Divide $$\frac{4x^4 + 6x^3 + 3x - 1}{2x^2 + 1}$$ How do you solve this problem? Where do I start?
0
votes
3answers
52 views

Rationals over an interval

Suppose $I$ is an interval $[a,b]$. It is noted that $a$ and $b$ are real integers. Divide the interval into $n$ parts with step size $h=(b-a)/n$. Clearly all the points $a$, $a+h$, ...
9
votes
4answers
1k views

Is the fact that there are more irrational numbers than rational numbers useful?

Although it is known that the cardinality of the set of irrational numbers is greater than the cardinality of the set of rational numbers, is there any usefulness/applications of this fact outside of ...
0
votes
1answer
126 views

Zero vs Infinity relation type

I'm not sure it should be asked here or in philosophy. Bertrand Russell in his book "Introduction to Mathematical Philosophy" in chapter 7 when discussing rational numbers on page 66 says: "It will ...
0
votes
1answer
55 views

Prove that a number is rational if and only if it has a finite or periodic representation on every base

How do we go about proving: $q$ is rational $\iff$ $q$ has a finite or periodic representation on every natural base $n>1$? In other words, we need to prove each of the following statements: ...