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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2
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3answers
53 views

Why are there opposite rules for dividing positive numbers and negative numbers?

I'm in confusion from some time about division of negative numbers. When we divide a positive number with a positive number, for example $$5/3 = 1.66 $$ we see what is biggest multiple of 3 which is ...
8
votes
4answers
370 views

Find the sum of reciprocals of divisors given the sum of divisors

Let $d_1, d_2, \cdots d_k$ be all the factors of a positive integer '$n$' including $1$ and $n$. Suppose $d_1 + d_2 + d_3+\cdots+d_k = 72$. Then find the value of $\frac{1}{d_1}+\frac{1}{d_2}+\cdots + ...
1
vote
1answer
42 views

Online math question didn't do a square of a square root properly?

I'm currently taking an online college math course, and I recently came across something that I can't make any sense out of. $(\frac{5x \cdot \sqrt{3}}{6})^2 = \frac{25x^2}{12}$ It looks like ...
0
votes
0answers
35 views

Differential equation - fractions, circular answer?

Hi this might seem like a really stupid question but then hopefully someone can asnswer it quite easily :) I have function $P{_t}$$=(E{_t}$ $(P{_t}{_+}{_1}+$ $δ{_t}{_+}{_1}$$ )-γΩx$${^*})/$$(1+rf+ψ_t ...
3
votes
1answer
29 views

How Many Rational Slopes?

Given an $N$ by $M$ grid with integer coordinates (e.g. like pixels in an image), how many slopes are defined by the set of lines passing through the each grid point pair? Note that because the ...
1
vote
2answers
240 views

How is N^2/3 equivalent to 1/(N^1/3)?

I've tried to look for similar things on StackExchange and elsewhere on the net, but can't seem to find anything, so thought I'd just ask for some help on here... Someone has kindly helped me with a ...
3
votes
2answers
76 views

Representing rational numbers on a number line

Though the cardinality of the set of natural numbers is the same as the cardinality of the set of rational numbers, when one looks at a number line this fact seems counterintuitive, since between ...
0
votes
1answer
86 views

why doesn't proof of sum of two rational number is rational not proving the irreducibility of fraction $\frac{ad+bc}{bd}$?

When I was comparing proof for $\sqrt{2}$ and sum of two rational numbers, I found that the proof of two rational number did not mention anything about common factor in the ratio. one proof I found ...
1
vote
1answer
38 views

How do I show that the equivalence relation defining the rational numbers is transitive?

I apologize if this is a super easy question, but there is something fishy about my proof. I was to show: $$(p,q) \sim (m,n) \wedge (m,n) \sim (a,b) \implies (p,q) \sim (a,b) $$ under the ...
0
votes
0answers
29 views

For which values of $\theta $ does this claim true?

According to the conjecture of , L. J. Lander, T. R. Parkin, and John Selfridge (1967):I suppose this claim : claim :let : $$\sum_{i=1}^n a^{\cos\theta}_i =\sum_{j=1}^m b^{\cos\theta}_j ,$$ for ...
2
votes
2answers
57 views

Presentation of the additive group of the rational numbers

We know that $\mathbb{Q}\cong\mathbb{Z}\times\mathbb{Z}/\sim$, where the isomorphism is a ring isomorphism and the equivalence relation is defined as $$(a,b)\sim(c,d)\Longleftrightarrow ad=bc$$ Then ...
0
votes
2answers
42 views

Rational number with rational exponent becomes rational

I'm looking for a proof to show when $p^q$ for $p,q \in \mathbb{Q}$ is again in $\mathbb{Q}$, without factoring. I'm not sure, if it's possible, given these two numbers to say if the result is again ...
21
votes
3answers
878 views

“Least trivial” function preserving rationality

Is there a "non-trivial" function $f(x,y)$ such that $$f(x,y) \in \mathbb{Q} \iff x,y\in \mathbb{Q}?$$ An example of a "trivial" function would be $$f(x,y) = \begin{cases} 0 & x,y\in ...
10
votes
1answer
183 views

What is known about this well-ordering of the rationals in a finite interval?

Given any interval $I=(a,b) \subset \mathbb R^+$, we may order the rationals in $I$ with a denominator-first lexicographic order, as follows: First, we list, in increasing order of numerator, all $q ...
0
votes
2answers
37 views

Is there such a thing as complex rational numbers and does it have the same properties as the usual complex numbers as extension of the real numbers?

I've been wondering if there is any use to defining a set that is isomorphic to $\mathbb{Q}^2$ (in the same way that $\mathbb{C}$ is isomorphic to $\mathbb{R}^2$). I immediately see a problem with ...
3
votes
2answers
96 views

Showing $r\in\mathbb Q\setminus \{0\}\implies e^r\notin \mathbb Q$

For a given $n>0$, let $\displaystyle J_n:x\to \frac{1}{n!}\int_{-x}^x(x^2-t^2)^ne^tdt$ a. Prove that there exists $A_n,B_n\in \mathbb R_n[X]$ such that $\forall x\in \mathbb R^+, ...
5
votes
1answer
131 views

Roots of a Cubic Polynomial with Elementary Symmetric Polynomial Coefficients

Let $R_n$ be a set of $n$ distinct nonzero rational numbers. Let $e_k$ be elementary symmetric polynomials over $R_n$---i.e. $e_0=1$, $e_1 = \sum_{1\le i\le n} r_i$, $e_2 = \sum_{1\le i<j\le n} r_i ...
8
votes
3answers
110 views

Minimal $ab$ for Rational Number $a/b$ in an Interval

Given rational numbers $L$ and $U$, $0<L<U<1$, find rational number $M=a/b$ such that $L \le M<U$ and $(a\times b)$ is as small as possible---$a$ and $b$ are integers. For example, If ...
3
votes
1answer
42 views

Real numbers and rationals - Decimal Expansion

How would one endeavor to show that A real number is rational if and only if its decimal expression ends in recurring digits?
5
votes
2answers
60 views

What automorphisms exist on the abelian group of positive rationals under multiplication?

Consider the abelian group $(\mathbb{Q}_{>0}, \times)$. What automorphisms exist for this group? I can only think of the trivial one and of $\phi(q) = \frac{1}{q}$. If we relax the problem to ...
4
votes
1answer
98 views

Is the set of real numbers really uncountably infinite?

The proof that the set of real numbers is uncountably infinite is often concluded with a contradiction. In the following argument I use a similar proof by contradiction to show that the set of ...
1
vote
3answers
91 views

Closed form of a sum of ratios of integers

I am computing in a program this sum (does it have a "name"): $$\sum_{\alpha=2}^{K} \frac{\alpha-1}{\alpha}$$ is there a way to avoid the sum, term by term, and use a more compact closed form ?
2
votes
1answer
38 views

Approximating non-rational roots by a rational roots for a quadratic equation

Let $a,b,c$ be integers and suppose the equation $f(x)=ax^2+bx+c=0$ has an irrational root $r$. Let $u=\frac p q$ be any rational number such that $|u-r|<1$. Prove that $\frac 1 {q^2} \leq |f(u)| ...
1
vote
1answer
48 views

Is every homeomorphism of $\mathbb{Q}$ monotone?

It is well known that every continuous injective map $\mathbb{R}\rightarrow\mathbb{R}$ is monotone. This statement is false for maps $\mathbb{Q}\rightarrow\mathbb{Q}$. (That is becaus $\mathbb{Q}$ is ...
5
votes
0answers
122 views

Prove that the square root of any irrational number is irrational.

The problem I'm having with this proof is that I'm not sure if my proof actually proves the theorem correct or if I'm using circular reasoning. Theorem: Prove that the square root of any irrational ...
1
vote
1answer
31 views

rational numbers as upper limit of a summation?

a quick question: Is it a legit way to use a fraction as the upper limit of a summation? Given is a frequency $f$ and a sample rate $f_s$. I want to use a sum like this: $\sum_{k=1}^{\frac{f_s}{2f}} ...
1
vote
1answer
46 views

Compare density of rationals to the density of integers

Is is possible to somehow quantitatively compare the density of rational numbers to the density of integer numbers, ascribing to the both a number characterizing the density?
3
votes
2answers
601 views

How can the decimal expansion of this rational number not be periodic?

I just noticed that dividing $1 \div 998$ gives me the apparently non-periodic $$0.001002004008016032064\ldots ,$$ which is $$10^{-3} + 2\times 10^{-6} + 4\times10^{-9} + 8\times 10^{-12} + \cdots = ...
2
votes
1answer
66 views

Rational Irrational Numbers

I know that a rational number can always be expressed as a fraction, but can't we also say that it is a number that follows a definite pattern? Like one-third for example; it is never ending as a ...
3
votes
2answers
37 views

For each rational $q$ there is an integer $n$ such that $q+n=271$, and vice versa

For all $q \in \Bbb Q$ there exists $n \in \Bbb Z$ so that $q + n = 271$. This is true? Because both $q$ and $n$ are rational numbers and $271$ is an integer thus it's a rational number? Also, ...
0
votes
2answers
460 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 ...
8
votes
3answers
488 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
33 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 $(- \sqrt{11}, \sqrt{11})$ ...
1
vote
1answer
46 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 ...
1
vote
1answer
62 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
138 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
57 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
61 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
137 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
85 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 ...
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1answer
36 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 ...
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vote
4answers
633 views

If $x$ and $y$ are irrational but $x + y$ is rational then $x - y$ is irrational [closed]

Prove that if $x$ and $y$ are irrational but $x + y$ is rational then $x - y$ is irrational. I can understand how it works in my head, I don't know how to prove it though.
1
vote
0answers
64 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
58 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
69 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
39 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
99 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 ...
1
vote
0answers
34 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
59 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
48 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 ...