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

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3
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7answers
341 views

Doubt on rational and real numbers

I am going through the numbers system from an analysis book. It is written that: 1) there is no rational number $p \ ( > 0)$ which satisfies $p^2=2$. 2) The set $\{p: p^2 < 2\}$ does not have ...
0
votes
1answer
44 views

Finding minimal period of rational

I'm trying to find the decimal representation minimal period of $1/n$ where $n$ is an integer. I'll clarify colloquially because I'm very noob with math terms: $$1/3 = 0,(3)$$ $$DP(3) = 1$$ $$1/7 = ...
3
votes
1answer
123 views

Extending the rationals using exponentiation

The set of integers can be constructed as an equivalence relation over the natural numbers using the the binary operation of addition, and a similar process yields the rationals from integers and ...
1
vote
1answer
70 views

Defining piecewise summation of continued fractions and rationality of sums

Let $a=[a_1,a_2\dots]$ and $b=[b_1,b_2\dots]$ be two real numbers and their continued fraction representations. They may be infinite or finite. Let us define a thing $+^*$ so that ...
0
votes
1answer
436 views

Inverse rational numbers counting function

I have found next formula of counting all rational numbers (or integer pairs, from which rational numbers can be builded): $N(i,j) = ((i - j - 1)(i - j) - 2j - 1)(1 + j / |j|)/2 + ((i + j - 1)(i + j) ...
2
votes
0answers
77 views

Two quartic polynomials to be made a square?

Given two generally non-square quartic polynomials that are to be simultaneously made squares for particular values of $x$, $$c_1x^4+c_2x^3+c_3x^2+c_4x+c_5 = y_1^2$$ ...
0
votes
0answers
75 views

Limits as a representation of the Dirichlet function

I read that the Dirichlet function (1 if Rational, 0 else) can be written as: What is the proof of that? Are those limits commutative? Is there any other closed formula for Dirichlet function? (With ...
0
votes
1answer
101 views

Ratio and Proportion - IV

If $a,b,c,d$ are continued proportion : Prove that : $(\frac{a-b}{c}+\frac{a-c}{b})^2-(\frac{d-b}{c}+\frac{d-c}{b})^2=(a-d)(\frac{1}{c^2}-\frac{1}{b^2})^2$ After solving LH.S I got : ...
1
vote
2answers
105 views

Proof by contradiction - help!?

I need to prove that the set of rational numbers in the closed interval 0,1 has a supremum and infimum. I know that they exist and I also know that I need to use proof by contradiction but I don't ...
6
votes
1answer
459 views

Faster arithmetic with finite continued fractions

I was curious about different representations of rational numbers and came across the finite continued fraction (see wp:Finite_continued_fractions ). Note: I will refer to traditional rational ...
2
votes
2answers
331 views

Is there a proof that $\mathbb{R}$ is connected?

Is there a proof that the set $\mathbb{R}$ of all real numbers is connected? I've been assuming that $\mathbb{Q}$ is discrete, with a (very small) gap existing between any two elements ...
5
votes
1answer
140 views

finite field to rational fraction

Suppose I have a number $n\in\mathbb F_p$, i.e. an element of the finite field obtained by arithmetic modulo some (odd) prime $p$. I'm looking for a way to find a simple description of $n$ as a ...
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vote
3answers
90 views

In what circumstances can $\dfrac{aA+b}{cA+d}$ be rational?

I am working on the chapter one practice problems in Hardy and cannot seem to figure it out. My attempt has actually left me with a result contrary to what the question is looking for. The Question ...
13
votes
1answer
265 views

Why is $x^3-5x$ injective on the rationals?

I've found the statement on the internet that the polynomial $x^3-5x$ is injective on the rational numbers, but without any comments on how to prove it. I think it means it must be easy, but I don't ...
8
votes
2answers
203 views

Conditions that $\sqrt{a+\sqrt{b}} + \sqrt{a-\sqrt{b}}$ is rational

Motivation I am working on one of the questions from Hardy's Course of Pure Mathematics and was wondering if I could get some assistance on where to go next in my proof. I have attempted rearranging ...
5
votes
1answer
168 views

Is this graph based on rationals familiar?

Has anyone come across a graph like this? The black circles represent rationals in $(0,1)$ and their heights are roughly proportional to the reciprocal of the square of their lowest terms ...
9
votes
4answers
663 views

How do I rewrite -100+1/2 as the mixed number -99 1/2?

This has been bugging me for some time now, so I ask you to try to help me realize what is going on here. I just can't get my brain around this. I have a proper fraction and a negative integer. The ...
0
votes
0answers
431 views

Mathematical problems having rational number solutions

Is there a well defined class of mathematical problems which produce only rational numbers as their solutions?
0
votes
2answers
72 views

Two reals with rational quotient, Z-span the integers

Given two real numbers $\alpha_1,\alpha_2\in\mathbb{R}\cap(0,1)$, with rational ratio $\frac{\alpha_1}{\alpha_2} \in \mathbb{Q}$, show there exist $m,n\in \mathbb{Z}$, such that $n\alpha_1 + ...
13
votes
2answers
1k views

Is sin(x) necessarily irrational where x is rational?

My friend and I were discussing this and we couldn't figure out how to prove it one way or another. The only rational values I can figure out for $\sin(x)$ (or $\cos(x)$, etc...) come about when $x$ ...
3
votes
1answer
231 views

Field containing all square roots of rational numbers

What is the smallest field which contains all square roots of positive rational numbers? I guess I mean “smallest” in terms of set inclusion, i.e. the minimal one with regard to the “$\subseteq$” ...
4
votes
2answers
5k views

Is a non-repeating and non-terminating decimal always an irrational?

We can build $\frac{1}{33}$ like this, $.030303$ $\cdots$ ($03$ repeats). $.0303$ $\cdots$ tends to $\frac{1}{33}$. So,I was wondering this: In the decimal representation, if we start writing the ...
4
votes
5answers
244 views

Will we get all real numbers if we add all limits?

Consider a set of all rational numbers from 0 to 1 inclusive. If we add to this set all limits of all convergent sequences of these numbers, will we obtain a set of all real numbers from 0 to 1?
3
votes
0answers
143 views

Must be rational number

Let $a$, $b$ positive rational number. Suppose that there exist two odd positive integers $p$, $q$ such that $\sqrt[p]{a}+\sqrt[q]{b}$ is rational. Prove that both $\sqrt[p]{a}$ and $\sqrt[q]{b}$ are ...
6
votes
3answers
163 views

How does one show that the set of rationals is topologically disconnected?

Let $\mathbb{Q}$ be the set of rationals with its usual topology based on distance: $$d(x,y) = |x-y|$$ Suppose we can only use axioms about $\mathbb{Q}$ (and no axiom about $\mathbb{R}$, the set of ...
3
votes
1answer
215 views

Any positive rational number can be expressed in one and only one way in the form …

I am attempting Miscellaneous Examples on Chapter 1, Number 2, from Hardy's Course of Pure Mathematics. Question Any positive rational number can be expressed in one and only one way in the form: ...
6
votes
0answers
151 views

Rational multiples of $\pi/2$ whose sines are also rational

Let $f(x)=\sin(x\frac{\pi}{2})$. Let $R$ the set of $x$ such that : $0\le x\le 1$ $x \in \mathbb Q$ $f(x) \in \mathbb Q$ Hence, $0\in R$ as $f(0)=0$. $1\in R$ as f(1)=1. And $\frac{1}{3}\in R$ as ...
2
votes
2answers
699 views

In every interval there is a rational and an irrational number.

When the interval is between two rational numbers it is easy. But things get complicated when the interval is between two irrational numbers. I couldn't prove that.
5
votes
1answer
186 views

Bézout's identity in higher dimensions?

I have an invertible rational matrix $C\in\text{GL}(n,\mathbb{Q})$ which works on lattice $\mathbb{Z}^{n}$. Can I write the resulting set in the following form $$C\cdot \mathbb{Z}^{n}=X\cdot ...
0
votes
1answer
170 views

Is $“2.1234… ”$ rational?

In my excercise book of math , I have found one problem . In that problem I have been asked to detect whether the number $2.1234....$ is rational or irrational? My concept is : "$2.1234....$ is ...
0
votes
1answer
135 views

$a_n(x):=nx-\lfloor nx \rfloor$

i have $a_n(x):=nx-\lfloor nx \rfloor$ where $x$ is real. i want to show that if $x$ is rational, then $a_n(x)$ has finitely many cluster points, if $x$ is irrational, then every real $a$ with $0\leq ...
21
votes
6answers
914 views

Axiomatic characterization of the rational numbers

We have the well-known Peano axioms for the natural numbers and the real numbers can be characterized by demanding them to be a Dedekind-complete, totally ordered field (or some variation of this). ...
7
votes
2answers
235 views

Chances of avoiding the diagonal

A circle of radius 1 is randomly placed in a rectangle $ABCD$ so that the circle lies completely inside the rectangle. Length and breadth of rectangles are 36 and 15 respectively. Let the ...
1
vote
1answer
34 views

Function of a rational number

Let a function exist such that $f(a+b)=f(a)+f(b)$. We have already shown that for any integer n, $f(nx)=n f(x)$. Now we must show that for any rational number $n/m$, $f(n/m)=n/m f(1)$. The problem is ...
3
votes
3answers
441 views

HINT: Prove there does not exist a rational number that satisfies $x^3=p/q$

I would like to request a hint for a problem I am working on form Hardy's a Course of Pure Mathematics. Question Prove generally that a rational fraction $p/q$ in its lowest terms cannot be the cube ...
2
votes
2answers
627 views

Rational numbers- sticks and stones

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. For all rational numbers, we will have a stick of variable length ...
2
votes
2answers
993 views

Hint for: Prove any terminating decimal can be represented as a rational number

I am currently working on a problem from Hardy and have been stuck trying to figure out what to do. I was wondering if someone could provide me with a hint that may help jump-start my thought process. ...
3
votes
1answer
111 views

Analytical function taking rationals to rationals.

Suppose $f:I \rightarrow \Bbb R$ is an analytic function defined on the interval $I\subset \Bbb R$ with the property that for every $q \in \Bbb Q:f(q)\in \Bbb Q$. Does this already imply that $f\in ...
4
votes
3answers
221 views

$n ^ 2 +5 $, $n ^ 2 +10 $ are rational number square

Seeking a nonzero rational number $n$, such that $n ^ 2 +5 $, $n ^ 2 +10 $ are rational number square。 This is a high school students asked the question, answer $n=\frac{31}{12}$, but no answer ...
2
votes
2answers
124 views

Proof that there is an infinite amount of rationals $t$, $0<t<1$ using the definitions of $=$ and $\leq$

This is a question to which the answer is of course intuitively obvious. We are asked to prove that there is an infinite amount of rationals $t$ with $0<t<1$ using the definitions of $=$ and ...
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vote
1answer
410 views

Prove that given any rational number there exists another greater than or equal to it that differs by less than $\frac 1n$

I am currently attempting to prove a claim in Hardy's Course of Pure Mathematics and am currently stuck. I was hoping that someone would be able to provide some assistance on how to go about this. ...
1
vote
3answers
157 views

Show that $p/q$ is the$ [(1/2)(p+q-1)(p+q-2)+q]$th term of the series

I am attempting to prove that given a series of rational numbers $p/q$ as presented below: $$ 1/1,\; 2/1,\; 1/2,\; 3/1,\; 2/2,\; 1/3,\; 4/1,\; 3/2,\; 2/3,\; 1/4,\; \ldots $$ That $p/q$ is the ...
2
votes
2answers
123 views

A question about rational.

Is that true : Every positive rational number $q$ can be written as $q = \sum_{i=0}^{k}1/n_i$ , where $n_i,k$ are positive intergers and $n_i\not=n_j$ if $i\not=j$.
0
votes
1answer
224 views

cross multiplication property

good morning guys,i have such question,when i was reading GRE book,there was such kind of property related to rational number,in shortly if we are trying to determine if $a/b$ is more then ...
2
votes
1answer
58 views

How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$?

I would like to prove the existence of $b \in \mathbb Q$ such that $a<b^2<c$ for any given $a,c \in \mathbb Q$ with $a,c>0$ I want to use the statement above to prove a statement in a link I ...
1
vote
1answer
42 views

How to represent fraction $\frac{1}{a+b\cdot \sqrt[3]{2} + c\cdot (\sqrt[3]{2})^2}$ as $a_1 + b_1 \cdot \sqrt[3]{2} + c_1 \cdot (\sqrt[3]{2})^2$?

What do I have to multiply both numerator and denominator with to get the representation is asked? $a, b, c, a_1, b_1, c_1$ are rational numbers.
9
votes
3answers
241 views

Half the rationals?

Let $\mathbb{Q}[n]$ be the set of rational numbers with denominator $\le n$ and for any $X\subseteq \mathbb{Q}$, let $X[n]=X\cap \mathbb{Q}[n]$. Is there a set of rational numbers, X, such that for ...
6
votes
2answers
241 views

For what algebraic curves do rational points form a group?

For what real algebraic curves do rational points form a group ? How does this relate to Jacobian Varieties ?
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vote
1answer
204 views

Let $r_n$ be an enumeration of the rationals in $[0,1]$, does the sequence $\{B_{\frac{1}{n}}(r_n)\}_{n=k}^m$ cover $[0,1]$ for $m-k$ finite?

Let $r_n$ be an enumeration of the rationals in $[0,1]$, does the sequence $\{B_{\frac{1}{n}}(r_n)\}_{n=k}^m$ cover $[0,1]$ for $m-k$ finite? This came up while trying to solve a different problem, ...
0
votes
0answers
19 views

Question on passing from rational to exponet

it's not a question itself, but I'd like to check if am I doing this passage from rational numbers to the exponent form right: From: $\sqrt{a\sqrt{a}}$ Evaluete to: $\sqrt{a*a^{\frac{1}{2}}}$ = ...