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

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9
votes
4answers
645 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
391 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
71 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 + ...
12
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
229 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
4k 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
206 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
146 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
622 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
161 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
886 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
232 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
421 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
621 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
948 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
110 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
220 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 ...
3
votes
1answer
654 views

How do I get the integer part of a number by using basic arithmetic?

While it is trivial to simply remove the fractional part of an irrational or rational number, and in programming I could just use the floor() or ...
1
vote
1answer
404 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
155 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
221 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
41 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
233 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 ?
1
vote
1answer
199 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
18 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: $\sqrt{a*a^{\frac{1}{2}}}$ = ...
6
votes
3answers
553 views

Why (directly!) does every number divide 9, 99, 999, … or 10, 100, 1000, …, or their product?

A curiosity that's been bugging me. More precisely: Given any integers $b\geq 1$ and $n\geq 2$, there exist integers $0\leq k, l\leq b-1$ such that $b$ divides $n^l(n^k - 1)$ exactly. The ...
2
votes
2answers
226 views

Constructing the reals from the rationals

Dr. H. Jerome Keisler, in his book Elementary Calculus: An Infinitesimal Approach, states on page 24: Just as the real numbers can be constructed from the rational numbers, the hyperreal numbers ...
0
votes
1answer
79 views

How to find out the probability of ordered pairs of rational or irrational number $(a,b)$ such that $1<a<50, 1,<b<50$, and $\log_b a$ is rational.

How to find out the probability of ordered pairs of rational or irrational or transcendental number $(a,b)$ such that $1<a<50$, $1<b<50$, and $\log_b a$ is rational? Uniformly ...
24
votes
5answers
1k views

Is the square root of -1 rational?

This is not a deep question, but if there is a definite answer then here is the place where I will find it. Is it justified to say that $i =\sqrt{-1}$ is rational? The origin of this question lies ...
2
votes
3answers
95 views

Corresponding numerator of the fraction

I am stuck at this question cant think of how to resolve this, sorry I did try to do working but can't think of any right solution. The question is If the denominator is $9900$, then what is the ...
0
votes
3answers
972 views

What is common and widely recognized abbreviations for *Numerator* and *Denominator* terms for Anglophone mathematicians? [closed]

I have a basic notation/convention question. I'm writing a program in Pascal programming language which does computations in the rational number field (ℚ). For that i defined a data type, which ...
5
votes
6answers
374 views

Prove that - for every positive $x \in \mathbb{Q}$, there exists positive $y \in \mathbb{Q}$ for which $y \lt x$

First my apologies if this question has been asked before. Exposition I'm new at learning how to prove theorems and among the given exercises from my reference material it is asked to prove the ...
-3
votes
1answer
85 views

Rational numbers and cardinality of some subset set of them.

Let $G$ be the set of rational numbers of the form $m/n$ , where $m,n$ are positive integers and $n \leq g $ for some possitive integer $g$. Suppose it is bounded by $1/k$ , k is a positive integer ...
4
votes
3answers
171 views

How do I write down a curve with exactly one rational point

Let $g\geq 1$. I would like to write down (for all $g$) a smooth projective geometrically connected curve $X$ over $\mathbf{Q}$ of genus $g$ with precisely one rational point. Is this possible? For ...
5
votes
0answers
103 views

Is there always a telescopic series associated with a rational number?

Here is something I thought up while I was bored and my, erm, fish were busy: Given a rational number $p\in(0,1)$, are there always positive integers $n$, $c_m$, and $w_m$ such that ...
2
votes
4answers
124 views

float result for two smallest integer division

I want to know the two integer number that division of them is this float. for example x / y = 1.333333333.... $x$ and $y$ can be ...
0
votes
2answers
485 views

Can dividing two rational numbers yield an integer?

I wanted to know if two non-int numbers (non-zeroes) when divided with each other can give an integer or not.I believe that's a NO. However I know they can only yield an integer $1$ provided both are ...
1
vote
4answers
973 views

Rationalize and simplify $\frac{x-1}{\sqrt{2\sqrt{x}} + 1 - \sqrt[4]{x}}$

For my exercise, I have been asked to rationalize and simplify this surd; $$\frac{x-1}{\sqrt{2\sqrt{x}} + 1 - \sqrt[4]{x}}$$ I don't know how to type it. The denominator is square root of 2 with ...
18
votes
3answers
1k views

GCD of rationals

Disclaimer: I'm an engineer, not a mathematician Somebody claimed that $\gcd$ only is applicable for integers, but it seems I'm perfectly able to apply it to rationals also: $$ ...
4
votes
1answer
86 views

How do I scale 3 fractions to 3 natural numbers?

Disclaimer: I'm an engineer, not a mathematician I have a set of three fractions (a/b, c/d, e/f). I can multiply them all by another fraction, so that their mutual ratios remain the same. I want to ...