For questions on rational numbers, numbers that can be expressed as the quotient or fraction $\frac pq$ of two integers.
4
votes
4answers
80 views
Definition(s) of rational numbers
The definitions of rational numbers are somewhat confusing for me. The definition of rational numbers on wikipedia and most other sites is:
In mathematics, a rational number is any number that ...
6
votes
1answer
71 views
Can someone clarify Example I.I.2 from Hardy's Course of Pure Mathematics?
"If $\lambda, m,$ and $n$ are positive rational numbers, and $m > n$, then $\lambda(m^2 − n^2), 2\lambda mn$, and $\lambda(m^2 + n^2)$ are positive rational numbers. Hence show how to determine any ...
2
votes
2answers
55 views
Proof f(x) is continuous given $x$ rational and irrational.
How can I resolve the task below:
Given $f(x)=
\begin{cases}
x, &x\in \mathbb{Q}\text{ }\\
1-x, &x\notin \mathbb{Q}\text{ (irrational)}
\end{cases}$, $0 \leq x \leq 1$.
Show $f(x)$ is ...
0
votes
1answer
29 views
Continuous variable defined over Rational numbers only?
Let $x(t)$ be a solution of some first order ODE, which is continuous over $t$. In this case, is the continuous $x(t)$ defined only over Rational numbers? what is the reason behind this? Please ...
4
votes
1answer
99 views
Linear equations; real solution; rational solution?
I saw this question
Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $B ∈ \mathbb{Q}^m$. Suppose
that the system of linear equations $AX = B$ has a solution in
$\mathbb{R}^n$. Does it necessarily have ...
-1
votes
3answers
135 views
I'm just curious, what exactly is $\mathbb{R}\setminus\mathbb{Q}$? [duplicate]
What exactly is $\mathbb{R}\setminus\mathbb{Q}$? How many different kinds of things live in this place?
For $n>1$ how does
$$ q_1x_1+\cdots+q_nx_n=p $$
have a solution for $q_i,p\in \mathbb{Q}$ ...
-1
votes
2answers
85 views
$\mathbb{R} \setminus \mathbb{Q}$:'a stamping tool' [closed]
What does it mean for the polynomial
$$ a_1x_1+\cdots+a_nx_n=b $$
to have solutions in $\mathbb{R} \setminus \mathbb{Q}$, where $a_i,b\in \mathbb{Q}$?
1
vote
2answers
52 views
Can all possible angles on a rational triangle be represented as a rational multiplied by Pi?
The reason I ask: I was wondering if it was possible to find the angle of a rational triangle by only using the lengths of its sides and knowledge of $\pi$ (that is, no inverse trig functions).
So, ...
1
vote
1answer
56 views
Can you produce a number like 1.01010101… by just addition and subtraction?
I'm working on a program in C# where a Decimal variable can hold negative and positive values including 0 and those values can only change by addition and subtraction.
I have a conditional where if ...
3
votes
2answers
63 views
Define two rational numbers $\alpha$ and $x$ such that $\sin( { \alpha }) =x$
Of course for $x\neq 0 $ and $\alpha$ in radians. Can you define them?
1
vote
1answer
46 views
Looking for name of theorem: “rational $\Leftrightarrow$ fractional part terminates or repeats”
I am looking for the name of the theorem that says that a number $x$ is rational if and only if its fractional part terminates or repeats (where "fractional part" refers to the representation of $x$ ...
2
votes
1answer
28 views
length of period
Each rational number (fraction) can be written as decimal periodic number. Does exists a method or hint that show how long will be the period of arbitrary fraction. For example $1/3=0.3333...=0.(3)$ ...
5
votes
1answer
75 views
Defining $\Bbb{Q}$ without the axiom of infinity
(TL;DR version: I want a meaningful definition of $\Bbb{Q}$ without $\sf{Inf}$.)
In the "conventional" construction of the rationals, we define $\Bbb{Q}$ as follows:
$\omega$ is the first limit ...
1
vote
0answers
34 views
“Rational grids” on manifolds.
Here is something which is bothering me a bit. You have rationals on a line. You can define a rational grid on R^n by taking points with all coordinates having rational values. Is there a ...
0
votes
3answers
52 views
Show that there is no rational number $r=m/n$ such that $r^3=3$ [duplicate]
How do I solve this by prime factorization?
I came across a similar problem on MSE just recently, but I can't find it and I thoroughly searched for it. If anyone can find it, please post it in the ...
2
votes
2answers
47 views
Every 10 years the population of a city is five-fourths of what it was 10 years before.
I am working through Serge Lang's "Basic mathematics", currently on chapter 1, Section 5 question 21.
The part that troubles me is, when I asked someone I know how to solve this, they suggested ...
1
vote
2answers
59 views
Approximating Euclidean geometry, restricted to $\mathbb{Q}$
I'm having trouble putting this into a fully coherent question, so I'll give the broad question, then a few bullet points to give you a better idea of what I'm asking.
I'm looking for a line of ...
1
vote
2answers
44 views
Understanding the boundary of a set
I try to understand the boundary of a set.
I know the definition (Let $A \subseteq \mathbb{R}
$: P is a point of the boundary, if for every small $\epsilon \in \mathbb{R}, \epsilon>0$ are points ...
3
votes
7answers
207 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
28 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:
$$1/3 = 0,(3)$$
$$DP(3) = 1$$
$$1/7 = ...
4
votes
3answers
98 views
On comparing fractions , fraction with smaller difference between numerator and denominator is greater than the other
A text book proposed that "when comparing fractions ,if the compared fractions's are such that numerator is smaller than denominator ,then fraction with more difference(absolute) between numerator ...
1
vote
1answer
30 views
Defining piecewise summation of continued fractions and rationality of sums
Let $a=[a_1,a_2\dots]$ and $b=[b_1,b_2\dots]$ be two real numbers and their continued fraction representations. They may be infinite or finite.
Let us define a thing $+^*$ so that ...
1
vote
2answers
65 views
Rational numbers
This number 2.962962 can be rational
$$x=2.962962$$
$$10x=29.62962$$
$$100x=296.2962$$
$$1000x=2962.962$$
$$1000x-10x=\frac{990x}{990}=\frac{2933}{990}$$
why is this wrong?
That way of getting the ...
2
votes
0answers
51 views
Two quartic polynomials to be made a square?
Given two generally non-square quartic polynomials that are to be simultaneously made squares for particular values of $x$,
$$c_1x^4+c_2x^3+c_3x^2+c_4x+c_5 = y_1^2$$
...
0
votes
0answers
46 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
42 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
74 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 ...
1
vote
1answer
38 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 ...
2
votes
2answers
148 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
60 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 ...
2
votes
3answers
68 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
...
10
votes
1answer
184 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 ...
7
votes
2answers
106 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 ...
9
votes
4answers
359 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
131 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
38 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 + ...
10
votes
2answers
267 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$ ...
2
votes
1answer
133 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
295 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 ...
0
votes
4answers
132 views
How can I explain $0.999\ldots=1$? [duplicate]
Possible Duplicate:
Does .99999… = 1?
I have to explain $0.999\ldots=1$ to people who don't know limit.
How can I explain $0.999\ldots=1$?
The common procedure is as follows
...
4
votes
5answers
218 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
124 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 ...
5
votes
3answers
139 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|$$
Suppose we can only use axioms about $\mathbb{Q}$ (and no axiom about $\mathbb{R}$, the set of ...
3
votes
1answer
102 views
Any positive rational number can be expressed in one and only one way in the form …
I am attempting Miscellaneous Examples on Chapter 1, Number 2, from Hardy's Course of Pure Mathematics.
Question
Any positive rational number can be expressed in one and only one way in the form: ...
4
votes
0answers
80 views
Rational multiples of $\pi/2$ whose sines are also rational
Let $f(x)=\sin(x\frac{\pi}{2})$.
Let $R$ the set of $x$ such that :
$0\le x\le 1$
$x \in \mathbb Q$
$f(x) \in \mathbb Q$
Hence, $0\in R$ as $f(0)=0$. $1\in R$ as f(1)=1. And $\frac{1}{3}\in R$ as ...
1
vote
2answers
88 views
In every interval there is a rational and an irrational number.
When the interval is between two rational numbers it is easy. But things get complicated when the interval is between two irrational numbers. I couldn't prove that.
5
votes
1answer
111 views
Bézout's identity in higher dimensions?
I have an invertible rational matrix $C\in\text{GL}(n,\mathbb{Q})$ which works on lattice $\mathbb{Z}^{n}$. Can I write the resulting set in the following form
$$C\cdot \mathbb{Z}^{n}=X\cdot ...
0
votes
1answer
115 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
121 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 ...
20
votes
6answers
562 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).
...
