For questions on rational numbers, numbers that can be expressed as the quotient or fraction $\frac pq$ of two integers.
4
votes
3answers
257 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 ...
16
votes
4answers
308 views
What do the cosets of $\mathbb{R} / \mathbb{Q}$ look like?
$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}$
Looking at the group of real numbers under addition $(\R, +)$ it contains the (normal) subgroup of rational numbers $(\Q, +)$. I am wondering how to ...
2
votes
3answers
108 views
Compute the period of a decimal number a priori [duplicate]
Possible Duplicate:
Upper bound/exact length of decimal expansion of simple fraction
I noticed that WolframAlpha given an operation like $\frac{n}{m},\;n,m \in N$ that result in a periodic ...
-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 $a_i,b\in \mathbb{Q}$?
14
votes
2answers
333 views
Solutions of $q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$ s.t. $q \geq 3$
Is it true that for every rational $q \geq 3$ , the following equation has a solution over $\mathbb N$ ?
$$q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$$
15
votes
3answers
576 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:
$$ ...
4
votes
2answers
149 views
For what algebraic curves do rational points form a group?
For what real algebraic curves do rational points form a group ?
How does this relate to Jacobian Varieties ?
4
votes
0answers
72 views
Is there always a telescopic series associated with a rational number?
Here is something I thought up while I was bored and my, erm, fish were busy:
Given a rational number $p\in(0,1)$, are there always positive integers $n$, $c_m$, and $w_m$ such that ...
4
votes
1answer
99 views
Linear equations; real solution; rational solution?
I saw this question
Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $B ∈ \mathbb{Q}^m$. Suppose
that the system of linear equations $AX = B$ has a solution in
$\mathbb{R}^n$. Does it necessarily have ...
3
votes
1answer
187 views
If a finite set of rational numbers sums to one, does one of the rationals have a denominator equal to the LCM of all the denominators?
I was experimenting with an algorithm for generating random numbers from a discrete distribution and came across an interesting observation. Suppose that you have any finite set of rational numbers ...
2
votes
4answers
115 views
float result for two smallest integer division
I want to know the two integer number that division of them is this float. for example
x / y = 1.333333333....
$x$ and $y$ can be ...
2
votes
1answer
100 views
Is there a rational univariat polynomial of degree 3 with 3 irrational roots?
The title pretty much asks my question: Does $f\in\mathbb{Q}[x]$ such that
$$ f(x)=(x-\alpha_1)(x-\alpha_2)(x-\alpha_3),$$
where $\alpha_1, \alpha_2, \alpha_3\in\mathbb{R}\setminus\mathbb{Q}\ $ ...
0
votes
3answers
647 views
Show that the ring of all rational numbers, which when written in simplest form has an odd denominator, is a principal ideal domain.
Show that the ring of all rational numbers $m/n$ with $n$ an odd integer is a principal ideal domain.
We haven't really discussed principal ideal domains. I've heard that this is easy, but I just ...
