# Questions tagged [rational-numbers]

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.

1,582 questions
Filter by
Sorted by
Tagged with
163 views

3k views

### How can we find and categorize the subgroups of $\mathbb{R}$?

$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}\newcommand{\Z}{\Bbb Z}$ What are all the subgroups of R = $(\R, +)$ and how can we categorize them? I started thinking about this question last night ...
1k 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 ...
7k views

74 views

### How many sieves are there on a given rational number $q$?

Consider the poset category $\mathbb{Q}^{op}$, i.e. where $p \rightarrow q$ iff $p \geq q$. Take any $q \in \mathbb{Q}$. Then how many sieves are there for $q$ in $\mathbb{Q}^{op}$? Supposedly, the ...
562 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 ...
3k views

### What is the ratio of rational to irrational real numbers?

There exists an infinite amount of rational and irrational numbers. But is there more irrational numbers than rational? And if so can a ratio of one to the other be calculated?
2k views

### How to find all rational points on the elliptic curves like $y^2=x^3-2$

Reading the book by Diophantus, one may be led to consider the curves like: $y^2=x^3+1$, $y^2=x^3-1$, $y^2=x^3-2$, the first two of which are easy (after calculating some eight curves to be solved ...
494 views

### When are $\theta$ and $\sin\theta^\circ$ both rational? [duplicate]

Possible Duplicate: Sine values being rational I'm guessing that if I look in Ivan Niven's elementary book on irrational numbers, I'll find the answer to this quickly, but I'm posting it here in ...