# 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.

1,567 questions
Filter by
Sorted by
Tagged with
71 views

111 views

### Does there exist a generating function for the rational numbers?

Since the rationals are countable, you can list them in a sequence $(a_n)_{n\geq 0}$ such that each rational appears at least once in the sequence. Is there such a listing $(a_n)_{n \geq 0}$ for which ...
83 views

52 views

### Density of $\mathbb{Q}$ in $\mathbb{R}$

Let $X$ be a topological space and $Y\subseteq X$ a subset. $Y$ is said to be dense in $X$ if $\overline{Y}=X$ (where $\overline{Y}$ denotes the closure of $Y$). Now consider $X=\mathbb{R}$ (with ...
86 views

### Countability of Sets with rational and real numbers [closed]

Determine whether it is finite, countably infinite, or uncountably infinite. Justify $$\Big\{\Big(\frac{m}{2}, \frac{n}{3}\Big) \in \mathbb{R}^2 \mid m,n \in \mathbb{Z}\Big\}$$ The set is ...
20 views

### Difference between rationals in a certain set is at least a certain amount

Define $A = \{\frac{p}{q} \in \mathbb{Q} \mid q \in \mathbb{N}, q < n, gcd(p,q) = 1\}$. I am trying to prove that the difference of any 2 distinct elements of this set is greater than $\frac{1}{n}$....
76 views

### Definition of the supremum in $\mathbb{Q}$

If $M=\sup \left( A \right)$ and $A\subseteq \mathbb{Q}$ (rational numbers) $\forall\ \varepsilon>0$ of $\mathbb{Q}, \exists\ x\in A$ s.t. $x > M–\varepsilon$. Is this property true for ...
55 views

### Mapping $\Bbb N\to\Bbb Q$

I want to set up a map from $\Bbb N\to\Bbb Q.$ Take $\Phi_S(x)=e^{(S/\ln(1-x))}$ and $M_T(1-x)=\Phi_S(x); S,T\in\Bbb N.$ Set $\Phi_S(x)=M_T(x)$ to obtain algebraic $x$ coordinates. If $x$ happens ...
30 views

30 views

50 views

### Finding rational numbers in an equation with two variables

How should we find two rational numbers $\alpha$, $\beta$ such that $\sqrt{7+5\sqrt{2}}=\alpha+\beta\sqrt{2}$? The answer I got alpha = 1 and betta = 1. If I'm wrong, please correct me. Thank you
28 views

### Some numbers represented by symbols

I am trying to find some numbers that are represented by symbols, such as π, e, i, φ. I couldn't find more. Can you guys help me? (English is not my main language and it is for school project.)
62 views

### Rational as series?

I was checking out a few things in the geometric series and realized all rational numbers can be shown as a geometric series.I was pretty sure I read something like that somewhere.Can anyone tell me ...
65 views

### Well-Ordering Irrationality

Let $D$ be a positive integer and the let the square root of $D$ be a real number. Assuming that the square root of D is not an integer (i.e. $D$ is not a perfect square), use Well-Ordering to prove ...
60 views

### Sum of two irrational numbers being rational or irrational

I am currently doing a project on irrational and transcendental numbers and part of this project requires me to look at sums and products of irrational numbers. I am aware that the sum of 2 ...
155 views

### For all integers $w, x, y, z$ with $w\neq{y}$ and $wz-xy\neq0$, prove that there exists a unique rational number $r$ such that $(wr+x)\div(yr+z)=1$

For all integers $w, x, y, z$ with $w\neq{y}$ and $wz-xy\neq0$, prove that there exists a unique rational number $r$ such that $(wr+x)\div(yr+z)=1$ How do I prove uniqueness? I know to show that ...
44 views

Set of integers is denoted by the symbol $\mathbb Z$, $\mathbb Q[x]$ stands for univariate polynomials over rationals, etc. Is there a symbol which indicates the set of dyadic rationals?
61 views

### Abstract Algebra Square Roots Are Irrational

For part (a), I begin by trying to prove $S$ is empty implies the square root of $D$ is irrational. If we take the contrapositive of this implication, this is equivalent to proving that if the square ...
86 views

### Sums and Products of Algebraic and Transcendental numbers

I am doing a project about irrational and transcendental numbers and part of this project involves looking at sums and products of various combinations of rational, irrational, algebraic and ...
65 views

### simple question : sequential criterion for continuity

Let $A$ be a subset of $\mathbb{R}$, and let $f:A\to\mathbb{R}$ be a function. Now, let $\{r_{n}\}$ be any rational sequence in $A$, and let $\{s_{n}\}$ be any irrational sequence in $A$. Suppose ...
232 views