# Tagged Questions

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.

1answer
89 views

### Is every homeomorphism of $\mathbb{Q}$ monotone?

It is well known that every continuous injective map $\mathbb{R}\rightarrow\mathbb{R}$ is monotone. This statement is false for maps $\mathbb{Q}\rightarrow\mathbb{Q}$. (That is becaus $\mathbb{Q}$ is ...
1answer
28 views

### Prove that If rational numbers $y$ and $z$ are $\epsilon$ close to $x$ then so is $w$ which lies between $y$ and $z$

By $\epsilon$ close I mean $|x-y| \leq \epsilon$ for some rational $\epsilon > 0$ I could prove it by representing $w$ as $w = \theta_1 y + (1-\theta_1)z$ where $0\leq\theta_1\leq1$, and then ...
1answer
29 views

1answer
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### Prove the sum of squares of 3 rationals cannot be 7

Prove there isn't $r_1, r_2,r_3 \in \mathbb{Q}$ so that ${r_1}^2 + {r_2}^2 + {r_3}^2=7 \tag1$ From (1) we get $a^2 + b^2 + c^2=7n^2 \tag2$ where $a,b,c,n \in \mathbb{N}$. I have tried playing ...
1answer
59 views

### How to generate primitive solutions to the equation $a^3 + b^3 = c^2$

The solution for this is that we are supposed to pick numbers x and y, then we can substitute them in the equation and obtain some z, which we then multiply the left side of the equation with to ...
0answers
46 views

### Non-constructive proofs for the rationality of a number

One of the key ideas in transcedental number theory is proving that a number is transcedental (i.e. not the root of any polynomial with integer coefficients) by showing a sequence of rational numbers ...
2answers
63 views

### Show that a subset $E$ $\subset \Bbb Q$ is not compact in $(\Bbb Q, d)$ and decide whether it is open or not

Assume $(\Bbb Q, d),$ $d(p, q):= |p -q|$ is a metric space and $E := \{p \in \Bbb Q : 2 < p^2 < 3\} = \{p \in \Bbb Q : \sqrt2 < p < \sqrt3\} \subset \Bbb Q.$ I have to show ...
1answer
41 views

### Show that $E \subset \Bbb Q$ is closed in $(\Bbb Q, d)$

Assume $(\Bbb Q, d),$ $d(p, q):= |p -q|$ is a metric space and $E := \{p \in \Bbb Q : 2 < p^2 < 3\} \subset \Bbb Q.$ I have to show that $E$ is closed. I see two ways of proving ...
1answer
54 views

### Assume $r,s \in\mathbb{Q}$. Prove $\frac{r}{s},r-s \in\mathbb{Q}$ [closed]

I have attempted this proof by contradiction. Beginning with assuming to the contrary that each a and b are irrational but was not sure if I did it correctly. Any help would be greatly appreciated. ...
6answers
2k views

### Visual representation of the fact that there are more irrational than rational numbers.

Would anybody know of a visual or even (preferably) geometric representation of this? To make it more specific: Text, symbols and written numbers are predominantly used as labels, and and less to ...
2answers
85 views

### (H.W question) In Mathemaical Analysis of Rudin example 1.1 Pg 2

The author went on and proved that 1) There exists no rational $p$ such that $p^2=2$ 2) He defined two sets $A$ and $B$ such that if $p\in A$ then $p^2 <2$ and if $p\in B$ then $p^2>2$ and ...
2answers
32 views

### Countable set of number rational, prove with $\mathbb{Z}$.

Good morning, I need to prove $\mathbb{Q}$ is a countable set, but I prove $\mathbb{Z}$ is a countable set, now, can I use this for proving $\mathbb{Q}$ is countable set? I was thinking about a ...
1answer
56 views

### What's the numerator and the denominator of a fraction called?

Just a quick question, is it right to call the numerator and the denominator of a fraction by "terms"? I don't think that "terms" is the right word here, but i don't know any alternatives. Can any ...
1answer
94 views

### For which $a,b\in \mathbb{N},$ is $\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$ is a rational number.

I found the following problem on a Olympiad question paper: For which $a,b\in \mathbb{N},$ is $$\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$$ a rational number. I am unable to solve it. Any help ...
0answers
40 views

### Is LCM of rationals used in higher math? [duplicate]

I read in this school book, an algorithm to find the LCM of rationals. It goes in the following manner. $[\frac{a}{b}, \frac{c}{d}]=\frac{[a,c]}{(b,d)}$. If you inspect as to why the formula is given ...
7answers
3k views

### $0.333333$ - a recurring or non-terminating decimal?

I have read like, 1.All terminating and recurring decimals are RATIONAL NUMBERS. 2.All non-terminating and non recurring decimals are IRRATIONAL NUMBERS. if the statements are right, then here ...
2answers
62 views

### How to prove the power set of the rationals is uncountable?

Recently a professor of mine remarked that the rational numbers make an "incomplete" field, because not every subsequence of rational numbers tends to another rational number - the easiest example ...
3answers
173 views

1answer
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### A sum of irrational numbers ending rational

Let $x$ be a positive irrational number I know that there exists $y$ such that: $$\begin{cases} y>0 \\ x+y\in \mathbb Q.\end{cases}$$ How would you construct explicitly such $y$ ? For instance ...
3answers
45 views

### is there any convergent sub-sequence of a sequence of all rational numbers?

Let $(a_n)$ be a sequence of rational numbers, where all rational numbers are terms. (i.e. enumeration of rational numbers) Then, is there any convergent sub-sequence of $(a_n)$?
1answer
51 views

### General Conic and its Rational Solutions

Suppose you have a rational conic $ax^2+bxy+cy^2+dx+ey+f=0$. There is a theorem that states if a conic has 1 rational solution it has infinitely many rational solutions. How can you prove this ...
2answers
42 views

### Irreducible fraction of a given rational

Given a rational $r \in \mathbb Q$, how to find the irreducible fraction $\frac a b = r$? Any direct formula based on the digits of $r$, instead of successive approximations by increasing ...
2answers
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0answers
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### for what x, is $\frac{1}{\pi} \cdot c\cos^{-1}(x) \in \mathbb{Q}$

While solving a question, I met the next problem, for what x, is: $$\frac{1}{\pi} \cdot \cos^{-1}(x) \in \mathbb{Q}$$ I found in this paper that for $0 \leq r \leq 1, r \in \mathbb{Q}$,  \...
2answers
509 views

### What exactly are those “two irrational numbers” $x$ and $y$ such that $x^y$ is rational? [duplicate]

It's possible to prove nonconstructively that there exists irrational numbers $x$ and $y$ such that $x^y$ is rational, but that proof only proves that such numbers exist and does not specify what they ...
5answers
268 views

### Complement of rationals has empty interior

This question refers to How to prove closure of $\mathbb{Q}$ is $\mathbb{R}$ I want to prove that the closure of $\mathbb{Q}$ is $\mathbb{R}$. I am trying to understand the accepted answer, but when ...