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.

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2
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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 ...
3
votes
0answers
43 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 ...
1
vote
2answers
62 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 ...
0
votes
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 ...
0
votes
1answer
53 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. ...
12
votes
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 ...
1
vote
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 ...
0
votes
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 ...
2
votes
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 ...
1
vote
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 ...
0
votes
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 ...
-2
votes
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 ...
1
vote
2answers
59 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 ...
7
votes
3answers
173 views

The equation $\{x^2\} + \{x\}=1$ has no solution over positive rationals

Prove there is no positive rational $x$ so that $$\{x^2\} + \{x\}=1 \tag1 $$ Let $x=\frac p q$ and $p=qc+r, p, q, c, r \in \mathbb{N}, 0 \le r \lt q$ From (1) $\{ 2c \frac r q + (\frac r q)^2\} +...
-1
votes
2answers
27 views

Proving by contradiction (6/9)

I have been given a statement that I need to prove using the contradiction method and I am just a little unsure of how to go about setting this up and executing. Here is the statement: If x is any ...
1
vote
2answers
43 views

How do you deduce the integer whose multiplicative inverse decimal has a digit sequence or repetend length of 3 digits?

A positive integer's, n, reciprocal, $\frac{1}{n}$, in which the decimal's repetend has a length of three digits which starts at the decimal mark. e.g. 0.037037... of the integer, 27 ,reciprocal $\...
0
votes
1answer
25 views

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 ...
4
votes
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 $(a_n)$?
0
votes
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 ...
1
vote
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 ...
1
vote
2answers
37 views

Show that there are at most two rational points on $(x - a)^2 + (y - b)^2 = r^2$ for $a, b$ irrational.

For any given irrational numbers $a, b$ and real number $r \gt 0$, show that there are at most two rational points (points whose coordinates are both rational numbers) on the circle $(x - a)^2 + (y - ...
0
votes
0answers
13 views

Should we expect any anomalies when dealing with rational differential equations?

Some time ago, I found myself reading a short article that proposed that the rational numbers where the 'appropiate' number system that we should use for most of mathematics (where we use the real ...
0
votes
3answers
33 views
2
votes
1answer
1k views

Proof by contradiction problem on rational numbers

Using proofs by contradiction, show that there is no smallest negative rational number and no largest positive rational number. Assume that there is a smallest negative rational number. Therefore, ...
5
votes
3answers
202 views

When can a set have an upper bound but no least upper bound?

So I'm taking real analysis and have noted that one of the benefits of the Dedekind cut is that 'if one of the sets made has an upper bound it also has a least upper bound'. I don't understand how a ...
1
vote
2answers
85 views

Minimizing rational solutions of $ x^3+y^3=9$

I´m trying to solve this problem: An old alchemist had two sphercial flasks, one with a circunference of 12 inches and the other with a circunference of 24 inches. He desired to transfer their ...
-2
votes
1answer
25 views

If $\cos\pi\theta$ is algebraic and $\theta$ is irrational, what is the set of possible $\theta$?

I know that $a= \cos \pi \theta$ is an algebraic number ($\theta$ is rational). I want to prove that if $\cos\pi\theta$ is rational, then the possible only possible values of $\theta$ are $0,±1/2,±1$ ...
6
votes
4answers
251 views

'Almost rational' integrals with no known closed form?

I recently stumbled upon an 'almost rational' integral, namely: $$\int_0^{\pi/2} x \frac{\sqrt{\sin x}-\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}} dx=0.231231222\dots \approx 0.231231231\dots= \frac{...
4
votes
1answer
42 views

Why do metric spaces that produce the same topology have different number theoretical difficulties?

Consider finding a a point with rational distance to the corners of unit square. Under the Euclidean metric this is very hard. (unsolved) Under the "city block" or taxicab metric this is very easy ...
2
votes
2answers
55 views

Rational Distance Problem triple — irrational point

Many points with rational coordinates are known with rational distances to three vertices of a unit square. For example, the following points are rational distances from $a=(0,0)$, $b=(1,0)$, and $c=(...
0
votes
0answers
21 views

Proof that the union of rational and irrational numbers sets is a set of real numbers [duplicate]

I see it all the time but is there a nice way to show that this is true? Or is this just a definition? I know that $\mathbb{Q} \subset \mathbb{R}$ and $\mathbb{I} \subset \mathbb{R}$, but how do we ...
1
vote
4answers
143 views

Approximation of $\sqrt{2}$

I got the following problem in a chapter of approximations: If $\frac{m}{n}$ is an approximation to $\sqrt{2}$ then prove that $\frac{m}{2n}+\frac{n}{m}$ is a better approximation to $\sqrt{2}.$(...
2
votes
1answer
48 views

If $a$ and $b$ are positive rational numbers with $a < b$, show that $\frac{1}{a} >\frac {1}{b}$

Since both $a,b\in \mathbb{Q}^+$ and $a<b$, then of course $\frac{1}{a}$ is greater than $\frac{1}{b}$. However, I don't know how to prove that. I suppose I could do the greater than property in an ...
3
votes
3answers
421 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
4answers
209 views

How do I write down a curve with exactly one rational point

Let $g\geq 1$. I would like to write down (for all $g$) a smooth projective geometrically connected curve $X$ over $\mathbf{Q}$ of genus $g$ with precisely one rational point. Is this possible? For ...
2
votes
1answer
36 views

Prove that the group of the rational points on the conic $u^2-Av^2=1$ is not finitely generated.

This is an exercise from Rational Points on Elliptic Curves by Silverman. Let $H$ be the conic $u^2-Av^2=1$ where $\sqrt{A}\notin \mathbb{Q}$. If $(u_1,v_1), (u_2,v_2)$ are two points in $H(\...
2
votes
1answer
71 views

A proof that $\frac{(2\phi)^n-(-1)^n}{\phi^{2n}-(-1)^n}\cdot\left(2^n-\phi^n\right)\cdot\sqrt5\in\mathbb Q$ for all $n\in\mathbb Z$

During computation of some series (with help of a CAS), at an intermediate step I encountered an expression, that after dropping non-essential parts looks like this:$$\mathcal E_n=\frac{(2\phi)^n-(-1)^...
1
vote
1answer
13 views

Let $H = \{2^m : m \in \mathbb{Z}\}$ & define a relation $R$ on the set $\mathbb{Q^{+}}$ of positive rationals by $a\mathbin{R}b$ iff $a/b \in H$.

Let $H = \{2^m : m \in \mathbb{Z}\}$ and define a relation $R$ on the set $\mathbb{Q^{+}}$ of positive rational numbers by $a\mathbin{R}b$ if and only if $a/b \in H$. Prove that $R$ is an equivalence ...
0
votes
2answers
34 views

Convert this into fractional number step by step?

3.41287548754875... Convert the above number to a rational number? I was reviewing some pre calculus on my own but couldn't figure this out.
1
vote
2answers
23 views

Let $S = { r \in \mathbb{Q} : r \lt 2}$. Prove that $S$ does not have a largest element.

Let $S$ = $[{ r \in \mathbb{Q} : r \lt 2}]$. Prove that $S$ does not have a largest element. My method: Assume to the contrary that $S$ does have a largest element, where $S$ = $[{r \in \mathbb{Q} :...
0
votes
0answers
47 views

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} $, $$ \...
5
votes
2answers
504 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 ...
4
votes
5answers
267 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 ...
5
votes
2answers
51 views

Rational Question for $a + b$ and Irrationality of $a^2 + b^2$

I have looked into the question and need help. Find some $a,b$ ${\in}$ $\mathbb{R}$ such that $a + b$ ${\in}$ $\mathbb{Q}$, $a^2 + b^2 \not\in \mathbb{Q}$, and $\frac{a}{2} < b < a$. Or prove ...
21
votes
2answers
373 views

Is $\frac{1}{11}+\frac{1}{111}+\frac{1}{1111}+\cdots$ an irrational number?

Obviously: $$\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\cdots=0.1111\dots=\frac{1}{9}$$ is a rational number. Now, if we make terms with demoninators in the form: $$q_n=\sum_{k=0}^{n} 10^k$$ Then ...
0
votes
2answers
26 views

Validity of certain arguments about the countability of infinite sets

I am trying to get an understanding, in layman's terms / on an intuitive level, why some arguments about the countability of infinite sets are valid, and some arguments which seem almost identical on ...
3
votes
7answers
145 views

Rational Expression equivalent form

EDIT: I know how to find the answer, but does anyone know why plugging in numbers for x does not work? The Question: If the rational expression $\frac {3x^2}{3x-1}$ is rewritten in the equivalent ...
1
vote
2answers
69 views

If a series grows more slowly than any geometric series, can it ever converge to a rational?

I was reading a proof of $e$'s irrationality which, in some sense, uses the fact that the series $\sum \frac{1}{n!} = e$ grows slowly. This got me thinking: can we generalize this and say "oh, $\sum \...