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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0
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1answer
21 views

Pugh's exercise on Dedekind cuts addition

I am trying to solve the following exercise: Let $x=A|B$ and $x'=A'|B'$ be cuts in $\mathbb{Q}$. Show that although $B+B'$ is disjoint from $A+A'$, it may happen in degenerate cases that $\mathbb{Q}$ ...
4
votes
6answers
264 views

How to Find a rational number between two irratonal number? [on hold]

Find the rational number between $\sqrt 2$ and $\sqrt3$. I try to solve by using some methods in my book but can not understand steeps.
-3
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0answers
22 views

Rational numbers problem [on hold]

I have a problem with rational numbers, how do i find the numbers behind the following equation: 0,a(b) = a/b, using the following rationale: a/b*10=a,(...
5
votes
4answers
151 views

Are there any natural proofs of irrationality using the decimal characterization?

Mathematicians typically define rational number to mean quotient of two integers. It is not hard to show that a number is rational by that definition if and only if its decimal expansion terminates ...
2
votes
1answer
20 views

If $\frac{a}{b}\in \left[\frac{p-1}{q},\frac{p}{q}\right]$, is then $b\ge q$?

Let $x=\frac{a}{b}$ be a rational number (in its lowest terms) in $[0,1]$. Let $x\in \left[\frac{p-1}{q},\frac{p}{q}\right]$ for some positive integers $p,q$ with $p\le q$. Is it true that $b\ge ...
0
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1answer
37 views

Number of Rational Solutions $\mathbf{x}\in[0,1)^n$ to the Matrix Condition $\mathbf{A}\,\mathbf{x}\in\mathbb{Z}^n$

Let $n$ be a positive integer and $\mathbf{A}$ an $n$-by-$n$ matrix with integer entries. Suppose that $k:=\big|\det(\mathbf{A})\big|$ is nonzero. How many $n$-by-$1$ column vectors $\mathbf{x}\in\...
1
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0answers
23 views

Algorithm that calculates decimal places of number

My background. I am a school student. Recently, we learned about rational numbers and irrational numbers. For example, we were told that rational numbers can always be written as a repeating decimal, ...
0
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0answers
55 views

Is $\pi e$ irrational? [duplicate]

During our ongoing research, we need to prove that $\pi e<\lceil \pi e\rceil$. Is $\pi e$ irrational? How to prove it? Thanks- mike
3
votes
3answers
61 views

Prove that there are exactly $k$ pairs $(x,y)$ of rational numbers with $0\leq x,y<1$ for which both $ax+by,cx+dy$ are integers.

Let $a,b,c,d$ are integers such that $(a,b)=(c,d)=1$ and $ad-bc=k>0$. Prove that there are exactly $k$ pairs $(x,y)$ of rational numbers with $0\leq x,y<1$ for which both $ax+by,cx+dy$ are ...
2
votes
0answers
71 views

Is $x^x$ rational for $x=\sqrt{2}^\sqrt{2}$

This might be naive. Is $x^x$ a rational number for $x=\sqrt{2}^\sqrt{2}$ ? I remember reading somewhere a long time ago that such $x^x$ is a rational number, as an example of issues with non-...
1
vote
3answers
60 views

Rational root coefficient

I saw this question in my exam recently, If a, b, c are distinct rational roots of $x^3+ax^2+bx+c=0$, find the values of a, b, c. Can someone give me a hint or answer? I tried factoring it and ...
0
votes
1answer
23 views

nearest approximation for a/b with denominator less than n

Given a rational number a/b what is the closest ration c/d such that d I would like a formula for c and d in terms of a,b and n if possible but if no mathematical solution exists, an algorithm for ...
5
votes
2answers
74 views

Rational numbers as vectors in infinite dimensional space with the basis $( \log 2,\log 3, \log 5, \log 7, \dots, \log p, \dots) $

Since every natural number can be represented as $a=2^{n_1}3^{n_2}5^{n_3}7^{n_4}\cdots p_k^{n_k}\cdots$ it makes sense to represent natural numbers by vectors, using the properties of logarithms: $$\...
2
votes
2answers
66 views

Pairs of irreducible fractions that add up to a given irreducible fraction

Given the irreducible fraction $\frac a b$, with $a, b \in \mathbb N$, what is the expression that enumerates all the irreducible fractions of integers that add up to $\frac a b$? Namely, an ...
0
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2answers
43 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 ...
0
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1answer
30 views

Prove that for rational $x$,$y$ and $\epsilon$ if $|x-y| \leq \epsilon$ , $ \forall \epsilon > 0$ then $x=y$

I know that this is a repeated question, but I wanted to show my attempt. Suppose $x \neq y$ and (wlog) $x > y$, then $x$ can be written as $x = y + \delta$, for some ($\delta > 0$ and $...
2
votes
2answers
29 views

For $\forall m \in \mathbb{N}, m \ge 3$ there are m elements of S in arithmetic progression.

Let $S=\{[n\pi], n \in \mathbb{N}\}$. Prove $\forall m \in \mathbb{N}, m \ge 3$ there are $m$ elements of $S$ in arithmetic progression. I don't know how to prove it, but I have the feeling the ...
1
vote
1answer
45 views

Proof that Epicycloids are Algebraic Curves?

Epicycloids are most commonly described by the parametric equations, $x(t) = (R + a)\cos(t) – a \cos \left(\frac{R + a}{a} t \right),$ $y(t) = (R + a)\sin(t) – a \sin \left(\frac{R + a}{a} t \right)...
6
votes
1answer
68 views

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 ...
3
votes
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 ...
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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 ...
0
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1answer
43 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
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. ...
0
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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
58 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 ...
12
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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 ...
0
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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 ...
1
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2answers
65 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\} +...
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2answers
28 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
77 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
26 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
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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
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1answer
54 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
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2answers
45 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
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2answers
39 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
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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
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3answers
33 views

How does one find a rational number in fraction form, knowing the repeating decimal?

For example, I have 0.786786786... How do I find the fraction equivalent?
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2answers
81 views

Is the value of $\log_27$ a rational number?

Is $\log_27$ a rational number?
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2answers
86 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 ...
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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 $0,±1/2,±1$ ...
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 ...
1
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1answer
96 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 ...
2
votes
2answers
58 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
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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 ...
2
votes
1answer
49 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 ...
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
37 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(\...
6
votes
4answers
252 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{...
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 ...