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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1answer
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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
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6answers
256 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 ...
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9answers
11k views

Proof that every repeating decimal is rational

Wikipedia claims that every repeating decimal represents a rational number. According to the following definition, how can we prove that fact? Definition: A number is rational if it can be written ...
3
votes
2answers
706 views

Why must the decimal representation of a rational number in any base always either terminate or repeat?

Wikipedia makes the following statement about rational numbers. The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same ...
2
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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 ...
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 ...
3
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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 ...
0
votes
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\...
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0answers
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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, ...
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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
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: $$\...
7
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4answers
7k views

How can I prove that all rational numbers are either terminally real or repeating real numbers?

I am trying to figure out how to prove that all rational numbers are either terminally real or repeating real numbers, but I am having a great difficulty in doing so. Any help will be greatly ...
2
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3answers
2k views

Show that the ring of all rational numbers, which when written in simplest form has an odd denominator, is a principal ideal domain.

Show that the ring of all rational numbers $m/n$ with $n$ an odd integer is a principal ideal domain. We haven't really discussed principal ideal domains. I've heard that this is easy, but I just ...
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 ...
3
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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 ...
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
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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 ...
2
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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, ...
1
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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
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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 ...
2
votes
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
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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
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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. ...
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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 ...
1
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2answers
86 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
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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 ...
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 ...
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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 ...
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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 ...
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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 ...
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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
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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
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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 ...
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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 ...
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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 ...
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3answers
33 views
5
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3answers
212 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 ...