For questions on rational numbers, numbers that can be expressed as the quotient or fraction $\frac pq$ of two integers.

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4answers
783 views

How to prove this is a rational number

I'm not sure how to prove this is a rational number $\frac{q}{m}$, can some one show me? $$\frac{q}{m}=\frac{(\frac{1+\sqrt5}{2})^n - (\frac{1-\sqrt5}{2})^n}{\sqrt5}$$
0
votes
1answer
63 views

representation of rational field

I want to know how is represented general form of rational field, for example definition of ${\mathbb Q}(\sqrt{2})$ is represented as $p+q \sqrt{2}$, where $p$ and $q$ are rational numbers, for ...
-2
votes
3answers
47 views

simplify this expression 2-√(2+√3)/√(2+√3)

2-√(2+√3)/√(2+√3) I need to simplify this. Can I multiplay with 2-√3 the numerator and dominator? I need your help.
2
votes
1answer
188 views

How to find simple rational numbers close to the decimal representation

It is a simple practical question. I am reverse-engineering poorly documented calculations made by someone else. I frequently find a mysterious number 0.0329. I'm quite certain it is some kind of ...
1
vote
2answers
71 views

equivalence classes and cardinality

I need to prove that every equivalence class created by the equivalnce relation $\sim$ on $\mathbb{R}$, that is defined by: $a\sim b \Leftrightarrow (a-b) \in \mathbb{Q}$, is $\aleph_0$. Furthermore, ...
3
votes
1answer
86 views

Prove that the Rationals are Countably Infinite [duplicate]

I was assigned to Prove that $\mathbb{Q}$ is countably infinite I did the following: We define $\mathbb{Q}= \lbrace \frac{a}{b} \mid a, b \in \mathbb{Z}_{>0} \rbrace$. Also define $\mathbb{Q} ...
4
votes
2answers
59 views

How to contruct such a sequence of rational?

How to order all rational numbers from $(0,1)$ in a sequence $(x_n)_{n=1}^\infty$ in such a way that $$|x_n-x_k| \geq \frac{1}{(n+1)^2}$$ for $k<n$ ?
12
votes
4answers
4k views

Is there a rational number between any two irrationals?

Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 < i_2$. Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$? How would you construct such ...
0
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2answers
64 views

Build the function by its values. Only combination of +, -, *, abs() allowed for this function.

I've decided to open a new, more common question about the simplest function f(1)=-1; f(2)=0; f(3)=1; f(4)=0.. So, here is the question. Let's say we have some function $y=f(x)$ we'd like to find by ...
0
votes
3answers
57 views

the simplest function f(1)=-1; f(2)=0; f(3)=1; f(4)=0.

I'm looking for a function that gives f(1)=-1; f(2)=0; f(3)=1; f(4)=0. The other values are undefined and I don't pay any attention on them. The prefered ...
0
votes
1answer
102 views

Epsilon Neighborhoods of the Rationals

What is meant when someone discusses an epsilon neighborhood of $\mathbb{Q}$?. Naturally the rationals are dense in $[0,1]$, so what is the epsilon neighborhood? More importantly, what does the ...
2
votes
2answers
91 views

Decimal to fraction conversion

We write software for managing recipes and are working on moving from an approximation based decimal to fraction conversion, for example, anything between 0.03125 and 0.09375 becomes 1/16 to a math ...
0
votes
1answer
67 views

How can we prove that every rational number has a terminating or periodic decimal form? [duplicate]

The title says it all. I'm aware of the proof of the converse of my statement, but how do I go on about proving this. Any help would be appreciated.
4
votes
3answers
253 views

Prove that $\sqrt[3]{5-\sqrt{2}}$ is not a rational number

My attempt: Consider the polynomial $ (x^3-5)^2 - 2 = x^6 -10x^3 + 23 = 0 $. By the rational zeros theorem, we can conclude that $ \pm 1$ and $ \pm 23 $ are the only possible rational solutions*. ...
7
votes
4answers
490 views

How to obtain all the rational numbers without repetitions?

Some days ago I've seen Cantor's diagonal argument, and it presented a table similar to the following one: $$\begin{matrix} ...
0
votes
3answers
129 views

When proving that there is not rational number $m/n$ equal to $\sqrt{2}$, why does $m$ and $n$ must be not both even?

I've read this on Rudin's Principles of Mathematical Analysis: 1.1 Example We now show that the equation $$p^2=2$$ is not satisfied by any rational $p$. If there were such a $p$, we ...
1
vote
1answer
226 views

Modulo over rational numbers?

Consider two irreducible fractions: $r_1 = \frac{p_1}{q_1}$ $r_2 = \frac{p_2}{q_2}$ with $r_1 \ge 0$ and $r_2 \ge 0$. How the modulo $\%$ is defined over rational numbers (I think that is $r_3$ ...
1
vote
3answers
50 views

Irreducibility of gcd/lcm or lcm/gcd

Consider two irreducible fractions: $r_{1} = \frac{p_{1}}{q_{1}}$ $r_{2} = \frac{p_{2}}{q_{2}}$ Are these two fractions: $r_{3} = \frac{\text{gcd}\left(p_{1}, p_{2}\right)}{\text{lcm}\left(q_{1}, ...
1
vote
1answer
120 views

Evaluate the expression

Evaluate the expression $$\frac{1^2}{1^2-10+50}+\frac{2^2}{2^2-20+50}+\cdots+\frac{80^2}{80^2-80+50}$$ What should I do after that ? $\sum\frac{n^2}{n^2-10n+50}$ I'm not seeing anything to find ...
2
votes
3answers
711 views

What is the ratio of rational to irrational real numbers?

There exists an infinite amount of rational and irrational numbers. But is there more irrational numbers than rational? And if so can a ratio of one to the other be calculated?
0
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3answers
78 views

Finding non-zero rational numbers to fit $a^2+b^2=C$

This is a question on my homework. Specifically, find non-zero rationals $a,b$ such that $a^2+b^2=9$. I think that this is related to work that Diophantus did, but I'm not really sure and I just don't ...
1
vote
1answer
85 views

For what values of $b\in \mathbb R$ is $\pi-b\in \mathbb Q$ true?

Just a simple short question. I'm looking for values $b$ such that $\pi-b$ is a rational number. Obviously $\pi$ is such a number, but are there more? Edit: $b$ is in $\mathbb R$
0
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1answer
109 views

Need help with proving that group is not finitely-generated [duplicate]

I need to prove that $(\mathbb{Q}^*, \times)$ (i.e rationals, zero excluded, under multiplication) is not finitely generated. So, suppose that G is finitely-generated. That means there exist a ...
0
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1answer
76 views

Dimension of the rationals over the integers

What is the dimension of the $Q$, when they are seen as a vector space over the integers $Z$ (with the usual definitions of addition and multiplication)? Initially I thought that the dimension ought ...
1
vote
1answer
103 views

An example of a group operation on the rationals, which is not isomorphic to the additive group

I'm looking for an example of a group operation on the rationals, which is not isomorphic to the rational additive group. Can you find such an example?
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votes
2answers
282 views

Do irrational numbers really exist?

Isn't it possible that an irrational number is in reality the quotient of two infinitely long integers that even if there were repeating sections in it, it would take infinite digits to find the first ...
18
votes
2answers
557 views

Order preserving bijection from $\mathbb Q\times \mathbb Q$ to $\mathbb Q$

Do you have a simple example of order preserving bijection from $\mathbb Q\times \mathbb Q$ to $\mathbb Q$? On $\mathbb Q$, I use the usual order and on $\mathbb Q\times\mathbb Q$, I define the ...
2
votes
1answer
321 views

Is every injective rational function $f:\mathbb Q\to\mathbb Q$ a polynomial?

I thought this might be quite easy to show, and then realized that the tools I know from real analysis aren't going to help here. Suppose we have a rational function: $$ f(X)=\frac{P(X)}{Q(X)} $$ ...
13
votes
6answers
825 views

Is there a function that gives the same result for a number and its reciprocal?

Is there a (non-piecewise, non-trivial) function where $f(x) = f(\frac{1}{x})$? Why? It would be nice to compare ratios without worrying about the ordering of numerator and denominator. For example, ...
5
votes
5answers
370 views

Odd divided by even is a fraction

How can we prove that an odd number divided by an even number is a fraction? I started with odd $=2m+1$ and even $=2n$ and get left with with $(m+2)/n$.
1
vote
1answer
52 views

Complex Numbers, Complicated Powers

We know there are two non-real imaginary numbers like $a$, $b$ such that the power $a^{b}$ is a real number. For example we have $i^{i}=\frac{1}{\sqrt{e^{\pi}}}$. Question: Are there two non-real ...
6
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1answer
611 views
0
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2answers
89 views

Rational numbers in base 6

Express the fractions , for several small values of , in base 6. Determine which rational numbers have terminating expressions in base 6. I am unsure how to reduce fractions in base 6. For example, ...
0
votes
4answers
249 views

How can I explain $0.999\ldots=1$? [duplicate]

Possible Duplicate: Does .99999… = 1? I have to explain $0.999\ldots=1$ to people who don't know limit. How can I explain $0.999\ldots=1$? The common procedure is as follows ...
3
votes
2answers
73 views

Existence of five real numbers satisfying a given condition.

Let $a_1,\dots,a_5$ be five distinct non-zero real numbers. Suppose that for $i\neq j$ either $a_i+a_j$ or $a_ia_j$ or both are rational numbers, does it implies that $a_i^2$ are rational numbers for ...
0
votes
1answer
119 views

subring of rational numbers and its ideal

Let $p$ be a prime number. For any $p$ the subring $\mathbb{Q}_p$ of of the field of rational numbers is defined: $\mathbb{Q}_p=\{\frac{a}{b}|a,b\mbox{ are integers, $p$ does not divide $b$}\}$ Let ...
4
votes
2answers
142 views

Rational solutions of $x^3+y^3=2$

I came along the problem of finding three perfect cubes that are consecutive numbers of an arithmetic progression, i.e: $a^3-b^3=b^3-c^3$, where $a>b>c$ (to avoid trivial solutions). Clearly it ...
3
votes
0answers
137 views

Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number.

Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number. I thought this question might have been asked by someone else, but I couldn't find any. Currently I'm studying ...
5
votes
1answer
145 views

If $q>1$ is not an integer, can $q^n$ be made arbitrarily close to integers?

This question arose when I heard about Mill's constant: the number $A$ such that $\lfloor A^{3^n} \rfloor$ is prime for all $n$. It made me wonder whether $A^{3^n}$ could be made arbitrarily close to ...
3
votes
1answer
46 views

Is the fraction of the irrational exponentiations of two coprime integers by a rational an irrational?

Consider two strictly positive integer coprimes $n, m\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. Consider furthermore that the three number statifies the following condition: ...
2
votes
2answers
94 views

Can the exponentiation of an integer by a rational be a non-integer rational?

Consider a strictly positive integer $n\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. My question is the following: what is the nature of $n^r$? My first guess is that $n^r$ is an ...
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vote
0answers
23 views

Rational exponentiation?

Consider the following operation: $\left(\frac{a}{b}\right)^\frac{n}{m}$ where $a, n\in\mathbb{Z}$ and $b, m\in\mathbb{N^*}$. My question is: when the result is a rational number, how (formula or ...
2
votes
7answers
449 views

Why is a repeating decimal a rational number?

$$\frac{1}{3}=.33\bar{3}$$ is a rational number, but the $3$ keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ...
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vote
3answers
108 views

Wrong Answer - Rewrite Rational Number as a Fraction.

This number 2.962962 can be rational $$x=2.962962$$ $$10x=29.62962$$ $$100x=296.2962$$ $$1000x=2962.962$$ $$1000x-10x=\frac{990x}{990}=\frac{2933}{990}$$ why is this wrong? That way of getting the ...
2
votes
4answers
354 views

Show that the integers have infinite index in the additive group of rational numbers.

Show that the integers have infinite index in the additive group of rational numbers. Anybody in a good enough mood to tell me how this is done?
1
vote
2answers
176 views

Does the set of rational numbers between 0 and 2 have the least upper bound property?

Let $A = \{ a \in Q : 0 < a < 2\}$ Does A have the least upper bound property? Definition: $A$ has the least upper bound property if $\forall$ nonempty $B \subseteq A$, if $B$ has an upper ...
21
votes
2answers
458 views

Is $\{\tan(x) : x\in \mathbb{Q}\}$ a group under addition?

A student asked me the following today : Is $S:= \{\tan(x) : x\in \mathbb{Q}\}$ a group under addition? I am quite perplexed by it. Clearly, the only non-trivial part is to check For any $x, ...
6
votes
4answers
310 views

Is there a bijection $f: \mathbb{Q} \to \mathbb{Q}_{>0}$?

For $\mathbb{R}$, we have the exponential function. Is there also a bijection $f: \mathbb{Q} \to \mathbb{Q}_{>0}$ or to $\mathbb{Q}_{\geq 0}$?
5
votes
1answer
152 views

Is $\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$ a rational number for $m,n\ge 2\in\mathbb N$?

Question : Is $$\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$$ a rational number for $m,n\ge 2\in\mathbb N$ where $\zeta (s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$? Motivation : We know that $$\zeta ...
1
vote
1answer
113 views

Does this equation have a rational point? (Elliptic curve?)

Can someone check pls if, $$852 + 3017 x - 1104 x^2 + 2009 x^3 - 3362 x^4=y^2$$ has a rational point? (This arose in an equal sums of like powers problem.) P.S. I've checked $x=p/q$ for ...