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

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91 views

Given a basis for $\mathbb{R}$, show that it constructs the standard topology on $\mathbb{R}$

Let $q_1, q_2, ...,$ be the rational numbers enumerated. Consider the countable collection $$\mathcal{B} = \{ B_{\frac{1}{n}}(q_i) \ | \ i,n \in \mathbb{N} \}$$ of open balls centered at rational ...
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1answer
69 views

Difference between density and measure

In terms of definition, I know the difference between the two. However, the set of rationals $\mathbb{Q}$ has measure zero but is dense in $\mathbb{R}$. Whenever I envision this, I see a set of ...
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2answers
43 views

Show that if S=a+b√2 : a,b are rational numbers and T=r+s√3 :r,s are rational numbers, then$S \cap T$ = rational

Someone please correct a formatting error in the problem [still a newbie] ; "S&T" (And = upside down U) Here's a bonus question that was on a test we received that I couldn't figure out. I'd ...
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2answers
178 views

Proof Involving Rational Numbers [duplicate]

I asked this same question last night and got some answers but still can't make sense of this, normally I'd move on but since I know how to do everything else for the test I'm going to try to get this ...
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1answer
675 views

Rational Number Proof [duplicate]

Stuck on a tutorial question trying to study for a test. The question is : Consider the following statement: "Between any two different rational numbers, there are at least two different rational ...
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1answer
61 views

What Will Happen Without Decimal Expansion?

After a discussion on the complexity of decimal expansion (such as $0.\bar{9}=1$), some of my students (middle school) decided to throw away the decimal expansion of some numbers! Namely, the numbers ...
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2answers
69 views

Is it possible to do elliptic curve cryptography over $\mathbb{Q}$ instead of a finite field?

Whenever I read about elliptic curve cryptography (ECC), the writer always works over a finite field. But as I understand it there is no group-theoretic reason not to use $\mathbb{Q}$ as the ...
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3answers
1k views

Is a cube root of a prime number rational?

The question is: if $P$ is prime is $P^{1/3}$ rational? I have been able to prove that if $P$ is prime then the square root of $P$ isn't rational (by contradiction) how would I go about the cube ...
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0answers
93 views

I am trying to prove this problem by induction, how can can i prove the following?

I am given $$ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}} $$ where, $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - \sqrt{5}}{2}$ The textbook states that it's equal to the n-th Fibonacci ...
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1answer
98 views

Can the rational numbers be specified as an ordered field with <order property>?

In other words, (the opposite of my question is) does there exist an ordered field which is isomorphic as (as an ordered SET) to $\mathbb{Q}$? If not, does there exist an order property which ...
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1answer
64 views

Can integers be defined in the first-order theory of the rationals?

Can integers be defined in the first-order theory of the rationals with addition, multiplication, and order?
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3answers
50 views

Is there a polynomial $f\in \mathbb Q[x]$ such that $f(x)^2=g(x)^2(x^2+1)$

I was asked the following question: $g\in \mathbb Q[x]$ is a polynomial (not the zero polynomial). Find $f \in \mathbb Q[x]$ such that $f(x)^2=g(x)^2(x^2+1)$ or show that such an $f$ does not exist. ...
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1answer
57 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 ...
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4answers
711 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? $$\frac{q}{m}=\frac{(\frac{1+\sqrt5}{2})^n - (\frac{1-\sqrt5}{2})^n}{\sqrt5}$$
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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.
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3answers
577 views

What is a rational number? What is the quotient of two integers?

How can we form an idea of the result of the operation of dividing a by b with a and b integers and b not equal to zero?
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2answers
69 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, ...
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1answer
78 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} ...
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2answers
61 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 ...
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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 ...
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2answers
58 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 $k<n$ ?
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1answer
97 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 ...
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1answer
64 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.
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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*. ...
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4answers
484 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} ...
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1answer
187 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 $r_3$ ...
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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}, ...
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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 ...
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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 ...
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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 $\mathbb R$
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1answer
95 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 ...
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1answer
71 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 ...
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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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261 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 ...
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2answers
70 views

How to represent the square root of 90 as a fraction?

How to represent the square root of 90 as a fraction? I can usually do this with a calculator but it's not wanting to play nice. I need it to find the distance between two points but, ...
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1answer
302 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: $$ f(X)=\frac{P(X)}{Q(X)} $$ ...
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308 views

If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$?

Question : For every even $k\ge 4$, is the following $(\star)$ true? $$\begin{align}\text{If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb ...
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6answers
822 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, ...
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1answer
51 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 ...
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2answers
87 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 ...
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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, ...
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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 ...
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1answer
111 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 ...
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0answers
131 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 ...
3
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1answer
43 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: ...
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2answers
93 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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0answers
22 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 ...
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3answers
123 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 ...
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7answers
397 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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2answers
514 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 ...