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

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5answers
180 views

Why can't the reals be constructed from the infinitesimal?

If the infinitesimal gives an unlimited precision as 1/∞ --> 0 Which can be thought of as the decimal 0.000000.....00000... then Why can't the reals, which demands, simply, unlimited precision (this ...
1
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1answer
101 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, ...
4
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1answer
86 views

Do rational and irrational numbers flip-flop?

I have found out that between every 2 rational numbers there is an irrational number, and between every 2 irrational numbers, there is a rational number. Does this mean that the rational and ...
2
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1answer
61 views

Limit of function defined on the rational numbers

Let $f$ the function defined on $\mathbb Q$ by : $ f(n/m) = n $. I would like to know whether it is true that : $ \forall q\in \mathbb Q - \{ 0 \} \quad \forall R>0 \quad \exists \delta >0\quad ...
2
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1answer
40 views

Convergence almost everywhere

Let consider rational numbers $\{r_n\}_{n=1}^{\infty}$ on [0, 1]. How to prove, that such sum $$\sum_{n=1}^{\infty}\frac{1}{n^2|x-r_n|^{0.5}}$$ converges almost everywhere on [0, 1]. There are my ...
2
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1answer
42 views

Rational vs Irrational distribution

Imagine I draw a number line, and I took two points. What's the distribution of rational and irrational numbers between them? If I put it in a diagram where I color rational with a color and ...
0
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0answers
79 views

irrational, proving or finding counterexapmle [duplicate]

Prove or find a counterexample: For all real numbers x and y it holds that x + y is irrational if, and only if, both x and y are irrational. Can anyone give me a hint just about how to start cause i ...
20
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4answers
2k views

Can every irrational number be written in terms of finitely many rational numbers?

Consider the irrational number $\sqrt{2}$. It can be written in terms, i.e., in a closed form expression, of two rational numbers as $2^{\frac{1}{2}}$. Does it hold in general that every irrational ...
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0answers
27 views

Closure of divisibility in denumerator, under sum of fractions

I have to prove that for a fixed positive integer n, the subset A of Q consisting of rationals with denumerator that divide n under addition, forms a group under addition. I just did that it's ...
2
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2answers
174 views

Let a, b, c, d be rational numbers… [closed]

Let $a, b, c, d$ be rational numbers, where $\sqrt{b}$ and $\sqrt{d}$ exist and are irrational. If $a + \sqrt{b} = c + \sqrt{d}$, prove that $a=c$ and $b=d$.
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1answer
73 views

Confusing rational numbers

Question: If $$x = \frac{4\sqrt{2}}{\sqrt{2}+1}$$ Then find value of, $$\frac{1}{\sqrt{2}}*(\frac{x+2}{x-2}+\frac{x+2\sqrt{2}}{x - 2\sqrt{2}})$$ My approach: I rationalized the value of $x$ to ...
0
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0answers
43 views

The solutions to $x^2+y^2=5$ in $\mathbb{Q}$. [duplicate]

Consider the following equation: $$x^2+y^2=5.\tag{1}$$ What are the solutions to this equation if $x,y\in\mathbb{Q}$, where $\mathbb{Q}$ is the set of all rational numbers? My attempt: Because ...
2
votes
1answer
52 views

Logic verification: $x^3$ is irrational, then $x$ is also irrational

Prove, by contraposition, if $x^3$ is irrational, then $x$ is also irrational. Just a verification do I need to show that given $x$ is rational $x^3$ is also rational? Suppose $x \in \mathbb{Q}$ ...
1
vote
4answers
108 views

The solutions to $x^2+5=y^2$.

Consider the equation $$x^2+5=y^2.\tag{1}$$ If $x,y\in\mathbb{Z}$, what are solutions to (1)? If $x,y\in\mathbb{Q}$, what are solutions to (1)? Note: $\mathbb{Z}$ is the set of all integers and ...
2
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1answer
77 views

Distance between two points in the plane

my teacher asked in the class today the following question: There exists an infinite set M of points in the plane with the property that any three points are non-collinear and such that the distance ...
1
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4answers
70 views

Cancelling out square roots gives 2?

Question: If $$N = \frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}$$Find N (This is a subset of a larger question) My approach: After rationalizing the denominator, by ...
5
votes
4answers
294 views

Rational numbers - rationalization

Question: $$\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$$ equals: My approach: I tried to rationalize the denominator by multiplying it by ...
0
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2answers
51 views

Infinite primes and notation

While reading a book about algebraic number theory, the symbol for a rational prime $p$ $$p^\infty$$ often occurs and I was wondering, what the exact definition of this is. Also, what is the ...
2
votes
0answers
133 views

Irrational numbers to the power of other irrational numbers: A beautiful proof question

The following theorem has a very beautiful proof. Theorem: There exist two irrational numbers $x$ and $y$ such that $x^y$ is rational. Proof: If $\sqrt{2}^{\sqrt{2}}$ is rational then we ...
0
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1answer
93 views

Multiplication of positive fractional numbers.

I am reading this answer. We know that multiplication of two positive rational number $\dfrac{a}{b}$ and $\dfrac{r}{s}$ respectivly is defined as follows: $\dfrac{a}{b}\times\dfrac{r}{s} = ...
2
votes
3answers
231 views

Equality of positive rational numbers.

I am reading the second article Rational Numbers of the book "A Treatise on Advanced Calculus" by Philip Franklin. I have mainly 3 questions regarding this article. I am writing all these $3$ ...
8
votes
4answers
382 views

Equation with an infinite number of solutions

I have the following equation: $x^3+y^3=6xy$. I have two questions: 1. Does it have an infinite number of rational solutions? 2. Which are the solutions over the integers?($ x=3 $ and $ y=3 $ is one) ...
7
votes
3answers
463 views

Sequences of Rationals and Irrationals

Let $(x_n)$ be a sequence that converges to the irrational number $x$. Must it be the case that $x_1, x_2, \dots$ are all irrational? Let $(y_n)$ be a sequences that converges to the rational number ...
1
vote
1answer
103 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 ...
1
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1answer
74 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 ...
0
votes
2answers
48 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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votes
2answers
194 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 ...
-2
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1answer
777 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 ...
2
votes
1answer
63 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 ...
4
votes
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 ...
5
votes
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 ...
1
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0answers
100 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 ...
3
votes
1answer
113 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 ...
1
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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
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 ...
4
votes
4answers
792 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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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.
0
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3answers
708 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?
1
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2answers
72 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
87 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} ...
0
votes
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
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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 ...
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 $k<n$ ?
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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 ...
0
votes
1answer
68 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
254 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} ...
1
vote
1answer
230 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}, ...