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

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2
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1answer
33 views

Confusing sum of fractions

Question is to find the sum of: $$(\frac{1}{2^2-1})+(\frac{1}{4^2-1})+(\frac{1}{6^2-1})+(\frac{1}{20^2-1})$$ I know that $a^2-b^2=(a+b)(a-b)$, and that with this I can find the LCM to be 1995, ...
2
votes
1answer
27 views

Rational Numbers and farey fractions

How can I go about proving the following problem: Prove that a number a is rational if and only if there exists a positive integer k such that $[ka]=ka$. Prove that a number is rational if and only ...
0
votes
1answer
37 views

Show that two field extensions are the same

Can you help me with showing that these two field extensions are the same: $\mathbb{Q}(\sqrt{3}, \sqrt[3]{5})$ $\mathbb{Q}(\sqrt{3} + b\sqrt[3]{5})$, where $b\neq0$ is any rational number. Thanks ...
2
votes
2answers
66 views

show that if $|x-\frac{m}{n}| \leq \epsilon$ then $n$ is very large

I am working on my calculus homework currently, and in order to solve a question, I need to prove this more simple statement: if $|x-\frac{m}{n}| \leq \epsilon$ for all $\epsilon>0$ then $n$ has ...
0
votes
0answers
28 views

Rational doubt ( doubt in rational number) [duplicate]

If there is a prime number x, if we reciprocate it we will get 1/x. Reciprocal of prime number will be a rational number , Except 1/2 and 1/5 , every number which is reciprocal of prime number is a ...
5
votes
2answers
424 views

Is the Nested Radical Constant rational or irrational?

Given the sequence $A_n=\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{\dots+\sqrt{n}}}}}$: Are there any known rational elements in $A_n$, or has it been proved that all are irrational? Is there any proof for ...
6
votes
3answers
1k views

number between 17 and 18, and has a rational square root

"number between 17 and 18, and has a rational square root" Is there even one? They all keep coming up irrational for me
1
vote
1answer
45 views

Prove that if $t \in T$ and $q \in Q$, but $q \neq 0$ then $qt \in T$ (where $T$ = transcendental numbers)

Question: Prove that if $t \in T$ and $q \in Q$, but $q \neq 0$ then $qt \in T$. This is Exercise 2.7.13(a) from Mark E. Watkins, Jeffrey L. Meyer: Passage to Abstract Mathematics. I'm currently ...
6
votes
3answers
129 views

Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$.

Question : Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$. Hint: consider decimal expansions ...
2
votes
2answers
78 views

Prove that if $y,z \in\Bbb Q$ then $y^z \in\Bbb A$

Question : Prove that if $y,z \in\Bbb Q$ then $y^z \in\Bbb A$. My attempt: Definition 2.7.8 states that a number $s$ is an algebraic number when there exists some $p \in\Bbb Z[x]$ such that ...
3
votes
1answer
34 views

Prove $\forall u,v,x,y,z,w \in \mathbf{R}^+, \frac{u}{v} < \frac{x}{y} \wedge \frac{x}{y} < \frac{z}{w} \implies \frac{u + z}{v+w} < \frac{z}{w}$

This is a question from a past exam that I can't seem to figure out. Any tips or hints? Prove $$\forall u,v,x,y,z,w \in \mathbf{R}^+, \frac{u}{v} < \frac{x}{y} \wedge \frac{x}{y} < \frac{z}{w} ...
0
votes
4answers
158 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
vote
1answer
53 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
votes
1answer
71 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
votes
1answer
54 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
votes
1answer
36 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
votes
1answer
29 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
votes
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 ...
19
votes
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 ...
0
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0answers
26 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
votes
2answers
136 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 $b=d$.
1
vote
1answer
67 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
votes
0answers
42 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
50 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
106 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
votes
1answer
70 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
vote
4answers
64 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
272 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
votes
2answers
50 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
120 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
votes
1answer
87 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
207 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
377 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
455 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
82 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
vote
1answer
68 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
41 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 ...
-3
votes
2answers
161 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
votes
1answer
483 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
58 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
67 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
931 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
vote
0answers
90 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
73 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
vote
1answer
63 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?
2
votes
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. ...
0
votes
1answer
52 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
561 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? $$\frac{q}{m}=\frac{(\frac{1+\sqrt5}{2})^n - (\frac{1-\sqrt5}{2})^n}{\sqrt5}$$
-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.
0
votes
3answers
426 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?