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Questions tagged [irrational-numbers]

Questions about real numbers not expressible as the quotient of two integers. For questions on determining whether a number is irrational, use the (rationality-testing) tag instead.

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Proof that $\sqrt2$ is irrational using the Fundamental Theorem of Arithmetic [duplicate]

I'm working through Lang's Basic Mathematics, which gives an idea of my level. The text includes the common proof that $\sqrt2$ is irrational using reduction of the rational number to least terms plus ...
Tsai Jaldun's user avatar
0 votes
0 answers
19 views

Is ${}_3 F_2(a,b,c;d,e;x)$ irrational for $ a,c,d,e,x \in \mathbb{Q} / \{0 \} $

My understanding of G-functions is simply nonexistent but I do know that they can assume algebraic values at nonzero rational arguments. But could those assumed values be rational? Specifically I was ...
Loading - 146 Complete's user avatar
1 vote
2 answers
33 views

A relationship between the initial precision of an irrational number with the computation of its continued fraction?

Warning: this might be trivial. Suppose I want to compute the simple continued fraction $$ [a_0;a_1,a_2,a_3,\ldots] $$ for a (non quadratic) irrational number $x$. I can do this numerically very ...
Alasdair's user avatar
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2 votes
0 answers
71 views

Proving that sine is irrational at rational arguments with infinite fractions

In his proof of irrationality of $\pi$, Lambert uses the following continued fraction for tangent: $$\tan x=\cfrac{x}{1- \cfrac{x^2}{3-\cfrac{x^2}{5-\cfrac{x^2}{7-\cdots}}}}$$ He notes that for any ...
Loading - 146 Complete's user avatar
6 votes
0 answers
82 views

Is the area enclosed by p(x,y) always irrational?

Take a polynomial $p \in \mathbb{Q}[X,Y]$. Now draw the graph of $p(x,y)=0$. If, like $X^2-Y^2-1$, this turns out to enclose a finite area, is the area enclosed always irrational? There are some ...
Zoe Allen's user avatar
  • 5,488
3 votes
1 answer
90 views

Is $\sqrt{2}$ an element of the set $\{k \bmod 2\pi \mid k \in \mathbb{N}\}$? [closed]

I'm exploring the properties of the set formed by taking the modulo $\pi$ of natural numbers, specifically $\{k \bmod 2\pi \mid k \in \mathbb{N}\}$. This set includes all values $k - 2n\pi$ where $0 \...
hans's user avatar
  • 213
1 vote
2 answers
65 views

A subtlety about a proof that if $n$ is not a perfect square, then $\sqrt{n}$ is irrational?

I am trying to understand a subtlety in the following proof: When they write: It follows that $p^{2k+1}$ divides $r^2$. The factorization of $r^2$ into prime numbers involves only even powers of ...
Red Banana's user avatar
  • 24.2k
2 votes
1 answer
82 views

Complement of any dense countable subset of reals is homeomorphic to irrationals

I recently stumbled upon this: For any infinite countable subset $A\subseteq\mathbb R$ such that $\overline A=\mathbb R$, the complement $\mathbb R\setminus A$ is homeomorphic to the Baire space. (Or,...
Martin Sleziak's user avatar
0 votes
2 answers
101 views

Why could $e$ + $π$ be rational

Really just what the title says. I have checked this old post Why is it hard to prove whether $\pi+e$ is an irrational number? but I am left none the wiser. I am trying to understand why there is a ...
lightningjay's user avatar
1 vote
1 answer
132 views

If $\alpha \in \mathbb R$ is algebraic over $\mathbb Q$, then there exists $n \in \mathbb N$ such that $\alpha^n \in \mathbb Q$. [duplicate]

I wish to disprove the following proposition: If $\alpha \in \mathbb R$ is algebraic over $\mathbb Q$, then there exists $n \in \mathbb N$ such that $\alpha^n \in \mathbb Q$. I'm trying to give a ...
RatherAmusing's user avatar
4 votes
1 answer
507 views

Are the vast majority of irrational numbers, transcendental? [duplicate]

It is often stated that the vast majority of real numbers are irrational. Does it also follow that the vast majority of irrational numbers are transcendental?
Larry Freeman's user avatar
1 vote
1 answer
34 views

What is the relationship between the silver ratio and the postion of a circle in the corner of a triangle?

I was recently trying to figure out how much to offset a circle in the corner of a right-angled triangle and found empirically that the x-offset needed to be around 2.414 which I later found to be the ...
TyghtMo's user avatar
  • 13
0 votes
2 answers
70 views

Get rid of irrationality in the denominator $\frac{9-40\sqrt[3]{6}-6\sqrt[3]{36}}{1-\sqrt[3]{6}-3\sqrt[3]{36}}$

Get rid of irrationality in the denominator $$\cfrac{9-40\sqrt[3]{6}-6\sqrt[3]{36}}{1-\sqrt[3]{6}-3\sqrt[3]{36}}$$ $$\left ( 1-\sqrt[3]{6}-3\sqrt[3]{36} \right )\left ( 1+\sqrt[3]{6}-3\sqrt[3]{36} \...
Dmitry's user avatar
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12 votes
1 answer
217 views

Is there a $f: \mathbb{N} \to \mathbb{N}$ such that $\sum_{n=1}^{\infty} \frac{1}{n^2f(n)} \in \mathbb{Q}$?

Take by convention $0 \not \in \mathbb{N}$, and let $f: \mathbb{N} \to \mathbb{N}$. Define the real number $N(f)$ by $$N(f) = \sum_{n=1}^{\infty} \frac{1}{n^2f(n)}.$$ $N(f)$ is well-defined because, ...
Robin's user avatar
  • 3,940
3 votes
3 answers
73 views

Notable algebraic numbers with high minimal polynomial degree

In this question I'll be referring to certain numbers as "notable". To remove the possible objection of this being opinion-based, we may define "notable" to mean someone has ...
Robin's user avatar
  • 3,940
0 votes
1 answer
93 views

are there 2 or more irrational numbers between any 2 rationals?

… in general, but also related to a calculus problem I have before me which is about continuity. The question regards continuity wrt the function $$ f(x) = \begin{cases} x, x \in \mathbb{Q} \\ 0, ...
El Jfe's user avatar
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6 votes
3 answers
789 views

Where is the mistake in the argument in favor of the (erroneous) claim "every Dedekind cut is a rational cut"?

A cut is a set $C$ such that: (a) $C\subseteq \mathbb Q $ (b) $C \neq \emptyset $ (c) $C \neq \mathbb {Q} $ (d) for all $a, c \in \mathbb Q $ , if $c\in C$ and $a\lt c$ , then $a\in C $ (e) for all $c\...
Vince Vickler's user avatar
7 votes
1 answer
1k views

Irrational numbers as "periods" in discrete sequences based on complex exponentials

Main string description To keep the core of the question short, I leave the context introduction at the end of this text. Here I describe a construction that is related to that introduction, but to ...
phionez's user avatar
  • 350
0 votes
1 answer
61 views

Are over half of the positive rational numbers in the interval [0, 1]?

There is a problem in my calculus textbook that wants me to draw/look at the following function $f:\Bbb{R}\to\Bbb{R}$ $$f(x)=\begin{cases} 1 & : x \in \Bbb{Q} \\ 0 & : x \in \Bbb{P} \\ \end{...
volticus's user avatar
0 votes
0 answers
83 views

Proof that a base 2 logarithm of a rational number is irrational

How can I prove that if $a = \log_2 b, b \in \Bbb Q, b \neq 2^c$ and $c \in \Bbb Z$ then $a \notin \Bbb Q$ ? And could the proof be easily adapted to differently-based logarithms? I am familiar with ...
fedsavi's user avatar
  • 35
1 vote
2 answers
64 views

Are singletons in $\mathbb{R} \setminus \mathbb{Q}$ both closed and open?

For context: I am looking at $\mathbb{R} \setminus \mathbb{Q}$ as a subset of $\mathbb{R}$ with usual topology. I know the singletons are closed. But are they open? I thought about something along the ...
metamathics's user avatar
5 votes
2 answers
148 views

Irrationality of numbers that are the sum of reciprocal factorials, like $e.$

I wish to prove that for any infinite $A\subset \mathbb{N},\ \displaystyle\sum_{k\in A} \frac{1}{k!}$ is irrational, in the same manner that Rudin proves $e$ is irrational. I should mention the ...
Adam Rubinson's user avatar
2 votes
2 answers
117 views

$\int_0^{\pi} ((\sin{x})^3+(\cos{x})^3)^n dx$ is rational iff $n$ is odd

Define the integral $$I_n = \int_0^{\pi} ((\sin{x})^3+(\cos{x})^3)^n dx$$ for any natural number $n$. I am trying to show that $$I_n\text{ is rational } \iff n \text{ is odd}$$ My first idea was to ...
Sam's user avatar
  • 3,360
0 votes
1 answer
68 views

How to generalise the argument in Chap. 1 in Baby Rudin to show that these sets $A$ and $B$ have no largest and smallest elements, respectively?

Let $n$ be a positive integer that is NOT a perfect square, and let the sets $A$ and $B$ be defined as follows: $$ A := \left\{ p \in \mathbb{Q} \colon p > 0, p^2 < n \right\} $$ and $$ B := \...
Saaqib Mahmood's user avatar
1 vote
1 answer
72 views

The sum of a sequence where any term is equal to the absolute value of the difference between the prior two terms.

In a sequence, any term is equal to the absolute value of the difference between the prior two terms. Will the sum of the sequence converge? It's known that if the first two given terms are rational, ...
718281828's user avatar
0 votes
0 answers
46 views

Approximate irrational numbers using rational numbers from up and below

Let $\alpha\in\mathbb{R}$ be an irrational number. It is well-known that for any integer $n$, there exists a rational number $\frac{p}{q}$ with $|q|\leq n$ such that $$|\alpha-\frac{p}{q}|<\frac{1}{...
Yang's user avatar
  • 1
3 votes
2 answers
76 views

Show a counterexample to the converse of the Intermediate Value Theorem [duplicate]

Consider the function $f : [0, 1] → [0, 1]$ defined by $$\begin{cases}f(x) = x \text{ for } x\in \mathbb{Q}\\ f(x) = 1 − x \text{ for } x \notin \mathbb{Q}\end{cases}$$ Prove for any $y ∈ [0, 1]$, ...
rlaivsezlt's user avatar
1 vote
1 answer
44 views

Approximation of $1/2$ by fractional parts

Let $\{u\}$ denote the fractional part of a real $u \geq 0$, and $\mathbb{N} = \{0, 1, 2, \dots\}$. For positive integer $N$, define $$ T_N(x) = \sup_{k \in \mathbb{N}} \min_{m \in [kN, (k+1)N) \cap \...
Drew Brady's user avatar
  • 3,744
0 votes
0 answers
66 views

Weird Property of Irrational Numbers and their "Conjugate"

Let us have a irrational number $a$. Let $a'$ be $\frac{1}{a'} + \frac{1}{a} = 1$. If we let $A$ and $B$ be sets such that $A =$ {$\lfloor a \rfloor, \lfloor 2a \rfloor, \lfloor 3a \rfloor, \lfloor 4a ...
bob john's user avatar
2 votes
1 answer
77 views

Approximation of $\pi$ by integral and rational number

Via WolframAlpha, I observed that $$\int_0^1\frac{x^{4n}(1-x^2)}{1+x^2}dx=\frac\pi2-\frac pq \to 0$$ when the integer $n\to\infty$. This gives an approximation of $\pi$ by a rational number. It is not ...
Bob Dobbs's user avatar
  • 11.8k
0 votes
2 answers
96 views

Cards with specific order

We have a deck of 99 cards each of which has a distinct real number written on it, and the sum of the all numbers is irrational. A specific order of the cards is said to be good if sum of the k top ...
Fermat's user avatar
  • 5,254
2 votes
2 answers
184 views

Finding an irrational number between two given irrational numbers constructively

Statement: Let $$X=\{(a,b) \in \mathbb{R} \setminus \mathbb{Q} \times \mathbb{R} \setminus \mathbb{Q}:a<b\}$$ There exists a function $f:X \rightarrow \mathbb{R} \setminus \mathbb{Q}$ such that for ...
Mohammad tahmasbi zade's user avatar
3 votes
2 answers
123 views

Might there be an $n^{\text{th}}$ digit of $\pi$ where the sequence becomes palindromic?

Assuming $n>1$, would it be reasonable to think there is an $n^{\text{th}}$ digit of $\pi$ where stopping there would yield a palindromic number $(3.14159...951413)$? Would it be more likely that ...
Pickelhaube808's user avatar
0 votes
0 answers
39 views

Let $a_n\geq a>0$, for any $x\in\Bbb R$, can we find a rational sequence $r_n$ such that $r_na_n\to x$?

Let $a_n\geq a>0$, for any $x\in\Bbb R$, can we find a rational sequence $r_n$ such that $r_na_n\to x$? My attempt: if $x/a_n$ is rational for any $n$, then $r_n=x/a_n$ is OK$. If $x/a_n$ is ...
xldd's user avatar
  • 3,603
1 vote
2 answers
97 views

Prove that $\sqrt{18} - \sqrt{12}-\sqrt{45} + \sqrt{6}$ is irrational [duplicate]

Prove that $\sqrt{18} - \sqrt{12}-\sqrt{45} + \sqrt{6}$ is irrational I tried to let $x = \sqrt{18} - \sqrt{12}-\sqrt{45} + \sqrt{6}$, and assume for the sake of contradiction that $x$ is rational. ...
fds-mqdwqmqkdwl's user avatar
0 votes
2 answers
111 views

Question about an inequality during proof that $e^2$ is irrational.

I am reading a proof of the irrationality of $e^2$ and I am stuck on the following inequality: Let $S := -a\underbrace{\left(\frac{1}{n+1} - \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} \mp ...\...
Felix Gervasi's user avatar
1 vote
1 answer
113 views

Does there exist an irrational number $x$ such that both $x^π$ and $π^x$ are also irrational? [closed]

Can we find a number, let's say $x$ that's not a simple fraction, and if you raise it to the power of $π$ or raise $π$ to the power of $x$, both results end up being not simple fractions too?
Mods And Staff Are Not Fair's user avatar
4 votes
5 answers
131 views

Why must rational $x$ and $y$ satisfy $x+y=2$ and $2\sqrt{3xy}=3$ in order for $2\sqrt{3}-3=(x+y)\sqrt{3}-2\sqrt{3xy}$ to be true?

I am going through a maths book and unable to understand the following: Given that $x$ and $y$ are rational numbers, then if the equation $$2\sqrt{3}-3=(x+y)\sqrt{3}-2\sqrt{3xy}$$ is true, then we ...
mrnakumar's user avatar
  • 185
4 votes
1 answer
150 views

Can this proof that $\sqrt{2}$ is irrational be rewritten using only integers?

Can this proof that $\sqrt{2}$ is irrational be rewritten using only integers? Most proofs that $\sqrt{2}$ is irrational start with assuming that $2=\dfrac{a^2}{b^2}$ and derive a contradiction. For a ...
marty cohen's user avatar
-1 votes
2 answers
50 views

My question is whether given function is periodic or not [closed]

F(x) = 1, if x is rational 0, if x is irrational So what i am really confused about is because all professors are telling me its periodic whose period is not defined, i mean it doesn't make any ...
MultiUniverseExplorer's user avatar
0 votes
1 answer
71 views

Is my proof of this fact about pi correct?

I recently have thought of a proof but I can’t tell if it is correct or not. The proof is of $n \pi$ being irrational if n is an integer and non zero. The proof is below: We assume that $n \pi$ is not ...
Evon Z's user avatar
  • 3
4 votes
2 answers
145 views

Which numbers can be expressed as $a+b\pi$, with $a$ and $b$ being integers?

Since $a$ and $b$ are both integers, there is only a countable amount of numbers $a+b\pi$. Thus not every real number can be expressed as such. But is there a way to determine if $x$ can be expressed ...
Quintium's user avatar
  • 162
6 votes
0 answers
131 views

Is there an "elementary irrational number" without a certain digit in its decimal presentation?

In this question, I define an "elementary irrational number" as an irrational number which is built up of a finite combination of integers, field operations (addition, multiplication, ...
Alex-Github-Programmer's user avatar
0 votes
0 answers
52 views

Show the values of $x$ such that $g$ is continuous.

Let $g(x)$ be: \begin{cases} 0 \space\space \text{if}\space\space x \in \mathbb Q \\ x\space\space \text{if} \space\space x \in \mathbb R - \mathbb Q \\ \end{cases} That's what I've ...
LightL96's user avatar
2 votes
0 answers
32 views

Are the semiballs induced by even p-norms the integrands of integral representations of trascendental numbers?

$\frac{π}{2}$ is a trascendental number, given by the integral: $$\frac{π}{2}=\int_{-1}^1\sqrt{1-x^2}dx$$ The integrand is also the curve of a semiball of the form: $$||(x,y)||_2=1$$ Induced by the 2-...
Simón Flavio Ibañez's user avatar
4 votes
2 answers
203 views

Can $x\sin(x)$ be algebraic when it is not $0$?

It's easy to show (using the Lindemann-Weierstrass theorem) that, for $x\ne 0$, at least one of $x$ and $\sin(x)$ must be transcendental. But what about $x\sin(x)$? After all, the product of two ...
Ted Hopp's user avatar
  • 635
1 vote
0 answers
188 views

Closed form for $A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$

Consider the double sums : $$A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$$ $$A_4 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^4 + b^4}$$ Is there a closed form for $A_3$ ...
mick's user avatar
  • 16.4k
4 votes
1 answer
241 views

Hard time grasping Dedekind cuts and real numbers [duplicate]

I have been trying to understand some elementary set theory recently, and am trying to understand how the real number line can be defined using the set of rational numbers. In particular, I am trying ...
Alice's user avatar
  • 508
0 votes
2 answers
94 views

Considering Rectangles Whose Perimeter/Long Side Ratio is Equal to Long Side/Short Side Ratio

Let us consider rectangles. Among the three numbers associated with rectangle is the perimeter, the long side, and the short side. If a rectangle is a square, the ratio of the perimeter to the long ...
Michael Ejercito's user avatar
3 votes
2 answers
111 views

An alternative way of looking at countable/uncountable infinities

Consider the decimal expansion of a rational number. This will either terminate, or repeat forever a finite number of its final digits. Thus, any rational number can be expressed with a finite amount ...
user19642323's user avatar

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