Questions tagged [infinity]
Use this tag for questions on the concept of infinity. Don't use this tag merely because $\infty$ appears somewhere in the question.
0
votes
3answers
37 views
why 1/ infinity isn't indeterminate like other indeterminate?
$1/\infty$ tends to 0.
$\mathbf {It\ doesn't \ satisfy\ the\ inverse \ process\ of\ multiplication \ and \\division\ i.e} $
$\infty * 0$ is undefined or indeterminate.
So why $1/\infty$ is not ...
0
votes
1answer
25 views
Why is $\mathbb{R}$ unbounded, despite being equinumerous to various bounded sets? Is there a name for this “distinction”?
$[0, 1] \approx (0,1) \approx \mathbb{R}$, for example.
Intuitively, it seems that the infinity of $\mathbb{R}$ is of a different nature than that of the intervals; with $\mathbb{R}$ I can “explode” ...
0
votes
1answer
19 views
A problem with an infinite multitude of numbers that follow some rules
We call the number n a "special number" if there are three distinct natural numbers divisors (of n) so that the sum of their squares is equal to n. We know that n is a natural number and n is diffrent ...
-2
votes
0answers
67 views
Can we have complete closure for the known six arithmetical operators even with inclusion of the transfinites?
[EDIT] Lets define an implementation of the integer numbers as ordered pairs of cardinals with zero, so we stipulate that:
$\langle 0,x \rangle = +x$
$\langle x,0 \rangle = -x$
Where $x=|X|$ for ...
2
votes
0answers
51 views
What kind of infinity is the number of different infinities? [duplicate]
Cantor’s theorem states that $|\mathcal{P}(A)| > |A|$ for any set $A$.
As a special case of this, we have $|\mathcal{P}(\mathbb{N})| > |\mathbb{N}|$. If we denote the power set of the naturals ...
0
votes
3answers
51 views
Does $\frac {1}{∞} = 0$? [duplicate]
Actually , I am not believing that $\frac {1}{∞} = \frac{0}{1}$ because simply $0 ≠ 1$ (we get this if we multiple numerator by denominator) , In the other hand it is very very very small , So I am ...
0
votes
2answers
45 views
Biject all points on a plane to the real line [closed]
I understand that the continuum hypothesis implies (since there are only two infinities; discrete and continuous) that the set of points on an n-dimensional plane is equal to (can be bijected to) the ...
0
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0answers
10 views
Example of the infinite point density of continua
There is an interesting example that I've seen, possibly attributed to Hilbert. One draws a ray from the center of a circle to every point in the circle, and then one increases the radius of the ...
1
vote
2answers
36 views
Is there a notion of a smallest possible number?
Firstly, yes, you can divide any number and get even smaller one but hear me out.
My logic goes as follows:
Let there be a number X. This number behaves similiarly to infinity, except it is not ...
0
votes
1answer
61 views
Evaluating $\int_1^A x^{-\alpha} dx$ as $A\to\infty$
Suppose $\alpha$ is a constant, $A$ is a positive real number.
(a) Find the value of $\int_1^A x^{-\alpha} dx$, for $A > 1$.
(b) What is the answer of (a) when $A$ tends to infinity?
...
1
vote
1answer
54 views
Real Analysis: viewing infinity as the end
Alternating harmonic series, $\frac11-\frac12+\frac13-\frac14+\frac15-\frac16+ \ldots$ converges to $\log(2)$,
and the rearranged series, $\frac11-\frac12-\frac14+\frac13-\frac16-\frac18+\frac15-\...
-3
votes
0answers
33 views
Real Analysis, the way to treat infinity. [duplicate]
I was reading Real Analysis book, and found a problem that contradicts with my intuition.
When I looked up the google, 1/1-1/2+1/3-1/4+... converges to log(2), and the rearranged one, 1/1-1/2-1/4+1/3-...
0
votes
0answers
17 views
How to take limitation of a function at infinity
Does anyone please tell me how to take limitation of following function with $\omega \to \pm \infty$, with M positive parameter and $\omega$ real variable?
I guess we have to separate $M$ < 2 or $M&...
1
vote
2answers
40 views
Given series is convergent or divergent?
$$\sum_{n=2}^{\infty} \frac{1}{(\ln(n))^2}$$
I thought I can compare it with $\frac{1}{n}$ but I couldn't prove it (and be satisfied with it). Therefore, how should I approach these type of series ? ...
1
vote
1answer
38 views
Does it make sense to talk about rationality and countability when dealing with nonstandard quantities?
I have two closely related questions:
Firstly, suppose that I can find two hyperintegers $P$ and $Q$ s.t. $\frac{P}{Q}=\sqrt{2}$. Obviously, both $P$ and $Q$ lie in an extension of the integers. ...
4
votes
1answer
104 views
I do not understand how the continuum hypothesis can be undecidable
I have been thinking about what it means for the continuum hypothesis to be undecidable.
In short, here is my understanding
$\aleph_{0}$ is the cardinality of the set of integers.
$\aleph_{1}$ is ...
1
vote
1answer
36 views
What goes wrong when we try to make a binary partition of a countable set?
Let $f$ be a function that maps an interval $[a, b]$ to some irrational number $r \in [a, b]$ that roughly splits the interval in half (e.g. both sides have at least 1/3 of the mass).
Suppose we then ...
3
votes
2answers
50 views
What are space filling curves used for in the real world?
I recently watched 3Blue1Brown's video on the Hilbert Curve and Fractals, and I was wondering if the concept behind space-filling curves like the Hilbert Curve and Flow Snake could be applied in real ...
0
votes
1answer
26 views
Optimal algorithm to guess any random integer without limits?
Guessing Game In Range $[1, n]$
The classical guessing game goes something like this...
Our friend thinks of an integer between $1$ and $100$ (let's say they pick $42$).
We try to guess that number ...
2
votes
2answers
757 views
Is infinite sequence of irrational numbers digits mathematically observable?
I have a little question. In fact, is too short.
Is infinite sequence of irrational numbers digits mathematically observable?
I would like to explain it by example because the question seems ...
-2
votes
3answers
67 views
infinity mathematics
what is the result of (approaching infinity)/(approaching zero) ?
I think its approaching infinity, but if it is approaching infinity, then multiplying both sides by approaching zero, it became: (...
1
vote
3answers
54 views
Is it valid to take $|- \infty | = \infty$?
Is it valid to take $|- \infty | = \infty$?
or is the absolute value e.g. not defined for infinity?
Particularly,
if one wishes to argue that operator $f(x)=x$ is not bounded below on $\mathbb{R}_{-...
0
votes
1answer
39 views
Singularity and behaviour at infinity for complex function
I'm suppose to check the singularities and behaviour at infinity. However, I've never seen that and couln't find something about it online.
So i have a function
$ f(z) = \frac{1}{\exp{(z)} -1)}-\frac{...
0
votes
0answers
59 views
how to prove that $\lim \limits_{x \to \infty}[f(x+1)-f(x)] = L \implies \lim \limits_{x \to \infty} [f(x)/x] = L$ [duplicate]
I don’t know how to formally prove that
if $$\lim_{x \to \infty} \left(f(x+1)-f(x)\right) = L,$$ then $$\lim _{x \to \infty} \frac{f(x)}{x} = L$$
where L is a constant and the function is limited ...
1
vote
1answer
101 views
Using union of countably infinite sets, I tried to prove that set of all real numbers in [0,1) is countable
Cantor's diagonal method shows that the set $S=\{x\in \Bbb R|x \in [0,1)\}$ is uncountably infinite, because there is no bijection between the set $S$ and the set of natural number $\Bbb N$.
I came ...
-1
votes
1answer
64 views
Evaluate $\lim_{x\to {\infty}} \frac{\int_1^x (t^2(e^{1/t}-1)-t)\,dt}{x^2\ln\left(1+\frac{1}{x}\right)}$
Calculate and evaluate the limit:
$$\lim_{x\to {\infty}} \frac{\int_1^x (t^2(e^{1/t}-1)-t)\,dt}{x^2\ln\left(1+\frac{1}{x}\right)}$$
When plotting the upper and the lower part of the fraction ...
2
votes
0answers
31 views
Probability for (in)finitely many hits
Suppose I have a set $A \subset \mathbb{N}$ and I want to pick elements at random and check whether they belong to $A$ or not.
I know that the probability for a given $k$-digit number $n \in \mathbb{...
0
votes
0answers
46 views
Do we ever distinguish between $0^+$ and $0^-$?
I know the IEEE standard does for numerical precision reasons, but I’m asking mathematically. I ask because I’ve been thinking about how:
(VAGUE LANGUAGE BEGINS HERE) the sum of uncountably many ...
6
votes
3answers
104 views
How can I define an infinitely small positive value?
I have a question about infinity. (I'm in 8th grade so please just let me know if this is a stupid question, and please ask if you want any clarification).
How do you define an infinitely small value ...
1
vote
1answer
85 views
Is there a notion differentiating $\frac1{\mbox{countable }\infty}$ and $\frac1{\mbox{uncountable }\infty}$?
It is accepted belief $\mbox{countable }\infty$ is not $\mbox{uncountable }\infty$. Is there notion differentiating $\frac1{\mbox{countable }\infty}$ and $\frac1{\mbox{uncountable }\infty}$ (latter ...
2
votes
3answers
122 views
Why $\infty×0=-1$ from multiplication of two slopes of two lines perpendicular to each other and how do we define infinity?
Here is given $A(x_1,y_1), B(x_1,y_2), C(x_2,y_3)$ and $D(x_3,y_3)$. I have recently read that, multiplication of two perpendicular lines is always $-1$.
From the above graph, the slope of $AB, m_1 = ...
0
votes
1answer
56 views
Sum to 5 is greater than the Sum to infinity. How is this possible?
I've reached the correct answer but I don't see how the sum of the first 5 terms in the following geometric sequence is greater than the sum to infinity:
$$a= 9, \, r= \frac{-1}{3}$$
Given: Sum to ...
1
vote
2answers
71 views
What is $-\infty \cdot - \infty$?
What is $-\infty \cdot - \infty$?
And what do I do if I come across it in an assignment when taking the limit?
E.g.
$$\lim_{x \to - \infty} (x \cdot x) $$
Since $x^{2}$ is continuous, can you just ...
2
votes
3answers
195 views
Does $\lim_{x \to - \infty} \left(\frac{\pi}{2} + \arctan{x} \right) \cdot x = - \infty$?
Does $$\lim_{x \to - \infty} \left(\frac{\pi}{2} + \arctan{x} \right) \cdot x = - \infty$$?
My logic is that “something“ times "negative infinity" equals negative infinity. Am I right?
2
votes
1answer
65 views
Is infinity the reciprocal of zero/is zero the reciprocal of infinity?
Is infinity the reciprocal of zero? Is zero the reciprocal of infinity? It would make sense that they would be--they behave in a similar way (anything multiplied by zero or infinity results in zero or ...
0
votes
1answer
51 views
Decomposition of an $\aleph_1$ set into $\aleph_1$ sets $\aleph_1$ [closed]
Can we justify the claim
Any set of cardinality $\aleph_1$ can be expressed as the disjoint union of $\aleph_1$ sets of cardinality $\aleph_1$
in a simple yet reasonably-correct way?
0
votes
2answers
51 views
Difference between infinite and undefined [closed]
In questions, sometimes I see these terms being used interchangeably. But also some lecturers say that they are different. I am pretty confused with these. Please explain.
2
votes
0answers
42 views
How to determine how fast something is going towards infinity?
Just for a little bit of information I'm more of a programmer and less a mathematician so if some of my terms seem out of place it is due to a lack for formal training in Math.
While working on my ...
6
votes
7answers
373 views
Error evaluating $ \lim_{x\to 0}\frac{x-\tan x}{x^3} $
Evaluate the limit:
$$
\lim_{x\to 0}\frac{x-\tan x}{x^3}
$$
I solved it like this,
$$
\lim_{x\to 0} \left({1\over x^2} - \frac{\tan x}{x^3}\right)
=\lim_{x\to 0}\left({1\over x^2} - {\tan x\over x}...
4
votes
1answer
46 views
Billiards in a holey square
Suppose you start a point-billiard (or light ray) in a square at a random location, shooting off at a random angle, reflecting with angle-of-incidence equals angle-of-reflection. In general, because ...
-1
votes
1answer
36 views
Could infinity have a numerical value?
For example, $$\frac{1}{1}=1\quad \frac{1}{2}=0.5\quad \frac{1}{3}=0.\overline3\quad \frac{1}{10}=0.1$$ so the larger the denominator is, the smaller the number is.
Would this mean that $\frac{1}{\...
0
votes
2answers
63 views
Is $\infty$ undefined? [duplicate]
I am confused at $\infty$ in a lot of ways. First, we sometimes say that ${1\over 0}=\infty$. That gives us these confusing calculations.$${1\over0}=\infty$$
$$1=0*\infty$$
$$2=(0*2)*\infty$$
$$2=0*\...
0
votes
2answers
29 views
limit as x approaches infinity from the left (Notation)
This question doesn't involve a specific problem, instead focuses on notation. With limits that approach infinity, is it incorrect to say that the limit approaches infinity from the positive/negative ...
0
votes
1answer
24 views
Approximating when variable to infinity
In a book on algorithms I read that $n^2 (1+\log n)$ as $n$ approaches infinity is approximated to $n^2 \log n$.
I am not sure if I understand reasoning in this. Is it because $1+\log n$ grows so ...
0
votes
1answer
39 views
Infinite solids similar to Gabriel's Horn
Do any solids of revolution exist with properties similar to Gabriel's Horn (i.e. a geometric solid with finite volume but infinite surface area)? Please restrict your answers to functions not in the ...
0
votes
1answer
13 views
Is the following statement about matrix norms of matrix products correct?
Let $A$ be a real-valued, square matrix and define its 2-norm as:
$$||A||_2 = \sqrt{\max_i\lambda_i(AA^T)}$$
where $\lambda_i(AA^T)$ denotes the $i^{th}$ eigenvalue of the product $AA^T$.
Now ...
1
vote
1answer
39 views
Integral as a limit of an infinite sum of infinitely narrow rectangles
I tried to approach integrating by filling the space with infinitely many infinitely narrow rectangles.
$a$ is the left bound
$b$ is the right bound
$n$ means the number of rectangles, approaches $\...
5
votes
2answers
132 views
Why can’t you reassign the ‘mystery number’ in Cantor’s diagonal argument to a new number in the natural numbers?
I don’t want to claim that I have ‘refuted Cantor’ or something here, I just want to understand it adequately.
I do understand that the proof works something like this:
You assume that you can map ...
2
votes
1answer
855 views
What is the status of the Axiom of limitation of size? (adrift for almost a century now)
On reviewing the wiki article
Axiom of limitation of size
I come away with the impression that there are issues surrounding this 'maybe too powerful' principle/heuristic/doctrine/axiom that haven't ...
0
votes
1answer
31 views
Probability for an infinite set
The way probability is defined as the expected value works for finite sets. The probability of getting heads is out of two possible outcomes, heads or tails.
If we asked the probability out of an ...
