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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
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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
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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
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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
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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
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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
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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
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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 ...
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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
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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
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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
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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-\...
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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-...
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 ...
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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
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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
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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
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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
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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 ...