Somewhere beyond the numbers lies the concept of Infinity. But what exactly does "infinity" mean? What rules does it obey? What interesting properties does it have?

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Which of these sets is bigger?

I am a fourth year computer science student and I am taking second year level maths because they are very useful for computer stuff. At the end of the linear algebra lecture the Prof left us with a ...
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1answer
833 views

Does negative infinity squared = positive infinity?

I googled this question and saw this answer but I wasn't satisfied.
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3answers
141 views

Incorrect proof of the infinities between 0 and 1 and 0 and 2

In reading another question (Explaining Infinite Sets and The Fault in Our Stars) it got me thinking about the way that you can prove that the number of numbers between 0 and 1 and between 0 and 2 are ...
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Explaining Infinite Sets and The Fault in Our Stars

In watching The Fault in Our Stars I could not help but cringe at a line that flew in the face of mathematics and subsequently ruined the movie for me: "There are infinite numbers between 0 and 1. ...
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2answers
100 views

Could you explain Perron's paradox to me, please?

This is Perron's paradox: Let $N$ be the largest integer. If $N > 1$, then $N^2 > N$, contradicting the definition of $N$. Hence $N = 1$. What does it mean? I get from it that a very large ...
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0answers
69 views

Backward from infinity?

The following question has been raised and answered lately: Problem 6 - IMO 1985 Please take a look at the Reverse method part of the answer given by this author. What's happening there is that we ...
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82 views

closed-form term for this sum:

related to this question: Is there an easy closed-form term for $$\sum_{j=k}^{\infty} \frac{x^j}{j!}e^{-x},$$ thus when the sum starts at a constant $k$ instead of $1$? EDIT: Thanks for your help. ...
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5answers
133 views

Is $\infty = \frac{1}{0}$? [duplicate]

Is $\; \infty = \frac{1}{0}$? My teacher says no but he wouldn't explain it. My question is why $\; \infty \neq \frac{1}{0}\;?$ My thinking: Let $\frac{1}{x}=p$ Now as $x$ becomes smaller $p$ gets ...
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1answer
34 views

Highest Common Factor to Infinity

Imagine you have a set of integers of x.For example: 7 9 11 13 Let us imagine that y is 1. Then for each nth generation you added 1 to each member of the set, found the HCF of the set and set y to ...
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2answers
98 views

How can you compare the number of real numbers in the interval [0,1] and [0,10]?

There are infinite number of real numbers between 0 and 1,i.e in the interval [0,1]. So definitely there should be more numbers in the interval [0,10] because it includes the numbers in the first case ...
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0answers
43 views

How do you call a scale that starts at $∞$, has $1/n$ divisions and tends to $0$?

A linear scale $2n$ divisions: 0 2 4 6 8 Logarithmic scales $10^n$ divisions: 1 10 100 1000 10000 ...
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3answers
39 views

Search for two Real Valued functions.

Can we have two real valued functions $f_1$ and $f_2$ defined on $[a,b]$ such that $f_1(x)=f_2(x)$ for infinitely many points and $f_1(x)\neq f_2(x)$ for infinitely many points. ?
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118 views

Cauchy principal value with log

How can I obtain the result of the following integration? $$\int_{-\infty}^{+\infty} \log \left(1+\frac{a^2}{x^2}\right)dx$$ Thank you!
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1answer
63 views

Newbie approach to understand generalized continuum hypothesis

There is this theorem that size of power set constructed from infinite set is "more" infinite than the previous set: $$ \begin{eqnarray*} \aleph_0 &= |\mathbb{N}| \\ \aleph_{n+1} &= ...
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0answers
46 views

Dividing an infinite plane into regions

I am currently working on a computer program for computing layout of graph-based diagrams. Their content is placed in an "infinite" 2D plane with cartesian coordinates in the center of the diagram. ...
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1answer
85 views

Assigning values to divergent integrals

I'm interested in the (obviously divergent) integral $$ \int_{-\infty}^\infty dx e^{-x f}\ ,$$ where $f$ is real. Is there any way to meaningfully assign a value to this integral? I was thinking of ...
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4answers
382 views

What is your intuitive understanding of infinity? [duplicate]

What is your intuitive understanding of infinity? Mine is the following, I prepared it as image: Those were the main points I got to after thinking by myself about what infinity is, without ...
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2answers
117 views

$\infty + \infty = \infty$?

(The context is a measure-theoretic one.) I know that $\infty - \infty$ is indeterminate, but what about $\infty + \infty = \infty$? It seems this statement is true and if I input it into Wolfram ...
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1answer
32 views

What truly are length, area and volume? And considerations about divergence in normed spaces

All the "(?)" are parts when i'm not sure at all if what i'm saying is right or not, it's just my intuition. Part 1 In $\mathbb{R}$, we can define the length of a segment. In $\mathbb{R}^2$, the ...
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5answers
140 views

Is $0$ the midpoint of $(-\infty,+\infty)$?

Is $0$ the midpoint of $(-\infty,+\infty)$? Intuitively, I'd think so, and trying to refine my intuition as to why I'd think so, I would say that this is the case because there is a one-to-one ...
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14answers
10k views

How big is infinity?

This might be more philosophy than math, but it’s been bothering me for a while. Question: If there’s an infinite amount of real numbers between $ 0 $ and $ 1 $, shouldn’t there be twice the ...
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3answers
112 views

Why is $\frac{\sum_{n=1}^{\infty} n}{\sum_{n=1}^{\infty} n}$ indeterminate?

We all know that $\dfrac{f(x)}{f(x)} = 1$ (where $f(x) \neq 0$) and that $\sum_{n=1}^{x} n = \dfrac{x(x+1)}{2}$. So, given $f(x) \stackrel{\text{def}}{=} \sum_{n=1}^{x} n$, we show that ...
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2answers
57 views

Limis - what is zero times infinity [duplicate]

what is the solution to a limit function that result in 0 times infinity. Or is that not a possible solution?
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2answers
159 views

Is the inverse ackermann function the slowest growing function that goes to infinity?

Actually, this is not precisely my question. If $a(x)$ is the inverse ackermann function, then obviously $a(a(x))$ grows slower than $a(x)$, as does $\log(a(x))$, and so on. But is there a function f ...
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0answers
131 views

What does “radical cube zero” mean?

My hobby is taking comics way too seriously. And I just came across a math topic. In a certain comic (Fantastic Four 51, according to some polls the greatest comic issue ever) there's a machine for ...
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1answer
114 views

Are there any good books on infinity?

I am looking for a book that discusses the concepts of infinity like the actual infinite and related concepts. Any suggestions?
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64 views

Doubts about infinite nested root

Find $f(a)=\sqrt{a-\sqrt{a^2-\sqrt{a^4-\cdots}}}$ where $a\in\mathbb{R}$. My Attempt : I consider $\frac{f(a)}{a}=\sqrt{1-\sqrt{1-\sqrt{1-\cdots}}}$. Now to finding this limit is easy but I cannot ...
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4answers
310 views

If $y=x^{x^{x^{x^{x^{.^{.^{.}}}}}}}$ then how $y=x^y$?

In questions like, find the derivative of $f(x)=x^{x^{x^{x^{x^{.^{.^{.}}}}}}}$, how can we formally show that $y=x^y$? We use this technique for all type of iterations, e.g. ...
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2answers
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How can I define $\mathbb{N}$ if I postulate existence of a Dedekind-infinite set rather than existence of an inductive set?

Suppose in the axioms of $\sf ZF$ we replaced the Axiom of infinity There exists an inductive set. with the Axiom of Dedekind-infinite set There exists a set equipollent with its proper ...
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3answers
95 views

Applying L'Hôpital's rule infinitely

I tried to prove that $\int\limits_0^\infty t^{x-1} e^{-t} \, \mathrm{d}t$ satisfies the functional equation of the gamma function $\Gamma(x+1)=x\Gamma(x)$, so I partially integrated $\Gamma(x+1)$, ...
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2answers
73 views

How to explain indeterminations, and some aprpoaches to $+\infty$ or $-\infty$, for middle school students?

Question: how to explain the undefinitions $0^0$ and $\frac{0}{0}$ for Middle school students?? I am a math teacher and I don't know how to answer properly when studens ask me why some operations ...
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1answer
163 views

Is there a highest order of infinity?

Does there exist an infinite set of cardinality such that it can never be reached by taking power sets of a set with cardinality aleph-null. Please prove your answer, or include a link to a proof. I ...
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1answer
32 views

Prove that function f has a local minima and maxima

$f:R->R, f(x) = (x^2+mx)e^-x$ Show that, for every m in R, the function f has a local ...
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1answer
104 views

What is the point of (Compactness Theorem in the) Overspill Principle?

I am trying to understand the basics of computation theory. The Overspill Principle (also at google) basically says if you are cool you can do everything Let Г be a sentence of predicate logic ...
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2answers
57 views

Expected value of a the reciprocal of a random number

If I selected a real number at random from the interval (0.0,1.0), assuming a uniform distribution, the "expected value" would be 0.5. (I am not certain I am using the phrase correctly; I mean, if I ...
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2answers
326 views

Is infinity a real or complex quantity?

Since I was interested in maths, I have a question. Is infinity a real or complex quantity? Or it isn't real or complex?
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4answers
89 views

Calculate $\displaystyle \lim_{x \to \infty} x - \sqrt{x^2 + 2x}$ without derivations.

How can I calculte $\displaystyle \lim_{x \to \infty} x - \sqrt{x^2 + 2x}$? Here is what I´ve done so far: Multiplying by $\displaystyle \frac{x + \sqrt{x^2 + 2x}}{x + \sqrt{x^2 + 2x}}$ I got ...
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94 views

Limit of constant function

I was reading a proof using Markov Chains in a finite state space $E$. Denote $p_{ij}(n) = P(X_n = j | X_0 = i)$. Since the state space is finite, then probability of landing somewhere in the state ...
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2answers
73 views

Difference between limits $\infty$ and $+\infty$

Is there a difference between these two limits? $$\lim_{x\rightarrow\infty}f(x)=+\infty\text{ and }\lim_{x\rightarrow+\infty}f(x)=\infty$$
3
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8answers
315 views

How come $1^{\infty}$ = undefined, while $2^{\infty} = \infty$ and $0^{\infty} = 0$? [duplicate]

$1^\infty$ = undefined $2^\infty = \infty$ $0^\infty = 0$ Why is $1^\infty$ undefined? People were trying to explain to me that infinity isnt part of the Real numbers, yet, $2^\infty$ and ...
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248 views

New Axioms of Infinity

Axiom of Infinity says there is an inductive set (i.e. a set which includes $\emptyset$ and is closed under successor operator). Formally: $Inf:\exists x~(\emptyset\in x~\wedge~\forall y\in ...
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3answers
168 views

What's wrong with using algebra on infinite series?

I've recently found an article (referred somewhere on this site) criticizing the use of common rules of algebra on infinite series. To be honest, the video referred is one of the videos of Numberphile ...
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1answer
299 views

Infinite Series of the asymptotic expansion of Fresnel Integrals

I need to find the infinite series for the asymptotic expansions of the fresnel integrals as $x\rightarrow \infty$ and $x\rightarrow 0$. Now I have computed the asyptotic expansions to be as follows ...
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75 views

How can infinity be an accumulation point?

I can't wrap my mind around this: An accumulation point is a point of a set, which in every yet so small neighborhood (of itself) contains infinitely many points of the set, right? So if (in my case) ...
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1answer
35 views

limits for positive and negative infinity

It says we use the l'hospital's rule, however I don't understand because the limit for positive infinity and negative infinity are different. Please help!
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1answer
37 views

How do we evaluate this limit?

$$ A_N(x) = \lim_{N\to \infty}(\sin(x)/x)^N $$ The solution to this problem is given as, $$ A_N(x) = \exp( -Nx^2/6). $$ The problem is solved through Taylor series expansion for $\sin(x)$. And ...
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2answers
79 views

A solution to the equation $\frac{1}{x}=0$ [duplicate]

The number $i$ is defined as a solution to the equation $x^2+1=0$. How come no one has yet defined a number $j$ as a solution to the equation $\frac{1}{x}=0$? The purpose of course is to be able to ...
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2answers
32 views

Find any sequence that meets these criteria.

I'm struggling with this problem and don't know where to start looking: Is there any sequence $a_n$ such that $\lim\limits_{n \to \infty}a_n \neq 0$ and $\lim\limits_{n \to \infty}(n \sqrt[n]{|a_n|}) ...
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0answers
18 views

Is it possible to extend the complex plane affinely and projectively at the same time?

Is it possible to extend the complex plane affinely and projectively at the same time? That is by adding both the positive infinity (with based on it directed infinity) AND the unsigned complex ...
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4answers
179 views

What is the answer to the paradox of the infinitesimal?

I just read this article on npr, which mentioned the following question: You can keep on dividing forever, so every line has an infinite amount of parts. But how long are those parts? If they're ...