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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Evaluate the limit: $\lim_{x\to \infty}$

Evaluate the limit: $$\lim_{x\to\infty} \frac{(2x^2 +1)^2}{(x-1)^2(x^2+x)}$$ The answer is 4 and I don't understand why, but why can't I just do something like:$$\frac{(\infty)}{(\infty)(\infty)} = ...
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1answer
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Are distance-related paradoxes limited by the size of an atom?

See these 2 paradoxes: Coastline paradox The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. ...
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38 views

Infinitesimal Unit of Measurement

This is just a question that popped into my head which I lack the knowledge to answer (or even to know whether there is an answer, honestly). Does the idea of an infinitesimal unit of measurement even ...
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2answers
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I need to understand why the limit of $x\cdot \sin (1/x)$ as $x$ tends to infinity is 1

here's the question, how can I solve this: $$\lim_{x \rightarrow \infty} x\sin (1/x) $$ Now, from textbooks I know it is possible to use the following substitution $x=1/t$, then, the ecuation is ...
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6answers
99 views

Help me understand infinity [duplicate]

I asked a math professor once about infinity and his answer puzzled me. I asked if i had two sets of numbers: A = all the whole numbers in infinity B = all the whole and half numbers (1, 1.5, 2, ...
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1answer
49 views

How is this definition of a constant divided by zero called?

I divide a constant by zero. One example is the following: 2/0 My father told me he learned at school earlier that the result is "not defined". If I enter this arithmetic problem in Wolfram Alpha, I ...
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6answers
2k views

A strange puzzle having two possible solutions

A friend of mine asked me the following question: Suppose you have a basket in which there is a coin. The coin is marked with the number one. At noon less one minute, someone takes the coin ...
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4answers
91 views

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
78 views

difference of infinities [closed]

So I haven't studied this subject In a while, and when i get home, I'm going to consult my book on the Theory of computation to reinforce my understanding, but here is my question. So I understand ...
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1answer
72 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
113 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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1answer
34 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
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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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4answers
72 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
101 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
28 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
78 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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41 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
32 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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0answers
24 views

Can a non-cyclic infinite proof tree with always-reachable provable nodes be used to construct a proof?

Suppose that I have a finite number of basic elements x,y,z ... and a finite number of operators +, * ... Terms X,Y,Z ... are created by combining basic elements and operators. For example, x+y, and ...
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3answers
85 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
55 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
27 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
58 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
336 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
107 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
28 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
129 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
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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
104 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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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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60 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
93 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
75 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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What if infinity does not exist? [duplicate]

I am trying to realize the importance of the infinity concept. Where will we find great problems if by axiom infinity does not exist?
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2answers
57 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
276 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
60 views

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
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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
51 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
151 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
24 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
70 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
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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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297 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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135 views

Why is 2 bigger than 1? [closed]

If you can split $2$ in infinite pieces, and $1$ in infinite pieces, why is $2 > 1?$ It's like saying infinit > infinit.
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4answers
84 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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0answers
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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
71 views

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

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