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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4
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2answers
634 views

Can the distance between 2 non-empty sets be infinite?

Intuitively I would immediately assume no, but that's not how things usually work in math and considering there are different kinds of infinities I haven't been able to find the answer. Here's my ...
2
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1answer
67 views

Method for computing limit of a sin function as x tends to zero

I have a question about computing $$ \lim_{x \to 0} \sin\left(\frac{\pi x}{4|x|}\right)$$ I found the limit of $\pi x$ and $4|x|$ seperately and ended with $\sin(\pi/4)$ which is equal to ...
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3answers
65 views

Method for computing limit of a function as $x$ tends to zero

I have a question about computing $$\lim_ {x \to 0} \dfrac{(2/x^3)+(1/x^2)+(1/x)+1}{(1/x^3)+1}.$$ I used a shortcut method of dividing by the highest power but I don't think that I can use this method ...
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4answers
126 views

May seem like a noob question: really, why can't we divide by 0? [duplicate]

Yes, I know, can't be answered, blah, blah, blah.... but here are a few of my theories. I know, plenty of other questions like this, but before marking this as a duplicate, consider this, my ...
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1answer
21 views

One set of functions larger than another set of functions?

This summer I've been slowly working through Halmos's Naive Set Theory. I'm not that far, but I know what lies ahead, which is proving that one infinite set is larger than another (the reals larger ...
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4answers
1k views

Can a set be infinite and bounded?

I don't understand a statement in my math book course, I was restudying the compact sets part of the chapter when at a certain moment there is a corollary saying : 'every infinite and bounded part of ...
2
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1answer
125 views

Teaching the Concept of Infinity to Children.

I was recently out with the family and we left it up to the children where we ate lunch (11 and 9 years old). They couldn't agree and were going back and forth calling each other names. This ...
2
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1answer
88 views

Is there a mathematical concept of fractions using transfinite numbers as numerators and denominators?

http://de.wikipedia.org/wiki/Cantors_erstes_Diagonalargument (German) http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument (English) While looking at Cantors method of proof, which he used to ...
8
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4answers
205 views

The nature of infinities

I have been thinking about the nature of infinity lately. I have no experience with higher mathematics or theorems regarding infinity, so please forgive me if my ideas on this topic are extremely ...
0
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1answer
25 views

Deffered annuity with perpetuity

An annuity immediate has $40$ initial quarterly payments of $20$ followed by perpetuity of quarterly payments of $25$ starting in the eleventh year. Find the present value at $4\% $ convertible ...
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1answer
47 views

Comparing density of countable infinite sets by examining the association

The two questions that i am asking are in bold. To be clear, i am talking about whole number here. Having seen 3 is everywhere by Numberphile that shows that almost 100% of the whole the numbers ...
0
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2answers
64 views

The limit of $(x^3+\cos x+e^{-2x})/(x^2 \sqrt{x^2+1})$ as $x\to\infty$

I have this infinity problem which I do not know the answer to: $$\lim_{x\to\infty}\frac{x^3+\cos x+e^{-2x}}{x^2 \sqrt{x^2+1}}$$ I thaught that because $x^3$ is the fastest growing part, this would ...
8
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3answers
607 views

Does 0% chance mean impossible? [duplicate]

Suppose we pick a random real number between 0 and 1 and call it $x$. There are $2^{\aleph_0}$ possible values, so the chance of picking any specific number (such as $x$) in that range is 0. But in ...
2
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1answer
71 views

Question about the cardinality of sets and infinity

Let's say we have $\mathbb{N}$, the set of natural numbers: $\{1, 2, 3, 4, 5...\}$ ...which has a cardinality of infinity, and the set $A_x$ which consists of the variable "$x$" (so $\{x\}$). If I ...
0
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0answers
45 views

Cardinality of infinite between the set of rationals and set of reals

I remember learning that whether or not there is a cardinality of infinity between the set of rational numbers and the set of real numbers is unprovable. Is this true, and if so, how do we know it to ...
3
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0answers
69 views

Is it possible for infinite sets to exist in ZFC with the negation of the Axiom of Infinity? [duplicate]

The Axiom of Infinity states that at least one inductive set exists. Inductive sets are infinite, but not all infinite sets are inductive. Suppose that we take ZFC with the negation of the Axiom of ...
29
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2answers
4k views

Are weird numbers more rare than prime numbers?

By taking a look at the first few weird numbers: $$(70, 836, 4030, 5830, 7192, 7912, 9272, 10430)$$ It is certain that prime numbers occurs more often within this range of numbers. But are weird ...
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4answers
101 views

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
52 views

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. ...
0
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3answers
69 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 ...
3
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2answers
127 views

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
114 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
57 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
97 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
648 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
134 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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8answers
4k views

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. ...
2
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1answer
78 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
65 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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4answers
80 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
113 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
31 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 ...
0
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2answers
94 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
42 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 ...
2
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3answers
38 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
32 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 ...
2
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3answers
99 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!
0
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1answer
62 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
43 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
80 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 ...
2
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4answers
376 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 ...
0
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2answers
116 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
137 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 ...
3
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3answers
108 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
50 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
137 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
121 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 ...