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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$\pi$ and $\ln4$ relations. Even and Odd alternating sums.

Tonight, playing around on WolframAlpha, I discovered that the alternating sum of the odd numbers is $\frac\pi4$ and the alternating sum of the even numbers is $\frac{\ln4}4$ Are there any known ...
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2answers
539 views

Comparing infinite binary fractions to infinite decimal fractions

I'm trying to understand the cardinality between the set of all infinite binary (base-2) fractions and the set of all infinite decimal (base-10) fractions. I can easily think of infinite binary ...
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4answers
2k views

Are irrational numbers completely random?

As far as I know the decimal numbers in any irrational appear randomly showing no pattern. Hence it may not be possible to predict the $n^{th}$ decimal point without any calculations. So I was ...
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7answers
549 views

When does it make sense to say that something is almost infinite?

I remember hearing someone say "almost infinite" on one of the science-esque youtube channels. I can't remember which video exactly, but if I do, I'll include it for reference. As someone who hasn't ...
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2answers
78 views

Limit approaching infinity-related question

Why is $$\lim_{x\to\infty}\frac{x^2}{1+x^2}=1?$$
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1answer
444 views

zero raised to infinity

I encountered a question where the only condition stated that $t>0$ and was then asked to compare these two quantities $0^t$ $t^0$ The scope of $t$ is $(0,\infty)$ and hence for infinity 1.) ...
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3answers
407 views

Equinumerosity of infinite sets

Key issue: For infinite sets, does the existence of a bijection mean they have the same number of elements? For example, does the set of natural numbers N = {1,2,3,4...} have the same number of ...
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4answers
147 views

Infinity and structures

Do you know any case (example) where an "infinite" object with a structure (say, an infinite group) cannot be extended (in the sense of adding elements) in any way without it no longer having the ...
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2answers
464 views

Proof of a Property of Vertical Asymptotes

I'm trying to understand a proof in my Calculus textbook of the following theorem: Let the functions $f$ and $g$ be continuous on an interval containing $c$. If $f(c) \neq 0$, $g(c) = 0$, and ...
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1answer
303 views

Are the number of terms in an infinite series even or odd?

This question arose after I saw a youtube-vid where Grandi's series was discussed.It seems that the sum of the series will be 0 for an even, and 1 for an odd number of terms, where a term is defined ...
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1answer
78 views

Generlized Büchi Games and Closed under superset Muller Games

For a unique infinite play $p$ in a 2-Player game $G=(V_0,V_1,E)$. Let $$ \inf(p) \subseteq V_0 \cup V_1 $$ be the set of vertices which occur infinitly often in $p$. Generlized Büchi (GB) Games ...
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230 views

How many centers in a infinite connected graph?

I find a very interesting concept "center" while learning basic graph theory. A center of graph $G$ is a vertex with the minimal greatest distance (eccentricity) to other nodes in $G$. Now I’m curious ...
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72 views

Regarding playing an infinite number of games that could last infinitely long amounts of time

So after watching the last Stanley cup game, a problem popped up in my head for which I have no solution. Say we have a game, like a hockey game, that has the possibility of going on forever. Of ...
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1answer
108 views

Infinite Parity Function

I was looking at this problem, and I have a solution for a finite board with $2^n$ squares, that I want to extend to a countably infinite board. Label the squares from $0$ to $2^n-1$. Consider the ...
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2answers
157 views

Can you prove that function is positive for certain values using limit of function?

Here's example: Prove that $x-\frac{1}{x^2} \geq 0$ for every $x \ge 1$ I know that this can be done using elementary algebra, but in other cases it's not that simple. Can I prove this inequation ...
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441 views

Maximum Number in an Infinite set

Given an infinite set of random integers, is there a largest element? In other words is maximum as a concept inherently tied to finite sets?
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4k views

What's the difference between Complex infinity and undefined?

Can somebody please expand upon the specific meaning of these two similar mathematical ideas and provide usage examples of each one? Thank you!
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315 views

Solving for $x$: $1=\frac{1}{x}+\frac{1}{1+\frac{1}{x}}+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}+\cdots$

How can I solve for $x$: $$1=\cfrac{1}{x}+\cfrac{1}{1+\cfrac{1}{x}}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{x}}}+\cdots$$ Any clues?
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Why is $1 =1 $? why is it so why cant be $1 =$ something else? [closed]

It may sound stupid but why is $1=1$ or $n=n$ if thats the case does $1/0 = 1/0.$
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2answers
187 views

Are real numbers also hyperreal? Are there hyperreal $\epsilon$ between $-a$ and $a$ for any positive real $a$?

The set of all hyper-real numbers is denoted by $R^*$. Every real number is a member of $R^*$, but $R^*$ has other elements too. The infinitesimals in $R^*$ are of three kinds: positive, negative ...
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0answers
60 views

Meaningful measures for comparing infinite dimensional geometric objects

I have two infinite-dimensional convex polytopes, call them $A$ and $B$. I know that $B$ is completely contained within $A$, and I want to say something meaningful about their relative sizes. From ...
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7answers
420 views

Find $\lim_{x\to-\infty}{x+e^{-x}}$

I have this exercise in my worksheet: $$\lim_{x\to-\infty}{x+e^{-x}}$$ I am always ending up with $-∞+∞$ or $\frac{∞}{∞}$. It says the answer is $+∞$, but how can I get that?
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2answers
257 views

how to know the set is finite, countable or uncountable

I am trying to understand whether the set is finite, countable, or uncountable. $$\{x \in\Bbb Q \mid 1<x<2 \} \qquad\text{is countable. }$$ but i dont understand why though. is it countable ...
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1answer
323 views

Why does Michio Kaku say that $\frac{1}{0} = \infty$?

Why does Michio Kaku say that $\frac{1}{0} = \infty$? http://youtu.be/AJ4zlvqOtE8?t=4m43s Instead of $\frac{1}{0}$ that's not defined, so we don't know.
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234 views

Absolute value of infinite sum smaller than infinite sum of absolute values

A question emerging from an exercise in Ok, E. A. (2007). Real Analysis with Economic Applications. Princeton University Press. The exercise consists in showing that if $\sum_{i=1}^\infty x_i$ ...
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489 views

Taking the limit of an integral using residues, why is this wrong?

I have the integral $\lim\limits_{R\to\infty}\int_{-R}^{R} \frac{\cos(x)}{x^2+a^2} dx$ where $a$ is a positive real number. The strategy was to evaluate the limit of the integral on the boundary of a ...
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2answers
341 views

Singularities of $f(z)=z/\cos(z)$

Regarding complex functions (in complex variables), I was wondering why the function $g(z)= \cos(z)$ has a singularity at $z = \infty$ but $f(z)= \dfrac{z}{\cos(z)}$ does not. I am a bit confused ...
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191 views

What happens to the infinite monkey theorem when there are an infinite number of keys on the typewriter?

What happens to the infinite monkey theorem when there are an infinite number of keys on the typewriter? So what is the probability of a finite string of keys like the works of Shakespeare being typed ...
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2answers
194 views

asymptotic infinity

I am very very new to math as a whole, so please excuse my n00biness. I read: ...
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4answers
158 views

Dividing Two Infinities

I am Curious if the following is mathematically correct: Let $a$ be the infinite set of all nonnegative integers $0,1,2,3...$. Let $b$ be the infinite set of all nonnegative EVEN integers ...
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1answer
86 views

Is there an element in $^* \Bbb N$ is Dedekind-infinite?

One definition of a finite set is that it can be injected into an initial segment of $ \Bbb N$, thus any $n$ in $\Bbb N$ is finite. Accordingly, if it's legitmate to define every element in $^* \Bbb ...
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214 views

Can countability coexist with infinity?

This question concerns the countability of the real numbers. First I will show how I count the numbers between 0 and 1 on the real line. It is done by reversing digits behind the coma, so that e.g. ...
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457 views

A question with infinity

I am a sophomore in high school and my math teacher did a very short lesson on infinity, here's how it went: (Try to solve each part yourself the first two are easy) Part 1 You have an inf. number ...
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488 views

Subtracting two infinities

I am Curious if the following is mathematically correct: Let $a$ be the infinite set of all nonnegative integers $0,1,2,3...$. I take from $a$ some of its elements, say integers $10$, $11$, and $12$ ...
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152 views

What is wrong with this proof that $\pi=\infty$

Does this "proof" show that $\pi =\infty$??? http://www.academia.edu/1611664/Sum_of_an_Infinite_sequence_PAPER Is there something wrong with the subbing in of infinity at the end of the paper?
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Infinite shots fired in a lattice forest

A hunter is standing in the center of an infinite 2D forest. There are point trees at all the integer lattice points. The hunter fires a gun with a bullet of zero width in a random direction. He ...
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3answers
306 views

How big is the size of all infinities?

"Not only infinite - it's "so big" that there is no infinite set so large as the collection of all types of infinity..." What does exactly mean? How many infinities are there? I've heard there are ...
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1answer
79 views

What are the factors of $\aleph_0$?

Extend the system of positive natural numbers with $\aleph_0$. Then we have: $$\aleph_0 = \aleph_0\cdot n,\quad \forall n \in \mathbb{N}^+$$ Does it make sense to talk about factors of $\aleph_0$? ...
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Which infinity is meant in limits?

For example, when we write $\lim_{x\rightarrow \infty} f(x)$ - which infinity is meant and why? Countable? If uncountable - which and why?
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4answers
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infinity times infinitesimal - what happens?

So what happens if we multiply infinite number by. Infinitesimal number? Like $dx \times \infty$ where $dx$ is treated as in one-dimensional integration. Also, can we divide infinite number by ...
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225 views

Use of infinity as an “idealistic approximation”

There have been several recent posts about the work of N. J. Wildberger, a finitist who seems to think that mathematics should only focus on things that have some sort of "real world" connection, ...
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6answers
341 views

prove that , $2+2+2+2+2+ \cdots= 1+1+1+1+1+\cdots$

how can we prove that ?? i think they are equal but a friend say that they are not equal my argument is $$1+1+1+1+1 + \cdots = \infty$$ $$2+2+2+2+2+\cdots = (1+1) + (1+1) + \cdots = (1+1+1+1+1 ...
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4answers
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Non-existence of irrational numbers?

I realize the title of my question will probably cause the raising of some eyebrows, so let me explain. Not sure whether to file this under "math" or "philosophy". This also might be able to be ...
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3answers
377 views

Why the need of Axiom of Countable Choice?

Two theorems: $(1)$ Countable Union of Countable Sets is Countable $(2)$ Cartesian Product of Countable Sets is Countable Linked are the formal proofs on Proofwiki. I do not understand why they ...
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Is the proper class of all ordinals equivalent to the potential infinity of pre-Cantor times?

My understanding is that the class of all ordinals is, by definition a proper class. This in the end is done to avoid a paradox: the collection of all sets would be paradoxical if you allow it to be a ...
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334 views

Why I can't calculate $0*log(0)$ but can $log(0^0)$

I got this doubt after some difficult in programming. In a part of code, i had to calculate: $$ x = 0 * Log(0) \\ x = 0*-Inf $$ and got $x = NaN$ (in R and Matlab). So I changed my computations to ...
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“Infinito”, a combinatorial game with infinite width game-tree

I recently designed a combinatorial game (sequential game of perfect information) with an infinite branching factor, that is it has a game-tree of infinite width. I'm wondering how is it possible to ...
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Paradox of Infinity? [duplicate]

If a series such as '$a$' below adds to infinity: $a = 1 + 2 + 4 + 8 + 16 + \cdots\to \infty$ Multiplying '$a$' by $2$ yields: $2a = 2 + 4 + 8 + 16 + \cdots\to \infty$ However when I subtract ...
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143 views

Infinity/exponential problems.

I want to evaluate $$\int_{0}^{\infty} e^{(i\alpha-1)x}\,\mathrm dx,$$ where $i$ is the imaginary number. $$\left [ \frac{e^{(i\alpha-1)x}}{i\alpha-1}\right]_0^{\infty}$$ At this point, I beleive, ...
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192 views

Understanding limits at infinity with regard to the definition of a limit

This is sort of a follow up to my previous question Say you have $$ \lim_{x\to +\infty} f(x) $$ where $f : \mathbb{R} \to \mathbb{R} , x \in \mathbb{R}$ What exactly does this mean? From the last ...