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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14
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2answers
314 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?
-2
votes
6answers
176 views

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 $1/0 = 1/0.$
3
votes
2answers
185 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 ...
1
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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 ...
10
votes
7answers
419 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?
3
votes
2answers
249 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 ...
3
votes
1answer
319 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.
0
votes
1answer
223 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 $\sum_{i=1}^\infty x_i$ ...
3
votes
2answers
479 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 ...
4
votes
2answers
339 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 ...
7
votes
1answer
189 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 ...
2
votes
2answers
186 views

asymptotic infinity

I am very very new to math as a whole, so please excuse my n00biness. I read: ...
4
votes
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 ...
2
votes
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 ...
0
votes
3answers
210 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. ...
10
votes
6answers
453 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 ...
5
votes
2answers
470 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 $12$ ...
2
votes
2answers
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?
4
votes
2answers
113 views

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 ...
0
votes
3answers
302 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 ...
2
votes
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 $\aleph_0$? ...
30
votes
5answers
2k views

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?
2
votes
4answers
986 views

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 ...
6
votes
3answers
224 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, ...
2
votes
6answers
339 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 ...
3
votes
4answers
1k views

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 ...
4
votes
3answers
371 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 ...
6
votes
2answers
328 views

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 ...
0
votes
3answers
324 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 ...
12
votes
0answers
408 views

“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 ...
2
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0answers
82 views

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 ...
0
votes
2answers
140 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, ...
0
votes
2answers
191 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 ...
1
vote
2answers
224 views

Creating the set of natural numbers

I am not a mathematician but an engineer, so I can read some basics of the language proofs are written in. Second I am bad in probability and infinity and my question covers both. So I like to ...
2
votes
1answer
56 views

Statements true for all n Vs. statements true as n->infty

Let P be a statement. What are the necessary and sufficient conditions for the following statement to be true? (P is true $\forall n \in \Bbb N$)$\implies$(P is true as n$\to \infty$) As background ...
2
votes
1answer
135 views

Random infinite binary sequence

What I mean by random infinite binary sequence is an infinite sequence of $0$'s and $1$'s with probability of occurrence in this sequence equal to $1/2$ (all digits being equally likely). How is it ...
11
votes
4answers
18k views

1 to the power of infinity, why is it indeterminate? [duplicate]

I've been taught that $1^\infty$ is undetermined case. Why is it so? Isn't $1*1*1...=1$ whatever times you would multiply it? So if you take a limit, say $\lim_{n\to\infty} 1^n$, doesn't it converge ...
1
vote
0answers
145 views

Relationship between ordinals and rank of well founded relations on $\mathbb N$

I want to understand the relation between ordinals and well founded relations on $\mathbb N$. I found a nice starting point here cut-the-knot/ordinals. Ordinals start like this 0={}, 1={0}, 2={0,1}, ...
2
votes
1answer
204 views

What is the probability of guessing the right number $n$ from all numbers $\mathbb{N}$?

As we all know, $\mathbb{N}$ contains infinitely many numbers. What is the probability of guessing the right number $n \in \mathbb{N}$, i. e. what is $\frac{1}{\infty}$? It is clear that there is a ...
2
votes
1answer
3k views

Infinity = Undefined?

Let's start with the equation $y = |1/(x-1)|$. The positive and negative limit of $x$ at $1$ both approach $+∞$, but at $x = 1$, $y$ is undefined. I know this is because the denominator of the ...
1
vote
0answers
122 views

maximize an objective function with an infinite component

Suppose I have the following maximization problem: $\log\det(\alpha K_p)-c\alpha$ with respect to $\alpha$ with $c$ being a constant and $m$ being the dimension of $K_p$. Here, one of the eigenvalues ...
2
votes
2answers
107 views

Is there an infinite sequence AB, BC, CD, DX, …, YZ

Is it possible to construct an infinite set of ordered pairs of form S = {(A, B), (B, C), (C, D), (D, x), ..., (y, Z)}? Every element (B, C...) must appear only once as the first object in one of the ...
4
votes
1answer
191 views

How to prove that $0.01001100011100001111…$ is not periodic decimal number?

I have the following decimal number: $0.01001100011100001111...$ Notice how whenever we have one 0, we also have one 1, two 0's, two 1's, etc. How do you continue it to infinity and prove that this ...
0
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1answer
107 views
2
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2answers
170 views

Elaboration of infinite, finite and enumerable definition

I am starting to learn some of the basic concepts of math. The concept I am learning now is infinite, finite, and denumerable. I am having trouble understanding the book's definiton. I am hoping if ...
1
vote
3answers
284 views

Random Point on Infinite Line Paradox

I've invented a paradox, or at least I think I have. Here is how it goes: On an infinite line, a point is placed at random. You start at point 0 on the line, and your job is to find the point, but ...
1
vote
3answers
156 views

Is ∞ considered defined?

$\infty$ (Infinity) is not a number, but infinity is considered to be defined, right? There are expressions in mathematics such as: $\frac x0,0^0, \frac\infty\infty,$ which are not defined because ...
5
votes
3answers
2k views

Does the concept of infinity have any practical applications?

I know what you're thinking: "of course it has, for example, it can be used to tell you how many times you can go around a circle". But that isn't really true, now is it? You'd be dead or the world ...
3
votes
1answer
88 views

Are distinctions in definitions of “finite” material in, eg, topology or measure theory?

There are several definitions of "finite", like Dedekind's and Tarski's (Thanks to A.K. for point out the latter - first time I've heard of it): From the Wikipedia entry on Finite Set: (Richard ...
1
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1answer
62 views

Please help me understand this formula for Fourier analysis

I'm a programmer with a poor knowledge of math. Could anyone tell me how to read the infinity above the sigma, and the n=1 below the sigma?