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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2
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2answers
56 views

Is $\frac00=\infty$? And what is $\frac10$? Are they same? Does it hold true for any constant $a$ in $\frac{a}0$

I know that $\lim_{x\to0}\frac{x}{x}=$ 1. But in my text book, it is written that it is $\infty$ and even $\frac10=\infty$. But how is it possible? And are they both same? What is the difference ...
-4
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2answers
68 views

What application/deeper meaning do countable and uncountable infinities have? [on hold]

Georg Cantor proved that there are two different infinities but what application does this proof have? Is this result used in some other more useful theorem?
1
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2answers
42 views

What is the value of $0\times \infty$? (in $[0, +\infty]$)

In $[0,+\infty)$, $0^+\times +\infty$ can be any number in $(0,+\infty)$ so is undetermined; (in which $0^+$ means when a variable approaches to $0$). Because $\lim_{x\rightarrow ...
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1answer
89 views

Is 1 + 1 + 1 … a finite number? [on hold]

Just a simple question whether the "number" $x = 1 + 1 + 1 ...$ is a finite number. On one hand, you can think of x as the result of a never-ending process; $0$ is finite and $n + 1$ is finite when ...
5
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4answers
132 views

Can a number have an uncountably infinite amount of digits?

I'm an extreme mathematical layman, so please excuse the probable ignorance and awkward phrasing of this question. Is there such thing as a kind of number which has an uncountably infinite amount of ...
-5
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1answer
55 views

What is ordinal expression of $\infty$? [closed]

$\infty$ - is cardinal expression? Origin of a line ray is ordinal expression of $\infty$, if infinity distance from the origins to the infinity of line ray?
5
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7answers
1k views

One divided by infinity is not zero? [duplicate]

I know that $\frac{1}{\infty}$ is undefined. But my question is - can we say that $\frac{1}{\infty}\neq0$ ? I've got some idea how to explain that: Let's say we have a random-number generator that ...
-1
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2answers
97 views

Does 1/∞ = 0.0̅01? Can some tell why not? [closed]

im just an 8th grade math student going into geomotry 1, but im curious to know if 1/∞ = 0.0̅01 (or 0.000...1). If that is so, then because 0.9̅9=1, then 0.0̅01 would be 0, thus proving that 1/0=∞ its ...
1
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2answers
51 views

Third axiom of Kolmogorov axioms

Let us define for a countably infinite set $S$ of real numbers that can be enumerated as $x_1,x_2,\cdots$, $$P(S) = \sum_{x \in S}p(x) = \sum_{i=1}^\infty p(x_i) = \lim_{n \to \infty}\sum_{i=1}^n ...
-4
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3answers
85 views

If you remove an element from an infinite set, does the set remain infinite? [closed]

Say you have the infinite set of all positive integers. If you remove the number 2, is this set still infinite? If yes, why? If no, also why?
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2answers
78 views

How to find the limit of a sequence?

Question: If $0 < x < \frac{\pi}{2}$ and $f_k(x) = \tan(x)+\frac{1}{2}\tan(x/2)+ ...+\frac{1}{2^k}\tan(x/2^k)$. In Sigma Notation: $$f_k(x) = \sum_{n=0}^k \frac{1}{2^n}\tan\frac{x}{2^n}$$ ...
1
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2answers
93 views

$\ln{\left(\frac{1}{0}\right)} = -\infty$?

I have shown it using a theorem that I made, but I am not sure, as $\lim_{\alpha \to 0^{-}}{\left(\frac{1}{\alpha}\right)} = -\infty$, and $\lim_{\alpha \to 0^{+}}{\left(\frac{1}{\alpha}\right)} = ...
24
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2answers
841 views

W. Mückenheim claims a severe inconsistency of transfinite set theory; true? [closed]

The abstract for a paper on arxiv.org (http://arxiv.org/pdf/math/0408089v3.pdf) reads (with my emphasis): "Transfinite set theory including the axiom of choice supplies the following basic theorems: ...
0
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3answers
155 views

Is infinity an element of the natural numbers? [duplicate]

I am wondering, whether $\infty \in \mathbb{N}$.
15
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4answers
1k views

Are there more transcendental numbers or irrational numbers that are not transcendental?

This is not a question of counting (obviously), but more of a question of bigger vs. smaller infinities. I really don't know where to even start with this one whatsoever. Any help? Or is it ...
1
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1answer
25 views

Group action of $GL(2, F)$ on the projective line $P(F)$

I refer to section 8.3, page 119 of Algebra, A Computational Introduction by John Scherk. It is about group action of $GL(2, F)$ on the projective line $P(F) = F \cup \{\infty\}$. Given a matrix ...
1
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2answers
58 views

Calculating $\lim_{(n,x) \rightarrow (\infty, 1^+)}x^n$?

Obviously, when $|x| \leq 1$, $\lim_{n \rightarrow \infty}x^n$ has a value such that $|x^n| \leq 1$, and $|x^n| \rightarrow \infty$ for $|x|>1$. However, this led me to wonder, what would happen ...
0
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1answer
32 views

The probability of a number appearing in an approximation of an irrational number?

I was wondering if for the number Pi some numbers are more likely to appear than others, for example 3.141594 ... The number 1 appears twice does that mean that the probability for the number 1 ...
1
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1answer
94 views

Can we formally distinguish between actual and potential infinities?

Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the ...
5
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4answers
146 views

Circle revolutions rolling around another circle

I just watched this video, and I'm a bit perplexed. Problem: ...
8
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2answers
128 views

Does $\tan (x)$ equal $\frac{-1}{x-\frac{\pi}{2}}+\frac{-1}{x+\frac{\pi}{2}}+\frac{-1}{x-\frac{3\pi}{2}}+\frac{-1}{x+\frac{3\pi}{2}}+…$?

I set my Year 12 students a question involving the sums of rational functions $\frac{1}{x-n}$. The graph of a sum of these functions looks an awful lot like a tan graph. This led me to ask: Does ...
1
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1answer
97 views

Infinity xkcd style: can a turing machine exist?

I recently read this xkcd comic. It's about a guy who simulates a universe by a Turing machine (specifically, Rule 101, a cellular automaton), by laying down infinite rows of rocks, each row ...
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0answers
45 views

Does the value of infinity equals to negative one? [duplicate]

Consider a series, 1+2+4+8+16+32+64....... The next number is twice the previous number, The sum the series is clearly infinity. Now lets multiply the series by one. ...
1
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1answer
43 views

The real projective line and $1/\infty$

so I came up with this idea: the real projective line defines that $\infty = - \infty$. What if I divide any value $x$ (not equal to $\infty$) by infinity? Would that be 0? or "something" between ...
3
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7answers
2k views

After switching a lamp on and off infinitely many times in one minute, is it on or off? [duplicate]

So we have a lamp. It's switched on. let's represent its state of being switched on with associating it with $1$ and being off with $-1$. after half a minute passes, you turn it off, after another ...
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2answers
39 views

Possible proof for the multiple sizes of infinity [closed]

One can easily relate all regular polygons to the triangle: triangle = 1 triangle square = 2 triangles pentagon = 3 triangles hexagon = 4 triangles and so on and so forth… A circle is basically ...
0
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5answers
132 views

What we get if we add 1/2 infinite times [closed]

I want to know if this is correct We have this sums: $$S1=1-1+1-1+1-1+1-1+1-1...=\frac12$$ $$S2=1-2+3-4+5-6+7-8...=\frac14$$ $$S3=1+2+3+4+5+6+7+8...=-\frac{1}{12}$$ If we take ...
3
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7answers
151 views

The meaning of the symbol $\infty$ in Spivak's calculus book

Spivak in "Calculus" writes ... symbols of $\infty$ and $- \infty$ are purely suggestive: there is no number $``\infty"$ which satisfies $\infty \geq a$ for all numbers $a$. What is the meaning ...
1
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2answers
64 views

I am trying to find the limit of P(x)

When I am looking for a $\lim\limits_{x \to -1} P(x)$ where P(x)$= \sum \limits_{n=1}^\infty \left( \arctan \frac{1}{\sqrt{n+1}} - \arctan \frac{1}{\sqrt{n+x}}\right) $ do I have to ignore a ...
3
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4answers
720 views

Alternate method to calculate an infinite string of numbers that's not $\pi$, and contains any string

So, rather than using $\pi$, is there any way that isn't overly complicated, (and can be calculated on a computer without taking a year) in which I could generate an infinite string of numbers that ...
3
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3answers
174 views

Infinity indeterminate form that L'Hopital's Rule: $\lim_{x\to0^+}\frac{e^{-\frac{1}{x}}}{x^{2}}$

When I tried to find the limit of $$ \lim_{x\to0^+}\frac{e^{-\frac{1}{x}}}{x^{2}} $$ by applying L'Hopital's Rule the order of denominator would increase. What else can I do for it?
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1answer
48 views

How to show uniform convergence of series

Let $$f(t) = \sum_{k=0}^\infty ke^{-t\sqrt{k}}u_k$$ for $t \in (0,\infty)$, where the $u_k$ is such that $\sum \sqrt{k}u_k$ converges, but we know nothing about the convergence of $\sum ku_k$. How do ...
1
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2answers
40 views

How much information does learning this interval give you?

Let's say you have a number $x$, and a priori, you know that $x \in [0, 1)$ (each value from 0 to 1 is equally likely.) Then a wizard comes and tells you that $x \in [a, b) \subseteq [0, 1)$. How much ...
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2answers
43 views

Complex infinity when proving divergence

My calculus course book (Adams' Calculus) does not explain why $(-1)^n$ diverges (it just says "$(-1)^n$ simply diverges"), and I tried to see why it diverges by taking its limit as $n$ approaches ...
1
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2answers
54 views

Mandelbrot Set area

If there are an infinite amount of details that can be found in a Mandelbrot set, shouldn't the Mandelbrot Set have an infinite area? Supposedly the area of a Mandelbrot set is 1.5065918849 ± ...
6
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5answers
2k views

Are there fewer reals on $(0, 1)$ than on $(1,\infty)$?

I know that the cardinality of the sets of real numbers $(0, 1)$ and $(1, \infty)$ are equal. So what is the fallacy in this argument? For every real on $(0, 1)$, we can add any integer $n$ to it ...
1
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1answer
28 views

If $\dim(V_F)$ is Infinite, Does It Follow $\dim(\operatorname{Hom}(V_F, W_F)) \ge |F|$?

Part of the proof that $\dim(V^*_F) > \dim(V_F)$ for an infinite dimensional space is that $\dim(\operatorname{Hom}(V_F, F)) \ge |F|$ (i.e $\dim(V^*_F) \ge |F|$). See for example Dual space ...
2
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6answers
102 views

More numbers between $[0,1]$ or $[1,\infty)$?

There are infinitely many real numbers between any two real numbers, therefore there are infinitely many real numbers in the range $[0,1]$ as there are in $[1, \infty)$. In a mathematical sense, are ...
3
votes
1answer
55 views

Improper integral: why $\int_0^1(x^2+ x^{1/3})^{-1}\,dx$ is convergent and not $\int \frac{1}{x^2}\,dx$ ???

How do I show that $\int_0^1(x^2+ x^{1/3})^{-1}\,dx$ converges? I assume you show it on $(0,1]$. Can't seem to get my head around why this would be true.
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0answers
33 views

For Vector Spaces V and W with one Infinite Dimensional , is Hom(V, W) Isomorphic to Hom(W, V)?

If V and W are both finite then clearly Dim (Hom(V, W)) = Dim(V).Dim(W) = Dim(Hom(W, V)) so they are isomorphic. I'm not so sure if one is infinite. An "infinite matrix" construction for a linear ...
3
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2answers
67 views

What are some good reasonably rigorous texts on the mathematics of infinity?

The Infinite Book is too light and not focused enough on the mathematics of infinity, and Everything and More: A Brief History of Infinity has too much focus on the history of infinity instead of the ...
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2answers
42 views

What is the correct mathematical notation for something comprised of the sum of constituents n where n is infinite?

I am trying to figure out what the correct mathematical notation would be for something like the following: I want to describe that the value V of a company is equal to sum of parameters P at any ...
3
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2answers
59 views

Is my argument correct to solve this textbook problem?

The problem is from M.Bona's "A Walk through Combinatorics", Ch1 Prob 13: There are infinitely many pieces of paper in a basket, and there is a positive integer written on each of them. We know ...
1
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3answers
60 views

Find a limit without using L'Hopitals rule 9

Can someone please show me how to do this without using L'Hopitals rule: $$\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^x$$ I know the limit is $e^a$, but I would like to know the steps taken to ...
1
vote
1answer
39 views

What is the limit of the below functions when n tends to inifinity?

What is the value for the functions in the image when limit n tends to infinity?. Also what is the asymptotic complexity (big $O$ notation) for all the four functions?. $$\begin{aligned}f_1(n) &= ...
2
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2answers
110 views

derivative of x^x^x… to infinity?

I am a 12th grade student, and I am afraid that in realistic terms this question might not even make sense because of the infinities that have to be dealt with. However, in my attempt to calculate ...
0
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3answers
71 views

Why does $e^{\frac10}\neq e^{\frac1{-0}}$?

I was unable to explain why this fails? I asked to it many peers and they too can't. I faced this situation when solving a kind of integration problem. Consider $x=-x$ Then $x=0$ That is, $0=-0$ ...
0
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3answers
222 views

Do different infinities have different base representations?

I've been wondering about different types of infinity; e.g., $\aleph_0,\aleph_1$, e.t.c.; where $\aleph_0$ represents the smallest infinity, the countable infinity (e.g., the cardinality of the ...
4
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3answers
82 views

Proof of $\sum_{x = 1}^\infty \frac{1}{x}$'s divergence by absurdity?

(From this site.) The following argument purports to show that the series $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} \dots = 0$. It begins with the harmonic series. $$ \begin{aligned} \sum ...
2
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1answer
148 views

determinant of infinitely large matrix by decomposition

Read the too long didnt read version in bold before going into the finer detail. The overall point is that when I decompose this matrix to try and find its determinant I get an answer that doesn't ...