Use this tag for questions on the concept of infinity. Don't use this tag merely because $\infty$ appears somewhere in the question.

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90
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7answers
9k views

Why is $1^{\infty}$ considered to be an indeterminate form

From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are ...
72
votes
9answers
19k views

Is infinity a number?

Is infinity a number? Why or why not? Some commentary: I've found that this is an incredibly simple question to ask — where I grew up, it was a popular argument starter in elementary school ...
55
votes
11answers
80k views

Why is Infinity multiplied by Zero not an easy Zero answer?

I did a bit of math at school and it seems like an easy one - what am I missing? $$n\times m = \underbrace{n+n+\cdots +n}_{m\text{ times}}$$ $$\quad n\times 0 = \underbrace{0 + 0 + \cdots+ ...
54
votes
7answers
19k views

Infinity = -1 paradox

I puzzled two high school Pre-calc math teachers today with a little proof (maybe not) I found a couple years ago that infinity is equal to -1: Let x equal the geometric series: $1 + 2 + 4 + 8 + 16 ...
16
votes
3answers
2k views

The Aleph numbers and infinity in calculus.

I have a fairly fundamental question. What is the difference between infinity as shown by the aleph numbers and the infinity we see in algebra and calculus? Are they interchangeable/transposable in ...
164
votes
11answers
87k views

What is the result of infinity minus infinity?

What is $\infty - \infty$? Is it $\infty$ or $0$ or what?
214
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13answers
28k views

Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet?

Of course, we've all heard the colloquialism "If a bunch of monkeys pound on a typewriter, eventually one of them will write Hamlet." I have a (not very mathematically intelligent) friend who ...
81
votes
5answers
15k views

Is infinity an odd or even number?

My 6 year old wants to know if infinity is an odd or even number. His 38 year old father is keen to know too.
60
votes
5answers
11k views

Are all infinities equal?

A friend of mine was trying to explain to me how all infinities are equal. For example, they were saying that there are the same amount of numbers between $0$–$1$ as there are between $0$–$2$. The ...
7
votes
6answers
298 views

dirac delta integral with $\delta(\infty) \cdot e^{\infty}$

I have a question about this integral $ \displaystyle \int_{-\infty}^{+\infty} \delta'(x-3)e^{x^2}dx $ by integration by parts I get; $ \displaystyle ...
65
votes
6answers
6k views

Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?

In the book "Zero: The Biography of a Dangerous Idea", author Charles Seife claims that a dart thrown at the real number line would never hit a rational number. He doesn't say that it's only ...
73
votes
3answers
6k views

Are there any series whose convergence is unknown?

Are there any infinite series about which we don't know whether it converges or not? Or are the convergence tests exhaustive, so that in the hands of a competent mathematician any series will ...
66
votes
6answers
4k views

Why is $\omega$ the smallest $\infty$?

I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. ...
17
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4answers
13k views

Types of infinity

I understand that there are different types of infinity: one can (even intuitively) understand that the infinity of the reals is different from the infinity of the natural numbers. Or that the ...
34
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5answers
6k views

How can a structure have infinite length and infinite surface area, but have finite volume?

Consider the curve $\frac{1}{x}$ where $x \geq 1$. Rotate this curve around the x-axis. One Dimension - Clearly this structure is infinitely long. Two Dimensions - Surface Area = ...
17
votes
7answers
12k views

Are there more rational numbers than integers?

I've been told that there are precisely the same number of rationals as there are of integers. The set of rationals is countably infinite, therefore every rational can be associated with a positive ...
17
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6answers
2k views

Partitioning an infinite set

Can you partition an infinite set, into an infinite number of infinite sets?
4
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4answers
6k views

How does the sum of the series “$1 + 2 + 3 + 4 + 5 + 6\ldots$” to infinity = “$-1/12$”? [duplicate]

(I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have ...
48
votes
12answers
30k views

I have learned that 1/0 is infinity, why isn't it minus infinity?

My brother was teaching me the basics of mathematics and we had some confusion about the positive and negative behavior of Zero. After reading a few post on this we came to know that it depends on the ...
28
votes
3answers
2k views

Math without infinity

Does math require a concept of infinity? For instance if I wanted to take the limit of $f(x)$ as $x \rightarrow \infty$, I could use the substitution $x=1/y$ and take the limit as $y\rightarrow 0^+$. ...
32
votes
12answers
3k views

What exactly is infinity?

On Wolfram|Alpha, I was bored and asked for $\frac{\infty}{\infty}$ and the result was (indeterminate). Another two that give the same result are $\infty ^ 0$ and ...
19
votes
5answers
3k views

How many different sizes of infinity are there?

It's pretty straightforward to say that there is an infinite number of different sizes of infinity, but then I thought, "What size of infinity is that?" My thoughts are that the number of unique ...
16
votes
2answers
892 views

Is there an infinite number of primes constructed as in Euclid's proof?

In Euclid's proof that there are infinitely many primes, the number $p_1 p_2 ... p_n + 1$ is constructed and proved to be either a prime, or a product of primes greater than $p_n$. Trivially, we ...
8
votes
2answers
4k views

Why do the rationals, integers and naturals all have the same cardinality?

So I answered this question: Are all infinities equal? I believe my answer is correct, however one thing I couldn't explain fully, and which is bugging me, is why the rationals $\mathbb Q$, integers ...
13
votes
3answers
101k views

Whats infinity divided by infinity?

This should be a simple question but i just want to make sure. I know from infinity/infinity is undefined. However if we have 2 equal infinities divided by each other it would be 1? And if we have ...
20
votes
10answers
9k views

Does infinity and zero really exist?

I'm not going to prove something, this is just a question. From the first day which I went to University until now I had some root problems in some basic mathematical assumptions and concepts. Please ...
12
votes
3answers
11k views

Is $\tan(\pi/2)$ undefined or infinity?

The way I have understood, $0/0$ is undefined or indeterminate because, if $c=0/0$ then $c\cdot 0=0$, where $c$ can be any finite number including $0$ itself. If we also observe a fraction $F=a/b$ ...
6
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1answer
1k views

Cardinality of a set that consists of all existing cardinalities

What is the easiest way to prove (if possible, without using ordinals etc. as my current math understanding of set theory counts only cardinals, and countable & uncountable sets) that the number ...
17
votes
5answers
14k views

One divided by Infinity?

Okay, I'm not much of a mathematician (I'm an 8th grader in Algebra I), but I have a question about something that's been bugging me. I know that $0.999 \cdots$ (repeating) = $1$. So wouldn't $1 - ...
9
votes
2answers
40k views

Limit approaching infinity of sine function

I'd like to ask a question which I have been reflecting on for some time now. What is the limit of: $f(x) = \sin(x)$ as $x$ tends to infinity? As we know, the function has a definite value for each ...
9
votes
8answers
4k views

What is larger — the set of all positive even numbers, or the set of all positive integers?

We will call the set of all positive even numbers E and the set of all positive integers N. At first glance, it seems obvious ...
4
votes
4answers
818 views

What infinity is greater than the continuum? Show with an example

The diagonal argument establishes that the continuum is greater than countable infinity. What is an example of the next infinity (or any greater infinity) and how can it be shown that there is no 1:1 ...
1
vote
7answers
6k views

Why do we say the harmonic series is divergent? [duplicate]

If we have $\Sigma\frac{1}{n}$, why do we say it is divergent? Yes, it is constantly increasing, but after a certain point, $n$ will be so large that we will be certain of millions of digits. If we ...
42
votes
12answers
3k views

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
35
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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?
5
votes
3answers
3k 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 ...
5
votes
1answer
324 views

“Real” cardinality, say, $\aleph_\pi$?

Is there any meaningful definition to afford for $\aleph_r$ (as in cardinality) where $r\in\mathbb{R}^+$? $r\in\mathbb{C}$? What about $\aleph_{\aleph_0}$? Can we iterate this? ...
12
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2answers
3k views

Are there uncountably infinite orders of infinity?

Given a set $S$, one can easily find a set with greater cardinality -- just take the power set of $S$. In this way, one can construct a sequence of sets, each with greater cardinality than the last. ...
4
votes
3answers
527 views

Comparing infinite numbers

Suppose you have 2 infinite numbers, say $A$ and $B$. $A$ is an element of the hyperreals, so that $A$ is greater than every real number. $B$ is the size of the set of natural numbers, $\aleph_0$ ...
2
votes
3answers
462 views

Why can't you count real numbers this way?

Sorry but this is probably a naive question. Why can't you generate real numbers by a*10^b, the same way as rational numbers by a/b? a and b could be integers so that you would start counting real ...
1
vote
1answer
313 views

Division by zero [duplicate]

Possible Duplicate: Division by $0$ Everyone knows that $(x/y)\times y = x$ So why does $(x/0)\times 0 \ne x$? According to Wolfram Alpha, it is 'indeterminate'. What does this mean? ...
18
votes
3answers
843 views

What good is infinity?

I am becoming increasingly convinced that Wildberger's views are, if a little bizarre, at least not hopelessly inconsistent. When I was reading the comments in the video following (MF17), somebody ...
33
votes
8answers
5k 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. ...
13
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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 ...
18
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4answers
5k views

Two paradoxes: $\pi = 2$ and $\sqrt 2 = 2$ [duplicate]

Possible Duplicate: Is value of $\pi = 4$? Can anyone explain how to properly resolve two paradoxes in this YouTube video by James Tanton?
9
votes
2answers
6k views

Infinite expected value of a random variable

How can a positive random variable $X$ which never takes on the value $+\infty$, have expected value $\mathbb{E}[X] = +\infty$?
14
votes
1answer
643 views

Is there an absolute notion of the infinite?

Skolem's paradox has been explained by the proposition that the notion of countability is not absolute in first-order logic. Intuitively, that makes sense to me, as a smaller model of ZFC might not be ...
12
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1answer
2k views

Why does Cantor's diagonal argument not work for rational numbers?

If we map every integer to a string that represents a rational number, and produce a number different from all the ones listed, we are essentially following Cantor's algorithm. But why does it not ...
7
votes
3answers
512 views

Why $\frac{1}{\infty } \approx 0 $ and $ \frac{1}{0} = {\infty}$?

First I have checked the search option but found nothing relevant to my problem and also level of math. I just started learning the language of mathematics, on my own and I have trouble understanding ...
0
votes
1answer
227 views

Finding a limit using arithmetic over cardinals

What is the value of: $$\lim_{n \to \infty} \frac{n}{2^n} (n \in \mathbb{N})$$ It seems to me that I can use L'Hopital's rule, but does that rule take into account types of infinity? More precisely, ...