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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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 ...
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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 ...
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3answers
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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 ...
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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.
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10answers
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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+ ...
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6answers
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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 ...
174
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14answers
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Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet? [closed]

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 ...
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5answers
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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 ...
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4answers
8k 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 ...
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11answers
37k views

What is the result of infinity minus infinity?

What is $\infty - \infty$? Is it $\infty$ or $0$ or what?
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13answers
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 ...
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3answers
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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 $y\rightarrow 0^+$. ...
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5answers
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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 = ...
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Do 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 ...
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6answers
3k 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. ...
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5answers
3k 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 - ...
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7answers
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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 ...
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4answers
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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 ...
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3answers
2k 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 ...
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2answers
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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 ...
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3answers
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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 ...
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3answers
1k 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 ...
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1answer
796 views

Cardinality of a set that consists of all existing cardinalities

I have taken a look at the following topics: number of infinite sets with different cardinalities Cardinality of all cardinalities Are there uncountably infinite orders of infinity? Types of ...
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2answers
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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 ...
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2answers
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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. ...
3
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2answers
377 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, $\aleph_0$ ...
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1answer
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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? ...
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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 ...
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7answers
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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 ...
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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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6answers
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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
1k views

Partitioning an infinite set

Can you partition an infinite set, into an infinite number of infinite sets?
14
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1answer
527 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 ...
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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 $F=a/b$ ...
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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?
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1answer
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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 ...
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3answers
460 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 ...
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1answer
212 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, ...
16
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2answers
640 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 ...
3
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4answers
862 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 ...
3
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4answers
758 views

Polynomial of degree $-\infty$?

I'm reading E.J Barbeau Polynomials. I'm in a page where he asks a polynomial of degree $-\infty$. Then I thought about $77x^{-\infty}+1$, but when I went for the answers, the answer to this question ...
3
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4answers
615 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 ...
3
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0answers
193 views

Proof that $ -1 = \infty $ . [duplicate]

Possible Duplicate: Infinity = -1 paradox $1+2+4+8+16+\ldots = \infty$ $LHS=1(1+2+4+8+16+\dots)$ $LHS=(2-1)(1+2+4+8+16+\ldots)$ $LHS=(2+4+8+16+32+\ldots)-(1+2+4+8+16+\ldots)$ ...
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0answers
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An intuitive reasoning for 1+2+3+4+5… + ∞ = -1/12? [duplicate]

I was just watching this video: http://www.youtube.com/watch?v=w-I6XTVZXww In it, a professor working at the Nottingham University( Dr. Ed Copeland I think) shows how 1+2+3+4+5....+ ∞ = -1/12 Is this ...
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1answer
123 views

Are there any infinites not from a powerset of the natural numbers?

With the cardinality of the natural numbers as $|\mathbb{N}| = \aleph_0$ and its powerset as $|\mathcal{P}(\mathbb{N})| = 2^{\aleph_0}$, the continuum hypothesis and the axiom of choice says that ...
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7answers
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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 ...
21
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4answers
992 views

You are standing at the origin of an “infinite forest” holding an “infinite bb-gun”

I use stories like these to develop intuition... or perhaps to destroy it. I have my own answers in mind, but I want to see if I have made any mistakes... You are standing at the origin of an ...
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8answers
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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
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1answer
166 views

Infinite square-rooting

$ \lim_{n\to\infty} {\sqrt{1+{\sqrt{2+{\sqrt{\cdots +\sqrt{n}\ }\ }\ }\ }\ \ }\ } = ? $ Either closed answer or an upper bound would help.
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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 ...