Questions tagged [infinity]

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

A lay person might think they know what infinity is, but a mathematician will ask you "what kind of infinity are you talking about, and in what context?"

See the question Is Infinity a number and in particular this answer for a good discussion on a number of things that people sometimes mean when they talk about "infinity" in mathematical terms.

Good questions about infinity will need to be explicit about the context in which the question is being asked; many of the questions already tagged essentially have the answer "The answer is undefined, because infinity isn't an ordinary number" or "Your question can't be answered without giving more information." And there is probably already a different tag covering whatever field of mathematics the question is about.

Please check the questions What is the result of $\infty-1$? and What is the result of infinity minus infinity? before asking questions about doing arithmetic on infinity.

Questions about fallacies involving infinity are on-topic for this tag.

2415 questions
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A formal proof that a sum of infinite series is a series of a sum?

I feel confused when dealing with ininities of any kind. E.g. the next equation is confusing me. $$\displaystyle\sum^\infty_{n} (f_1(n) + f_2(n)) = \displaystyle\sum^\infty_{n_1=1} f_1(n_1) + \displaystyle\sum^\infty_{n_2=1} f_2(n_2)$$ How do people…
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How can we compare a number with inifinity?

we know that $a<\infty\space\space \forall a\in\mathbb{R}$ but why? how can order in $\mathbb{R}$ include this concept?
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Can one non-cardinal infinity be greater than other non-cardinal infinity?

As far as I know, there are two different notions to the word "infinity" in Mathematics. First notion of infinity has to do with the cardinality of a set: if a set contains infinite number of elements, the set is said to be an "infinite set"; this…
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How can a bijection be made from $\mathbb{N}$ to $\mathbb{Q}$ using diagonalization?

I'm studying Cantor's diagonalization, but something seems unclear to me. There is this table for diagonalization: ╔═══════════════════════╗ ║ X 1 2 3 4 5 ║ ║ 1 1/1 1/2 1/3 1/4 1/5 ║ ║ 2 2/1 2/2 2/3 2/4 2/5 ║ ║ 3 3/1 3/2 3/3 3/4 3/5 ║ ║ 4…
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Friend B and C have eaten zero apples. How many more apples has C eaten?

Friend $A$ claims that he has eaten $1$ apple today. Friend $B$ responds. Congrats, I have eaten $0$ apples, so that is $\infty$ more apples than me. Friend $C$ says, but I have also eaten $0$ apples. So how many times more than you have I done? $0$…
Kasper
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Probability of selecting a number in a repeating decimal series

For example in a infinitely repeating series such as $\frac{110}{111}=0.\overline{990}$, what would be the probability of selecting a 0 in the series generated by the infinitely repeating decimals? I thought that the answer seemed obvious seeing…
dsjoint
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Finding a finite integer in infinite space

An adversary selects an integer k from the set of non-negative integers. Does any algorithm exist that, using only tests for equality or inequality (<, =, >), is guaranteed to find k in finite time?
Eric J.
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Another flavour of Hilbert's Hotel

Many of you have probably heard about Hilbert's Hotel problem. Mr Hilbert owns a hotel with countably infinite amount of one-bed rooms. All the rooms are, of course, taken. A (finite or infinite) group of k people walks in and wishes for…
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How does one prove that two sequences are equal at infinity?

I came up with $e = \sum_{n=0}^\infty \frac 1 {n!}$ (see here) I am now trying to prove that this is equivalent to $\lim_{n\to \infty} {(1+\frac1 n)}^n$ In general, how would one go about such a task? Please do not prove what I stated above, I'd…
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Find the equation of a line parallel to the y-axis, that goes through the point $(\pi,0)$

I have been trying to do this problem and I am very confused. I know the gradient is infinity when any line is parallel to the y-axis, therefore, $y = \infty \cdot x + c$, right ($y = mx + c$ being the general equation a straight line)? We know $y =…
Gigabit
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Is it true that the slope of a vertical line times the slope of a horizontal like don't equal $-1$, even though they're perpendicular?

I know that the slopes of two lines that are perpendicular have a value of $-1$ when multiplied because they're opposite reciprocals (e.g. $5$ and $-{1\over 5}$), but what if there's a horizontal and a vertical line ($x=3$ and $y=-2$). They're…
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Is$\ \infty \times 0$ undefined in the extended real numbers?

And if it is, why? Is it a kind of postulate related to the fact that infinitely many points make a line?
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Comparing density of countable infinite sets by examining the association

The two questions that i am asking are in bold. To be clear, i am talking about whole number here. Having seen 3 is everywhere by Numberphile that shows that almost 100% of the whole the numbers have the digit three in them, i have been thinking of…
v010dya
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Help me understand infinity

I asked a math professor once about infinity and his answer puzzled me. I asked if i had two sets of numbers: A = all the whole numbers in infinity B = all the whole and half numbers (1, 1.5, 2, 2.5 etc...) Is B twice as large as A? He chuckled…
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Is $\infty = \frac{1}{0}$?

Is $\; \infty = \frac{1}{0}$? My teacher says no but he wouldn't explain it. My question is why $\; \infty \neq \frac{1}{0}\;?$ My thinking: Let $\frac{1}{x}=p$ Now as $x$ becomes smaller $p$ gets bigger. Ultimately when $x$ is smallest in…