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 there a scientific name for 0.infinity?

First of all I want to say that when coming to math - I know absolutely nothing - so please forgive me if my question is not "scientifically" correct, if it is not "syntax-correct" - or even too ...
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1answer
118 views

Terminology for a property that holds in the finite but not infinite case?

(I apologize if this is a duplicate, but I don't know what terms to search for. Please feel free to close this if this has already been asked.) There are some properties of finite objects that don't ...
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2answers
458 views

Infinite palindromic numbers

Today’s Abstruse Goose comic got me thinking: Does an “infinite palindromic” number (other than the obvious $x\times1.\overline1$) make sense? In any conventional number system, the answer (as far ...
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562 views

Sierpinksi like triangle construction. How to find the number of triangles in each iteration?

So here is the question: If we look at the Sierpinski triangle (left column of attached image) and think about how many triangle's it takes to make the shape at each iteration we can get the sequence ...
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1answer
918 views

speed of convergence to infinity

Lets take for example $\lim_{x\rightarrow\infty} \log(x)$, from a mathematical point of view this is $+\infty$, but from a logical point of view it's clear that $x$ converges to $+\infty$ much more ...
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Practical Use of Series Expansion at $x=\infty$

Asking WolframAlpha on certain functions, it happens that you get a series expansion at $\infty$. Thinking of the expansion as an approximation of the function in the vincinity of a point $a$, like in ...
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2answers
117 views

Subset of infinite set

A point. There is an infinite number of lines intersecting it, but this is less than the number of possible lines. How do we represent this in mathematical notation?
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4answers
2k views

Examples of continuous growth rates greater than exponential

I read on Wikipedia that growth rate of a function can sometimes be greater than exponential. Can you give me some examples of such functions (preferably continuous ones)? Obviously $x^x$ grows ...
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881 views

Repeating digits in $\pi$

As $\pi$ has infinite digits in its decimal expansion, one could argue that its digits will repeat after a finite number of digits. If so, it is a rational number. What's wrong with this argument?
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1answer
559 views

Am I thinking about infinitesimals correctly?

I was all set to begin Calculus 2 when I thought, "I should have a more intuitive sense of what's happening with differentials before I move on." I want to tell you what I've learned and ask you all ...
2
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1answer
212 views

$\frac{-1}{0}$ is infinity or -infinity? [duplicate]

Possible Duplicate: Division by $0$ I was solving a question for my brother today when i got this doubt, i I arrived at the answer as $\frac{-1}{0}$ Will the answer be infinity since ...
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4answers
972 views

Is a subset of infinity still infinity?

In looking at the post here which really got me thinking what infinity and how it is notated. As the question states, is a subset of infinity still infinity? Also, what area of the Math world covers ...
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1answer
598 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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1answer
209 views

Maximizing gambling performance over the long run

Background. We can play a game in which we can put one dollar and get out $X$ dollars, where $X$ is 2 dollars with probability $p>1/2$, or zero dollars with probability $1-p$. We also assume that ...
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2answers
253 views

Resolving a paradox concerning an expected value

We have a coin that has a probability $p>1/2$ of coming up heads (and probability $1-p$ of coming up tails). We now play the following game: We start with a fortune of one dollar. We toss the ...
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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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2answers
567 views

What is infinity times the reciprocal of infinity?

I was talking with a friend about interesting properties of numbers and their theoretical contradictions and solutions when we came up with this. What is the answer? So... $x * ∞ = ∞$ and... ...
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0answers
197 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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408 views

Set of finite and infinite $0$-$1$ sequences countability [duplicate]

Possible Duplicate: Is the set of all finite sequences of letters of Latin alphabet countable/uncountable? How to prove either? Is the set of all strings with countably infinite length ...
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2answers
436 views

How many times more than $0$?

If I have $10$ apples, but you have $5$ apples, then I have $2$ times more apples than you. But what if I have $10$ apples, but you don't have any apples? If you look at the graph ...
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53 views

Probability in infinitary logic

Let X be a random variable taking the value 0.2 with probability 0.2, 0.4 with probability 0.4 and 0.8 with probability 0.2 and 1.0 with probability 0.2. Using Infinitary logic I can ask the ...
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2answers
96 views

The limit of a rational function

$$ \lim _{x \to -\infty} \frac{3x^2+3x}{2x^2+2}$$ Is a good practice to do this? Change the $ -\infty $ to $ \infty $, and change the sign of the $ x $ variables: $$ \lim _{x \to \infty} ...
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134 views

Sequence of numbers with infinite number of primes

If I have an infinite sequence of positive integers with infinite number of primes and if I have an infinite number of distinct sequences with such properties may I claim that there is an infinite ...
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2answers
405 views

Proof of an infinite sum of probabilities [duplicate]

Possible Duplicate: Alternative Expected Value Proof If $X$ is a random variable that takes values in the range $\left \{ 1,2,3,4,5,6,\ldots \right \}$ how can I prove the following ...
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1answer
102 views

Divisibility with sums going to infinity

I can't quite wrap my head around this. Given the formula $(1-x)(1+x+x^2+...) = 1$ It seems clear to me why this is true. All the x terms cancel out and we are left with one. And this is clearly ...
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1answer
104 views

Solve an infinite sum

I need to find the sum of this series: $1, 2 \left ( 1 - \frac{1}{\sqrt{15}} \right ), 3 \left ( 1 - \frac{1}{\sqrt{15}} \right ) ^ 2, 4 \left ( 1 - \frac{1}{\sqrt{15}} \right ) ^ 3, 5 \left ( 1 - ...
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284 views

Does an infinite random sequence contain all finite sequences?

If we have a finite alphabet, with each letter having a non-zero probability of being selected, will an infinite sequence of letters selected from that alphabet contain all finite sequences of letters ...
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3answers
237 views

What is the definition of $\infty$

I saw in a note that say $\infty$ is not a real number, and there is no interval of the form $(a, \infty]$? So what is the definintion of $\infty$?
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1answer
260 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? ...
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2answers
347 views

Doubly infinite numbers

Real numbers are sequences of integers which are infinite in one direction. If I have a string which is infinite in both directions, say ...345123985..., then I can form an injection from these ...
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1answer
497 views

Can one compare negative infinity and positive infinity?

I have always wondered, is negative infinity less than positive infinity? Can I compare them?
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472 views

What is the result of $\infty - 1$?

I was wondering after reading "What is the result of infinity minus infinity", is there any logical result for $\infty - 1$ ?
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Rejecting infinity

I've heard about mathematicians who defend a strictly finite conception of mathematics, with no room for infinity. I wonder, how is it possible for these people to do this? Are there any concepts that ...
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2answers
355 views

One-to-One Mapping

I have a conceptual kind of a question in one of my lectures, where you have two line segments one is $(0,1)$ and the other $(0,2)$ which of course includes $1$ in the segment. It then ask what the ...
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3answers
228 views

How to define $-\infty$?

I think I understand the fundamental concept of infinity. Elementary mathematics define $\infty := \frac{x}{0}$, for every $x$. And also $\infty := \frac{-x}{0}$ for every $x$. I know only one ...
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The limit of binomial distributed random variable

Edit (As Robert pointed out, what I was trying to prove is incorrect. So now I ask the right question here, to avoid duplicate question) For infinite independent Bernoulli trials with probability ...
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3answers
154 views

Continuity of $f(x)$ involving infinity

$f(x)= \frac{\sin(\pi x)}{x(1-x)}$ How can I define $f(0)$ and $f(1)$ to make $f(x)$ continuous on $[0,1]$? I've found that the limit at $0 = \pi$, and the limit from the left at $1 = \infty$. I ...
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1answer
507 views

Is the Unicode designed assuming the Continuum Hypothesis?

The Unicode chart for "letterlike symbols" states that א 2135 ALEF SYMBOL = first transfinite cardinal (countable) ב 2136 BET SYMBOL = second transfinite cardinal (the continuum) I ...
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12answers
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What is the result of infinity minus infinity?

What is $\infty - \infty$? Is it $\infty$ or $0$ or what?
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Sum of infinite divergent series

I have learned that positive infinity plus negative infinity isn't equal to zero, it's an indeterminate form. However what happens if we subtract two infinite divergent series ...
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493 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
291 views

Infinity - a simple question

This is a simple question, and maybe stupid: Is this true: $\infty < 1000\cdot\infty$ ?
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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 ...
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3answers
456 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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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 ...
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346 views

The leap to infinite dimensions

Extending this question, page 447 of Gilbert Strang's Algebra book says What does it mean for a vector to have infinitely many components? There are two different answers, both good: 1) The ...
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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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Negative 1 to the power of Infinity

Can anyone explain me what the result of $$\lim_{n\rightarrow\infty} (-1)^n$$ is and the reason?
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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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Why is $\infty^0$ indeterminate?

In a recent test question I was required to us L'Hopital's rule to evaluate: $$\lim_{x\to 0^+} x\ln{(e^{2x}-1)}$$ I assumed that anything multiplied by 0 would give an answer of 0. This turns out ...