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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Why are irrational numbers uncountable and rationals contable?

Question 1: Why are irrational numbers uncountable and rationals contable? I really struggle to understand this. I initially thought it had something to with the fact that between any two numbers ...
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1answer
26 views

Probability on the plane

Problem. On the Cartesian plane with origin O and x- y-axes, I randomly pick a point P. What is the probability that the line segment OP has a slope at least 1? Is the answer 1/4 or 1/2? answer = ...
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6answers
518 views

why does commutativity of addition fail for infinite sums?

While discussing the sum of a particular series, $\sum\limits_{n=0}^{\infty}{\left(-1\right)}^n$ (a sum that I've heard is alleged to be equal to $\frac{1}{2}$), it was mentioned to me that addition ...
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0answers
142 views

The sum of all the natural numbers [duplicate]

I've watched this video: ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12 Now, I'm not quite familiar with infinite groups and such, but common sense says that claiming that the sum of all natural numbers ...
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2answers
38 views

Countably or Uncountably Many Discontinuities

I want to know why the following function has uncountably many discontinuities: $$f(x)=\left\{\begin{array} & x^2 & x \not \in \mathbb{Q} \\ 0 & \text{otherwise} \end{array}\right .$$ ...
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1answer
109 views

What is the limit of difference between harmonic series and natural logarithm of n+1?

I'm an undergraduate student in geology and I'm dealing with a project in math. The last question of the project gives me the harmonic series (An = 1 + 1/2 + ... + 1/n) and this natural logarithm L = ...
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1answer
42 views

map sum of square integers to a contiguous range of integers

Given a list $a$ of integers, $$n_{a_1}, n_{a_2}, ..., n_{a_d}$$ have $$N_a = \sum_{j=1}^d n_{a_j}^2.$$ The various $N_a$, $N_b$ etc. are integers, but are not contiguous: for example, if $d=2$, ...
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4k views

Are the integers closed under addition… really?

Okay so I'm a 3rd year undergraduate studying Mathematics. I've proved in group theory countless times that the integers are closed under addition. It's obvious to me that they are. However this has ...
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1answer
951 views

Is there any mathematical or physical situations that $1+2+3+\ldots\infty=-\frac{1}{12}$ shows itself? [duplicate]

I just saw the proof that $$1+2+3+\cdots=-\frac{1}{12}$$ and my brain still hurts. I completely understood the proof and my question is NOT actually about the proof itself. At the end of the proof, ...
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0answers
408 views

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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4answers
4k 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 ...
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3answers
112 views

Number of Infinities in complex numbers

How many infinities would be there for complex numbers? Like there are 2 infinities (+infinity and -infinity) for the real numbers, is there a way to prove the number of infinities in the complex ...
4
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2answers
156 views

Gram-Schmidt in Hilbert space?

EDIT: After some contemplation I decided to phrase the question better to avoid trivial answers. Consider a Hilbert space with a basis $\{v_{i}\}$ where $i\in I$ an index set, which could be ...
3
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1answer
69 views

Set theory, show a set is countable, homework. check my answer

I solved this question but there is something strange going on and I am unsure of myself. Would like someone to review it. We are given a total order (or linear order) $<^{*}$on group $A$ such ...
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3answers
591 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 ...
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1answer
56 views

Credit Given - Geometricly Modeling Infinity with 3 planes and 9 circles - Ratio of Circles

Refer to the attached diagram sketch to help visualize the equation. I am requesting help with an interesting math problem. Basically, I am diagraming infinity using three planes. These planes ...
2
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1answer
128 views

Largest infinite cardinal used in a proof

I've heard before that Knuth holds the record for the largest constant used in a mathematical proof. I was wondering what is the largest cardinal ever explicitly considered in set theory. I presume ...
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1answer
43 views

Laurent Seies and Res

Prove that for any Laurent series f(t) one has "Res(f') = 0"? I know for a Laurent series of a complex function f is a representation of that function as a power series which includes terms of ...
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4answers
145 views

random thought: are some infinite sets larger than other [duplicate]

I was in the shower today and I just thought of this so I'm asking it. I'm sure this has been thought of before. Let's say we have two sets, the set of all even numbers and the set of all natural ...
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3answers
74 views

How to evaluate this limit with l'hopital's rule

is it possible to use L'hopital for this or is there another method I'm missing? I have no idea how to even start this. $$\lim_{x\to \infty} \frac{(9x+1)^\frac12}{x+1} $$
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0answers
50 views

Number of real numbers between 0 and 1 vs number of all integers [duplicate]

Let a be the number of real numbers between 0 and 1 Let b be the number of all integers. Can I say ...
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3answers
147 views

many infinities, how many zeroes? [closed]

although i am aware of the term non-standard analysis i have, as yet, no clear idea what it signifies. but i have often wondered about the pseudo-equation $$ \frac1{\infty} = 0 $$ which one may ...
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1answer
36 views

information content of a quadratic surd

how much information is required to construct the equation: $$ X^2 - 2=0 \; ? $$ suppose, in a spirit of seasonal festivity, we squander a few further bits, and pamper ourselves with the additional ...
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2answers
29 views

Infinite expansion of non-linear expressions with 3 or more variables

I just realised that if we expand any of the non-linear expression with power of 3 or more we can't stop expanding them until we are dead. So for example: Expansion: $(a+b+c)^2 = a^2 + b^2 + c^2 + ...
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0answers
45 views

Cantor, longish lines and the Landau -o notations

in general terms this question is about the behaviour of functions of a real variable as their argument $\rightarrow \infty$. i will present the matter as concisely as i can, but my presentation will ...
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0answers
48 views

Number of ways to cut a square

How many ways are there to cut the unit square into two pieces? And how many ways are there if the two pieces must have equal area? Some special cases: A. If the cut is required to be a horizontal ...
2
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0answers
36 views

Fine-grained way to measure infinity

It is known that the cardinality of $R$ is equal to the cardinality of $R^2$, $R^3$, etc. But, intuitively these sets have different sizes. A possible way to formalize this intuition is to talk about ...
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0answers
42 views

A simple question on limits

Is it true that $$ \lim_{x\to+\infty} \mathbb{I}_{S=\{z\mid e^{-z}>0, z\in\mathbb{R}\}}(x) = 1,$$ where $\mathbb{I}_{S}(x)$ is an indicator function for $x\in S$?
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1answer
88 views

Proof that infinity $\notin \mathbb{R}$ [closed]

Proof that infinity $\notin\mathbb{R}$ How can I prove that? I don't know where to start! Thanxx!
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2answers
58 views

Is probability meaningful in cases of infinity?

Is it meaningful to speak of probability in cases of infinity? For instance, consider me having an infinite line of balls arranged in the manner: - Red, Green, Blue, Red, Green, Blue, Red....... ...
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2answers
99 views

structure of the full symmetric group on a countably infinite set

trying to get a handle on the full symmetric group $S$ of permutations on a countable set $X$. i had never really thought much about this group, but now i look at it for the first time it appears a ...
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3answers
75 views

Limit Problem - No clue where to start

$$\lim_{x\to \infty}\frac{x^{2011}+2010^x}{-x^{2010}+2011^x}$$ I'm not sure where to even start with this one. One idea I had was that perhaps it could be "split" up as: $$\lim_{x\to ...
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1answer
127 views

Understanding countable ordinals (as trees, step by step)

Even though ordinal numbers – considered as transitive sets – are perfect non-trees, it is worth (and natural) to visualize them as trees, starting from the finite ones which are given as ...
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2answers
72 views

$f:\mathbb{R} \to \mathbb{R}$ be differentiable and $\lim\limits_{x\to\infty}f'(x)=1$, is $f(x)$ unbounded? [duplicate]

Suppose $f:\mathbb{R} \to \mathbb{R}$ is a differentiable function such that $\lim\limits_{x\to\infty}f'(x)=1$,then is it true necessarily true that $f(x)$ unbounded? I think that it will always ...
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4answers
87 views

Is it valid to write $\displaystyle \lim_{x \to 0} \frac{1}{x^2} = \infty$?

AFAIK the limes of a term does not exist if that term does not converge, but I haven't found a suiting question here yet. This probably is a double of a similar question.
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5answers
202 views

Product of all primes

Is the product of all primes a natural number? In other words, is this true: $$ \prod\limits_{\text{primes}} p_i \in \mathbb{N} $$ And if so, what about just some of them: $$ ...
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1answer
125 views

How useful is infinity? [closed]

If infinity in one case is just something that cannot be capped. Does it really find its use in something? Speed is a number and when Dr.Math can assume it can be infinite, in reality universe even ...
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1answer
183 views

Are there different types of infinity? [duplicate]

Today in class my professor mentioned that there are different types of infinity. This confused me at first because I always thought infinity is just infinity. What are the different types of ...
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4answers
106 views

Limit for $\lim _{n \to \infty}(n+2)^{2}\sin\frac{1}{n}$

Can't prove the limit $$\lim_{n \to \infty}(n+2)^{2}\sin\frac{1}{n}=\infty.$$ by definition it should start: Let $M>0$. There exists an $N>0$ for every $n>N$: ...
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2answers
232 views

On the Continuum Hypothesis

Let me start out by saying that I am not a mathematician. I read an article over at Scientific American that discussed the Continuum Hypothesis. I developed the following thought experiment that would ...
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5answers
103 views

Stuck with infinities

I have heard this "some infinities are bigger than others" . How can this be ? The context was that the cardinality of the set of integers is less than that of the cardinality of th real numbers , ...
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1answer
58 views

Question about $f$ continuous function with these conditions?

Suppose I have a differentiable and bounded function $$f: [0, + \infty) \longrightarrow \mathbb{R}$$ such that $$\forall x \in [0, + \infty) \, : f(x) \cdot f'(x) > \sin x.$$ The question is: ...
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1answer
44 views

Reducing a double summation with infinite limits

I've been solving a Renewal theory problem and I end up with this function $m(t)=e^{-4t}\sum_{k=1}^{\infty}\sum_{i=2k}^{\infty}\frac{(4t)^i}{i!}$. How do I solve or reduce the double summation? Is it ...
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1answer
124 views

What is infinity in complex plane and what are operation with infinity extended to complex numbers?

For a real number $a$, $$\infty + a = \infty,$$ and if $a$ is positive, $$\infty \cdot a = \infty$$ What is $\infty + a$ and $\infty*a$ if $a$ is non-zero complex number, where $\infty$ is real ...
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2answers
109 views

How can we define infinitary proofs?

In the first order logic the usual notion of a formal proof for a sentence $\sigma$ from a theory $T$ is a "finite" sequence ($<\omega$ - sequeance) of sentences which each one of them is a valid ...
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1answer
63 views

n-Ball Volume and surface with $n \rightarrow \infty$

I am thinking about something I just read: The volume of the n-ball is given by $V_n(r) = \frac{\pi^{n/2}}{\Gamma (\frac n 2 + 1)}r^n$ and its surface area is $S_n(r) = \frac{\pi^{n/2}}{\Gamma (\frac ...
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1answer
101 views

Aren't two infinite graphs always identical?

Suppose you have an infinite graph $G$. I assume $G$ to be cubic and planar. No further conditions, so it will be irregular, maybe in the sense of cubic planar version of Rado's graph: Every possible ...
6
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3answers
282 views

$\frac{1}{\infty}$ - is this equal $0$? [duplicate]

I've seen that wolfram alpha says: $$\frac{1}{\infty} = 0$$ Well, I'm sure that: $$\lim_{x\to \infty}\frac{1}{x} = 0$$ But does $\frac{1}{\infty}$ only makes sense when we calculate it's limit? ...
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1answer
161 views

The proof of the infinity base of $\mathbb{R}^{\infty}$

We know that a finite basis of the finite-dimensional space $\mathbb{R}^n$ is $$ \{(1, 0, 0, 0,\ldots,0),\:(0, 1, 0, 0, 0,\ldots,0),\:(0, 0, 1, 0, 0, 0,\ldots, 0),\:\ldots,\:(0, 0, \ldots, 0, 0, 0, ...
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4answers
222 views

Hilbert's Hotel and Infinities for Pre-university Students

Hilbert's paradox of the grand hotel is a fun and exciting ground to base a talk on the set theoretic concept of infinity for interested students - even in middle- and high school. However, it does ...