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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Can an infinite set be transitive, irreflexive, total, and have an upper and lower bound?

I need an infinite structure that can be put into an order with the following properties: The order must... be transitive, be irreflexive, be total (i.e., every two things share some sort of ...
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2answers
60 views

Doubts about infinite nested root

Find $f(a)=\sqrt{a-\sqrt{a^2-\sqrt{a^4-\cdots}}}$ where $a\in\mathbb{R}$. My Attempt : I consider $\frac{f(a)}{a}=\sqrt{1-\sqrt{1-\sqrt{1-\cdots}}}$. Now to finding this limit is easy but I cannot ...
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2answers
59 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
80 views

How can you compare the number of real numbers in the interval [0,1] and [0,10]?

There are infinite number of real numbers between 0 and 1,i.e in the interval [0,1]. So definitely there should be more numbers in the interval [0,10] because it includes the numbers in the first case ...
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2answers
39 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
172 views

Can a curve be an asymptote?

$f(x)=x^3+\frac{3}{x-1}$ This was the question given to me.I replied that $f(x)$ will have only a single vertical asymptote of $x=1$. My teacher told that there'll be be two asymptotes.One is the ...
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1answer
74 views

Teaching the Concept of Infinity to Children.

I was recently out with the family and we left it up to the children where we ate lunch (11 and 9 years old). They couldn't agree and were going back and forth calling each other names. This ...
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1answer
68 views

Regarding playing an infinite number of games that could last infinitely long amounts of time

So after watching the last Stanley cup game, a problem popped up in my head for which I have no solution. Say we have a game, like a hockey game, that has the possibility of going on forever. Of ...
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1answer
65 views

Assigning values to divergent integrals

I'm interested in the (obviously divergent) integral $$ \int_{-\infty}^\infty dx e^{-x f}\ ,$$ where $f$ is real. Is there any way to meaningfully assign a value to this integral? I was thinking of ...
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1answer
31 views

How do we evaluate this limit?

$$ A_N(x) = \lim_{N\to \infty}(\sin(x)/x)^N $$ The solution to this problem is given as, $$ A_N(x) = \exp( -Nx^2/6). $$ The problem is solved through Taylor series expansion for $\sin(x)$. And ...
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1answer
27 views

Probability Distribution of Count of Factors for All Numbers

Is the following a known thing? Define "factor count" as the count of factors each number has, then subtract 1. Ignore the number "1" as a factor. For example: Prime numbers have a factor count ...
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1answer
140 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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1answer
59 views

Calculus 2 Integral Question

I've been trying to resolve a calculus question and seem to be having troubles understanding exactly how to approach it. Some hints are supplied, but they don't exactly seem to help. Thanks to anyone ...
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1answer
142 views

Finding limit of a function as it approaches infinity

How do i solve the below without using L'hopital rule. The final answer obtained is $2/3$ ...
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1answer
51 views

infinity between two points on a line

I remember from school that the number of points on a section of a line is infinite. On the other hand, when you reach the number two in a number sequence, that is a number and how big the number is, ...
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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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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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229 views

“Infinito”, a combinatorial game with infinite width game-tree

I recently designed a combinatorial game (sequential game of perfect information) with an infinite branching factor, that is it has a game-tree of infinite width. I'm wondering how is it possible to ...
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166 views

Limits of infinite processes that terminate in finite time - checking my understanding?

I am a computer scientist by training, but have a fair amount of math background that I've picked up through classes, teaching, and general interest. A student of mine posed a question to me. I think ...
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30 views

Countability of unions versus products

Let $D_{n}$ be a set with $2^{n}$ elements for $n=1,2,...$. Let $A = \bigcup_{n=1}^{\infty}D_{n}$, and let $B = \prod_{n=1}^{\infty}\{0,1\}$. Let $A_{k} = \bigcup_{n=1}^{k} D_{n}$, and let $B_{k} = ...
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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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38 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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63 views

Backward from infinity?

The following question has been raised and answered lately: Problem 6 - IMO 1985 Please take a look at the Reverse method part of the answer given by this author. What's happening there is that we ...
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96 views

What does “radical cube zero” mean?

My hobby is taking comics way too seriously. And I just came across a math topic. In a certain comic (Fantastic Four 51, according to some polls the greatest comic issue ever) there's a machine for ...
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86 views

Limit of constant function

I was reading a proof using Markov Chains in a finite state space $E$. Denote $p_{ij}(n) = P(X_n = j | X_0 = i)$. Since the state space is finite, then probability of landing somewhere in the state ...
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55 views

Prove infinity arithmatics

How do you prove $ \infty * (-\infty) = -\infty$ or $ \infty +\infty = \infty$? I thought it is an axiom, but have been there's is proof for that.
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46 views

Meaningful measures for comparing infinite dimensional geometric objects

I have two infinite-dimensional convex polytopes, call them $A$ and $B$. I know that $B$ is completely contained within $A$, and I want to say something meaningful about their relative sizes. From ...
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115 views

Relationship between ordinals and rank of well founded relations on $\mathbb N$

I want to understand the relation between ordinals and well founded relations on $\mathbb N$. I found a nice starting point here cut-the-knot/ordinals. Ordinals start like this 0={}, 1={0}, 2={0,1}, ...
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109 views

maximize an objective function with an infinite component

Suppose I have the following maximization problem: $\log\det(\alpha K_p)-c\alpha$ with respect to $\alpha$ with $c$ being a constant and $m$ being the dimension of $K_p$. Here, one of the eigenvalues ...
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85 views

Hadamard regularization isn't working out

As part of an exercise in a grad course on "mathematical methods" (always such a helpful name), I've been asked to evaluate $I=\int_0^{1/2}{(x^2-x+c)^{-2}dx}$ as a Hadamard finite part integral for $0 ...
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50 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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43 views

Cardinality of infinite between the set of rationals and set of reals

I remember learning that whether or not there is a cardinality of infinity between the set of rational numbers and the set of real numbers is unprovable. Is this true, and if so, how do we know it to ...
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41 views

How do you call a scale that starts at $∞$, has $1/n$ divisions and tends to $0$?

A linear scale $2n$ divisions: 0 2 4 6 8 Logarithmic scales $10^n$ divisions: 1 10 100 1000 10000 ...
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27 views

Can a non-cyclic infinite proof tree with always-reachable provable nodes be used to construct a proof?

Suppose that I have a finite number of basic elements x,y,z ... and a finite number of operators +, * ... Terms X,Y,Z ... are created by combining basic elements and operators. For example, x+y, and ...
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34 views

Dividing an infinite plane into regions

I am currently working on a computer program for computing layout of graph-based diagrams. Their content is placed in an "infinite" 2D plane with cartesian coordinates in the center of the diagram. ...
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17 views

Is it possible to extend the complex plane affinely and projectively at the same time?

Is it possible to extend the complex plane affinely and projectively at the same time? That is by adding both the positive infinity (with based on it directed infinity) AND the unsigned complex ...
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20 views

How to properly clamp Beckmann Distribution

I am trying to implement the Cook-Torrance Microfacet BRDF shading model and I am having some trouble with the Beckmann Distribution: Beckmann Distribution with width parameter ...
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0answers
18 views

How the extension of complex plane with complex infinity $\tilde{\infty}$ coexists with extension of real line with positive infinity $\infty$?

How the extension of complex plane with complex infinity $\tilde{\infty}$ coexists with extension of real line with positive infinity? Are there any paradoxes arizing? What are the rules when the ...
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42 views

Help with calculating infinite sum

I'm working on a problem, and I'm stuck in the calculations of finding $\sum_{0}^{\infty}\frac{1}{1+n^2}$ Suggestions on how to approach this calculation? Thanks! (Also, I used Fourier to get to ...
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27 views

infinite dimensional Cramer-Wold theorem

The Cramer-Wold theorem states that if every fixed linear combination of $d$ random variables converges to a normal distribution, then the $d$ variables jointly converges to a multivariate normal ...
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26 views

Infinite One-Time Pad

As you know, when used correctly, a one-time pad allows one to send a message, such that the only thing that can be found out about it is the maximum size (which is also the key length.) It is ...
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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 ...
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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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55 views

A question with infinity [Part 2]

If you haven't seen my first post regarding infinity you can find it here: A question with infinity Thanks for all the constructive comments on my first post, and creative answers to my questions. ...
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97 views

How to observe infinity?

In my calculus course, there's example stated on the book: Given that $M$ is an ordered set and the sequence $\{a_n\}\subset M$, prove that there's a (weakly) monotonic subsequence of $\{a_n\}$. ...
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Optimization of infinite series. Dynamic programming.

I wonder how to optimize an infinite series using Classical (Lagrangian) method. For example:$$\sum_{t=1}^{\infty}\beta^{t-1}\ln c_t$$ subject to the constraint $$c_t+b_t\leq y_t+(1-r)b_{t-1}$$ ...