What is Infinity: Definition and 983 Discussions

Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol shown here.
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
In physics and cosmology, whether the Universe is infinite is an open question.

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  1. N

    Can a Center of a Sphere be conceived at infinity

    Hello everyone...I was wondering if it is possible to conceive a 2nd center of a finite sphere in infinity...(I am not a mathematician and therefore, my words might look ridiculous) apparently, it looks like every point on the sphere is at the same distance from infinity...Anyway, this...
  2. N

    Series expansion of integral (ln(x))^2/(1+x^2) dx from 0 to infinity

    Hi everyone, once I again I turn to all of your expertise in complex analysis. Homework Statement Evaluate \int\frac{(ln(x))^{2}}{1+x^{2}}dx from 0 to +infinity by appropriate series expansion of the integrand to obtain 4\sum(-1)^{n}(2n+1)^{-3} where the sum goes from n=0 to...
  3. J

    Find the limit as x approaches infinity

    Homework Statement Find the limit as x approaches infinity of (the square root of x^2 + x minus the square root of x^2 -x) Homework Equations The Attempt at a Solution I do not know how to simplify the expression. I know that plugging in x = inifinity would be wrong. How...
  4. H

    Formal definition of a sequence limit which tends to infinity

    Homework Statement hey there I have been given a question that asks me to define a sequence Xn which tends to infinity that has a limit that is infinity! I am so cofused. I would assume to use an adapted version of the epsilon delta condition? Homework Equations It's for Analysis...
  5. H

    Dirac delta integrated from 0 to infinity

    I was wondering if I integrate the dirac delta function from 0 to infinity where the function it's integrated with is the constant 1, will I get 0.5 or 1? And why? This is not homework so I decided to post this here although I asked this question in class and the teacher (assistant) wasn't...
  6. R

    What is the value of 1 to the power of infinity?

    One of our senior teachers talked about infinity and said that 1∞ is not defined. On deeper probing, he said that it is a bit higher mathematics and it wouldn't be appropriate to go deeper here. Naturally, I could think of a inductive proof that it should be 1, if ∞ ∊ N. I can't think of a...
  7. K

    Integrals By Parts With Infinity As Limit

    \int_0^\infty \lambda x e^{-\lambda x} dx How do I use the limits (infinity and 0) after getting the equation from integration by parts?
  8. O

    FLINE: Solving Infinity Conundrum Problems

    I hope that this is thr right place to post this. If I walk down an infinite yellow brick road, flipping a fair coin with each step, and taking a step to the left if tails, a step to the right if heads, it would seem possible that I could end up a trillion miles from the road's centerline...
  9. H

    Limit of 2^n/n! as n Approaches Infinity

    Find the limit of \frac{2^n}{n!} when n approaches infinity.
  10. A

    Max Velocity Below c When Falling from Infinity to Black Hole?

    Assuming a system with relatively small but non-negligable mass falling from infinity to a maximally massive black hole; Would there be a maximum possible coordinate velocity below c ? Is it possible to do an approximate calculation for this? A ballpark figure?? If there is a...
  11. K

    How to Show lim n^{1/2}=\infty using the Definition of lim x_{n}=\infty?

    Homework Statement Show lim n^{1/2}=\infty using the definition of lim x_{n}=\infty. Homework Equations The Attempt at a Solution We want to show \left|x^{1/2}-\infty\right|,\epsilon. I get this far and i hit a blank.
  12. MysticDude

    Limit as x Approaches Infinity

    I am starting this thread because when I was doing a homework problem I got interested in something: \lim_{x \to +\infty} x^\frac{1}{x} I already know that the answer is one because \frac{1}{\infty} is the same as 0 and anything to the zeroth power is 1. The thing that got me interested is...
  13. W

    Limits at infinity of trigometric function

    Homework Statement lim (x→pi)〖sin(x-pi)/(x-pi)〗 Homework Equations i don't know if we should use trig identity sin(a-b)= sinA cos B- CosA sin B The Attempt at a Solution i use identities to solve that, i did not get the answer. i tried to multiply by conjugate, did not work also.
  14. H

    Exploring Cantor's Sweet Infinity: C vs. N

    Cantor's s(w)et :) Hi! I don't know if this makes any sense, but after reading the book "Pi in the Sky" this summer (It got really exceptionally hot here in high latitudes) I was let wondering the following: Let C be the number of the continuum of numbers (with decimals). Let N be the...
  15. C

    Can a^x be negative infinity with i and e?

    I think about on this and found a result, e^{i\pi k}=e^{-i\pi k}=-1\Rightarrow i\pi k=ln(-1) a>1 and k=1,2,3... y=(-a)^{i\alpha }\Rightarrow lny=i\alpha ln(-a)=i\alpha (ln(-1)+lna)=i\alpha (i\pi k+lna)\Rightarrow y=e^{-\alpha \pi k}e^{i\alpha \ln{a}} Then \alpha \ln{a}=-\pi...
  16. C

    Used infinity to predict that another ClamShell

    This post is in cosmology (not number theory, eg, Cantor) because our only (main) exposure to infinity is astronomical. Folks have used infinity to predict that another ClamShell has or is or will be typing this message...ballderdash...it would take an infinity of infinities for this to...
  17. B

    Applying the MVT to Show f(x)/x Goes to b When x Goes to Infinity

    Excuse the typing please, as I am posting from my phone. Let f have domain [0,infty) and range in R. Suppose as x goes to infinity, f'(x) goes to a constant b. I wish to show that f(x)/x goes to b as x goes to infinity. I have tried numerous applications of the MVT to solve this and cannot...
  18. V

    Limits and infinity problem

    Homework Statement Hi, I need to figure out what happens to this equation in the limits E = \frac{1}{4\pi\epsilon_0} \frac{2\lambda L}{z \sqrt{z^2+L^2}} in the two different cases that z>>L and when L -> infinity (note this equation was derived from finding the electric...
  19. A

    The Infinity of the Smallest Particle

    when the smallest particle is cut in half over and over again infinitely times, it will never disappear... since I used the word infinite, right?
  20. A

    Making sense of continuity at a point where f(x) = Infinity?

    Is there a way to make sense of the following statement: "f is continuous at a point x_0 such that f(x_0) = \infty?" The standard definition of continuity seems to break down here: For any \epsilon > 0, there is no way to make |f(x_0) - f(x)| < \epsilon, since this is equivalent to making...
  21. L

    Work to move charges from Infinity to origin

    Homework Statement Eight 3.0 μC charges are located at the corners of a unit cube centered about the origin with 1mm edges. How much work does it take to bring a 5.0 μC charge from infinity to the origin? Homework Equations U= k*q1*q2/r The Attempt at a Solution r=...
  22. fluidistic

    Lorentz factor equal to infinity + lots of questions

    I've had my second class of Modern Physics and I have some "obvious" doubts that many of you certainly had, although I didn't find anything answering the first doubt on a quick google search. 1)Say I am in an inertial reference frame. I can see that any photon going around me do it at a...
  23. T

    Understanding Lens Separation for Creating an Image at Infinity

    Homework Statement The lenses are separated by 250mm. Where should the object be places so that the final image is formed at infinity? I'm confused.
  24. M

    Is 1 Equal to Infinity? A Discussion

    or 1 = infinity Discuss
  25. A

    Potentail Energy beyond infinity

    I read in a book that if an electron is beyond infinity from another charge particle, then its potential energy will be positive :/ Please tell me which formulae are used for potential energy beyond infinity??
  26. N

    Wave equation boundary conditions at infinity

    Are there general boundary conditions for the wave equation PDE at infinity? If there is, could someone suggest a book/monograph that deals with these boundary conditions? More specifically, if we have the following wave equation: \[ \nabla ^2 p = A\frac{{\partial ^2 p}}{{\partial t^2...
  27. A

    Is Infinity Equal to Infinity According to Wolfram Alpha and Friends?

    The program Wolfram Alpha say "True" but my friends want the Proof. Could you help me please?
  28. L

    What does infinity potential mean?

    In both http://en.wikipedia.org/wiki/Dynamical_billiards" we have a potential who's value is infinity. Now if it were just a finite number (rather than infinity), I would be getting a dirac delta function in the equations of motion (I think..) but when it's infinity I can't really do...
  29. N

    I know that 1^(+/- infinity) is indeterminate.Is c^(+/- infinity)

    I know that 1^(+/- infinity) is indeterminate. Is c^(+/- infinity) indeterminate for real numbers c?
  30. I

    Infinity symbol inside of an absolute value Tattoo

    I've been thinking about getting a tattoo. I was considering getting the infinity symbol inside of an absolute value. Or the absolute value of infinity. Whichever you prefer. How mathematically correct is that? The absolute value of infinity simplified is infinity is it not? Will I completely...
  31. A

    Infinity and unique don’t mix. Theory

    Because space and time are infinite, doesn’t that make it impossible for anything to be truly unique. I propose there are an infinite number of this planet right now in this universe. The distance between each planet is probably unconceivable. Also there is an infinite number of this planet in...
  32. A

    Infinity and unique don’t mix.

    Because space and time are infinite, doesn’t that make it impossible for anything to be truly unique. I propose there are an infinite number of this planet right now in this universe. The distance between each planet is probably unconceivable. Also there is an infinite number of this planet in...
  33. H

    Does this reasoning ever reach infinity? 0<1<2<3<4<5

    Does this reasoning ever reach infinity? 0<1<2<3<4<5... What does it mean? Thanks!
  34. U

    Division By Infinity: Is It Defined?

    I was just thinking about how division by zero is undefined and began wondering if dividing by infinity is undefined as well. I understand the limits that accompany this idea...
  35. B

    Is Using Absolute Value for Infinity Common Practice in Limits?

    Hi, I came across a book which looks at a problem like \lim_{x \to 0}\frac{1}{x} So you approach from 0-, and get -∞, approach from 0+, get ∞ Then it would write the answer as \lim_{x \to 0}\frac{1}{x} = \left| \infty \right| It looks bizarre to me. How do you parse this? Is...
  36. I

    Black Holes and the Mysteries of r=0: Understanding the Concept of Infinity

    Ok so I was watching this program on TV and the guy ended up getting r=0 With this he put this value into the equation as so: 1/r = 1/0 = infinity What is up with that? Isn't it undefined. Also, this is about black holes.
  37. I

    Behavior at zero and at infinity

    Hi, I have a second order DE involving the first derivative and with singularities at zero and at infinity. I need the approximate behavior at zero and at infinity. I have the answer but I would like to know how to get there. I don't know how to start this search. Any help?
  38. E

    Asymptotical values as ω ->0 and infinity

    Homework Statement Its an electrical circuit problem. I have obtained the Transfer function. I have substituted in the inductance/capacitance and resistance values and am now stuck on this part of the question: d) Find |H(ω)|, the modulus of H(ω), and determine its asymptotical values for ω =...
  39. J

    Superposition principle and charges from infinity

    This is not a homework assignment but a question from a past exam paper as i am studying for an exam on monday. I am one of only 3 out of 150 who have never done physics before this, the first year of an electronics degree. I would really appreciate any help. 1) A charge q1=1e-4C is in a...
  40. M

    Limit at infinity that may or may not be e

    Homework Statement Compute lim as x goes to infinity of (1+1/x^2)^x Homework Equations I know that lim at infinity (1+1/x)^x=e I do not know if that is still valid with the x^2 there. I don't really think it is, but it's throwing me off. The Attempt at a Solution Beyond the...
  41. U

    Sequences - Assumption that I need to confirm about n approaching infinity

    Homework Statement Find the limit: an= 2n/(n2+1)1/2 Homework Equations n/aThe Attempt at a Solution Because n is approaching infinity, is it OK to disregard the +1 in the denominator and just consider the denominator to be n? This would then divide out the n in the numerator leaving 2...
  42. F

    Fourier Transform and Infinity Problem

    Homework Statement Hi, I need to find the Fourier Transform of: g(t)=\frac{1}{x}e^{\frac{-\pi t^2}{x^2}} Homework Equations G(f)=\int_{-\infty}^{\infty }g(t)e^{-j2\pi ft}dt \therefore G(f)=\int_{-\infty}^{\infty }\frac{1}{x}e^{\frac{-\pi t^2}{x^2}-j2\pi ft}dt The Attempt at a Solution...
  43. K

    Integral of e^(-ax) from evaluated between neg/pos infinity

    Hey guys I am working on a Physics problem and can't figure out a little but of math involved. What is this integral? I tried it out and got "1/a + a", assuming "1/a" is the positive half of the integral.
  44. 2

    Lim 1/2^x-1 as x goes to infinity.

    I can not understand what to do with this Lim 1/2^x-1 as x goes to infinity. i tried using l'hopitals rule but it does't work or I am not applying it right.
  45. D

    Trying to understand potential energy at infinity

    A difficulty I'm having is that PE, at least in the context of gravity, is said to go up with altitude, or distance from the gravitational source (the analogy of pushing an object upward against the force of gravity and thus adding PE to it). Yet PE drops to zero at infinity, which would seem to...
  46. R

    Is Infinity Real or Just a Concept?

    Is infinity merely a mental concept, or is it entirely meaningful and common? :confused: Infinity does not exist in the context of a number system. Example: what would infinity minus 1 be? It's not infinite since no finite number plus one equals infinity. We see that the rules of arithmetic...
  47. I

    Infinite limit as X tends to infinity

    Homework Statement I am required to express in \varepsilon - \delta way what I'm suppose to prove in case lim_\below{(x \rightarrow \infty)} f(x) = \infty Homework Equations None. The Attempt at a Solution So first, intuitively I thought that what this means is that f(x) is bigger...
  48. R

    Limit of an exponent when it goes to infinity

    how this thing works ? integration of e to the power of (-x) from 0 to infinity sorry , not good in using the symbols ..^.^ ~
  49. P

    Closed form expression for f(x) = sigma (n = 1 to infinity) for x^n / [n(n+1)]

    Homework Statement Consider the power series. sigma (n=1 to infinity) x^n / [n(n+1)] if f(x) = sigma x^n / [n(n+1)], then compute a closed-form expression for f(x). It says: "Hint: let g(x) = x * f(x) and compute g''(x). Integrate this twice to get back to g(x) and hence derive...
  50. P

    Integrating x^2 * e^-x^2 from -Infinity to Infinity

    Homework Statement Using the fact that the integral from -Infinity to Infinity of e^-x^2 is equal to Sqrt(Pi), find the integral from -Infinity to Infinity of x^2 * e^-x^2 Homework Equations The Attempt at a Solution I really don't know how to find this using the fact that...
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