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.

View More On Wikipedia.org
Recent contents Top users
  1. C

    Finding the limit of a quotient as x goes to minus infinity

    Homework Statement Find the limit $$\lim_{x\to-\infty} \frac{\sqrt{9x^6 - x}}{x^3 + 9}$$ Homework Equations N/A The Attempt at a Solution To solve this, I start off by dividing everything by ##x^3##: Numerator becomes ##\frac{\sqrt{9x^6 - x}}{x^3} = \sqrt{\frac{9x^6 - x}{x^6}} = \sqrt{9 -...
  2. A

    Inifinity limit with natural log

    Homework Statement Limx--> ∞ Ln(x^2-1) -Ln(2x^2+3) Homework EquationsThe Attempt at a Solution Ln(x^2-1)/(2x^2+3) Then I divided the top and bottom by x^2 so in the end I got (1/2). Is this right?
  3. A

    Limit at infinity with radicals

    Homework Statement lim as x tends to -∞ (x)^3/5 - (x)^1/5 Homework EquationsThe Attempt at a Solution The first thing I did was convert it into a radical so it becomes fifthroot√x^3 - fifthroot√x. Then I rationalized to get ( x^3-x)/(fifthrt√x^3+fifthroot√x) . I then divided the top by x^3...
  4. G

    I Dipole: Which field lines go to infinity?

    Hi. An electric dipole field (two opposite point charges separated by some distance) has fields lines from the positive to the negative charge, but also field lines reaching to and coming from infinity. Starting from the positive charge, is there a way to compute the opening angle of the cone...
  5. D

    B Can the space (or else measurable) be actually infinite?

    The (most popular) flat model of Universe is space-infinite. How the infinity is measured? Can you give me references to the papers about the actual infinity of space?
  6. J

    MHB Proving Countable Infinity

    I am trying to prove how this set is countably infinite: q∈Q:q=a/b where a is even and b is odd a needs to be even and b needs to be odd, so I thought this would prove that it would be countably infinite: q = a/b + x/x, where x is any even number. a always needs to be even and b always...
  7. T

    I Infinity vs Expansion: Is an Infinite Universe Compatible with Expansion?

    I had a question after watching a Discovery Channel show on the universe. They talked about how some speculate the infinitude of the universe as opposed to a finite sized universe and I have also heard the same on this forum...and it got me to thinking... Isn't an infinite universe...
  8. P

    I Gravity of a Baseball-Sized Universe: Reach to Infinity?

    When the observable universe was the size of a baseball, did its gravity (field?) extend to (as opposed to towards) infinity?
  9. D

    B Ratio of circumference to diameter for infinitely large circ

    If you divided the circumference of an infinitely large circle by its diameter, would the result be pi?
  10. T

    MHB Proving $5^n - 3^n \le 2^n$ as n approaches Infinity

    I have $$5^n - 3^n \le 2^n$$ (as n approaches infinity) but I'm not sure how to prove this to myself.
  11. P

    B Why do we use infinity in Physics?

    When we talk about a particular problem in Physics. For instance, let's say that light is coming from somewhere to hit the earth. We often say that the light is coming from "infinity." Let's say that we're tackling a black hole and we have a person somewhere as an example and we say that let's...
  12. W

    I Is Infinity Inevitable in Our Understanding of the Universe's Beginning?

    Since the success of the Penrose Hawking singularity theorems many people have claimed that the universe must have a beginning . In recent years though people have explored models of the universe that resolve the singularity and imply the universe may have existed before the big bang. In such...
  13. Rectifier

    Is ##\frac{0}{\infty}## equal to 0 or infinity in mathematics?

    Is ## "\frac{0}{\infty}"=0 ## ?
  14. newrd

    B Is the Universe Finite? Exploring Expansion and Existence | True or False?

    The universe- from our understanding, is expanding, thus the regions (for lack of a better word) particles have not yet reached do not exist. How far our universe can/ will expand is unknown, it may be infinite, but we can conclude at this time, as it is still expanding, that it is finite. True...
  15. karush

    MHB 8.8.16 LCC 206 Integral at infinity

    $\large{8.8.16} $ $\tiny\text{LCC 206 Integral at infinity}$ $$I=\int_{0}^{\infty}\frac{x}{\sqrt[5] {x^2 +1}} \,dx= \infty \\$$ $\text{presume just taking the limit makes the } \\ x\implies\infty \\ \text{thus the integral goes to } \infty$ $\tiny\text{ Surf the Nations math study group}$...
  16. T

    B From a 4D perspective would 3D infinity exist?

    From the perspective of a 4D observer would 3D infinity appear to exist? Why/why not?
  17. karush

    MHB What is the Integration Formula for x and p in Maxima?

    $\Large{§8.8.15} \\ \tiny\text {Leeward 206 Integration to Infinity}$ $$\displaystyle \int_{e^{2}}^{\infty} \frac{dx}{x\ln^p\left({x}\right)}\,dx \,, p>1$$ $\text{not sure how to deal with this} $ $\text{since there are two variables x and p} $ $\text{answer by maxima is:'} $...
  18. karush

    MHB How Does Integration to Infinity Work in Calculus?

    $$\Large{§8.8. 14} \\ \tiny\text {Leeward 206 Integration to Infinity}\\ \displaystyle I=\int_{2 }^{\infty} \frac{1}{x\ln\left({x}\right)}\,dx \\ \begin{align}\displaystyle u& = \ln\left({x}\right) & du&=\frac{1}{x} \ d{x} \end{align} \\ \displaystyle I=\int_{2}^{\infty}\frac{1}{u}...
  19. karush

    MHB Leeward 206 {8.13} Integral at infinity

    $\tiny\text{Leeward 206 {8.13} Integral at infinity}$ $$I=\int_{0} ^{\infty} e^{-ax} \,dx \ a>0 = \\ \begin{align}\displaystyle u& = -ax & du&=-a \ d{x} \end{align} \\ \text{then} \\ I=-\frac{1}{a}\int_{0} ^{\infty} e^{x} \,dx =-\dfrac{\mathrm{e}^{-ax}}{a}+C \\ \text{hopefully, wasn't...
  20. U

    MHB Limit of = (sin nx) / (sin x) as n goes to infinity.

    Hello everyone. I need help trying to calculate/ trying to realize what the limit function of (sin nx)/(sin x) as n goes to infinity is. from another topic here on MBH ("Show δn = (sin nx) / (pi x) is a delta distribution") and after research with Wolfram Alpha I know that the limit function...
  21. beamie564

    Work done in moving a charge to infinity

    Homework Statement (Not for homework/assignment. Just doing problems for practice) This is from Griffiths Introduction to Electrodynamics, 4th edition, p.112 Problem 2.60 " A point charge q is at the centre of an uncharged spherical conducting shell of inner radius a and outer radius b...
  22. A

    Biot-Savart Law: infinity wire

    Hey! 1. Homework Statement One must simply calculate the magnetic field at a distance s to the wire, which carries a steady current I Homework Equations Should I write the point vector as: \mathbf{r} = s\hat{s} + \phi \hat{\phi} + z \hat{z} or \mathbf{r} = s\hat{s} + z \hat{z} ? The Attempt...
  23. Alpharup

    I Proving limit theorems when limit tends to infinity

    Am using Spivak and he defines limit of a function f 1. As it approaches a point a. 2.As it approaches infinity. He also defines limit f(x)=∞ x->a But though in solving exercises, we can see that all the three definitions are consistent with each other, I am not...
  24. Bran

    B Expanding from and eventually to a singularity?

    I know this thread, about why the Universe can't expand inward, is fairly old; but I stumbled across it today and there was something mentioned here that sparked a question I feel like people here would be qualified to answer. What was mentioned, was that a singularity is a point at which our...
  25. N

    A Can infinity be observed in the real world?

    To what extent is the term infinity used in the physical world. When talking in terms of mathematics we can have a set of all natural numbers called an infinity, then we can have a value that comes after this set of infinity (lets call it 'a'). After 'a' comes 'a+1' then after this set of...
  26. Dopplershift

    I Determining the Rate at Which Functions approach Infinity

    With basic fractions, the limits of 1/x as x approaches infinity or zero is easily determine: For example, \begin{equation} \lim_{x\to\infty} \frac{1}{x} = 0 \end{equation} \begin{equation} \lim_{x\to 0} \frac{1}{x} = \infty \end{equation} But, we with a operation like ##\frac{f(x)}{g(x)}##...
  27. NaukowiecGirl

    Infinity Focal Points of Plane Mirror

    Hello! I've read on several pages that plane mirrors have an infinite amount of focal points. I don't understand? I thought plane mirrors have no focal points because the rays are parallel and don't focus in the first place. Why does a plane mirror have infinity focal points and what does it mean?
  28. G

    Limit of arccosh x - ln x as x -> infinity

    Homework Statement find the limit of arccoshx - ln x as x -> infinity Homework Equations ##arccosh x = \ln (x +\sqrt[]{x^2-1} )## The Attempt at a Solution ## \lim_{x \to \infty }(\ln (x + \sqrt{x^2-1} ) - \ln (x)) = \lim_{x \to \infty} \ln (\frac{x+\sqrt{x^2-1}}{x}) \ln (1 + \lim_{x \to...
  29. KarminValso1724

    B What are the odds of getting a number on a hypothetical infinity sided dye?

    Let's say for example, there was a dye in which any number with any amount of digits could be scored. You also had an equal chance of scoring every number. Which means that you have the same chance of rolling a 1 as you do 5 billion. If you rolled that dye, how many digits would that number...
  30. jedishrfu

    The Man Who Knew Infinity Movie

    A movie on the life of Srinivasa Ramanujan staring Dev Patel and Jeremy Irons: https://en.wikipedia.org/wiki/The_Man_Who_Knew_Infinity_(film) with a planned April 29, 2016 release date. The trailer looks pretty good. The producers are Manjul Bhargava and Ken Ono, two well-known and...
  31. H

    Infinity times zero, rotational symmetry

    To show that the Lagrangian ##L## is invariant under a rotation of ##\theta##, it is common practice to show that it is invariant under a rotation of ##\delta\theta##, an infinitesimal angle, and then use the fact that a rotation of ##\theta## is a composite of many rotations of...
  32. Matejxx1

    The limit of a function as x--> infinity

    Hi everyone, So we were writting our math test today and I am not completely sure about one concept. For the sake of simplicity let's say that f(x)=x2 and let's say we were asked to find, lim f(x) as x--->infinity = ? is the correct answer here undefined or infinity. Thanks for the help
  33. K

    Calculating Minimal Distance between Two Protons in Motion

    Homework Statement A proton is moving at speed v from infinity toward a second stationary proton, as shown below. Determine the minimal distance between them. http://s27.postimg.org/lmw3d21j7/Untitled.png Homework Equations W = \frac{kq_1q_2}{r} E_k = \frac{mv^2}{2} The Attempt at a...
  34. B

    Angle of electric field line heading to infinity

    Homework Statement Charges 2q and -q are located on the x-axis at x=0 and x=a respectively. (a) Find the point on the x-axis where the electric field is zero, and make a rough sketch of some field lines. (b) You should find that some of the field lines that start on the 2q charge end up on...
  35. Justin LaRose

    Confused about why wave function is from zero to infinity

    Homework Statement I am trying to solve a problem from a popular quantum mechanics text. I am learning on my own. I am trying to calculate the variance, which is <x^2>-<x>^2 = variance in x. I posted a photo of the problem as a picture that is linked below as well as the solution, I simply...
  36. loreberto911

    What is the oblique limit of a function with a hard limit to infinity?

    Hi everybody, I have this function to study ##\frac{(x+1)}{arctan(x+1)}## I need the limit to infinity,it's oblique and I have to find q,from y=mx+q. so q=lim(x->inf) ##\frac{(x+1)}{arctan(x+1)} -2x/\pi## I don't know how to solve it.the limit gives infinity to me.but calculators online give...
  37. E

    What is the connection between Ramanujan and the value of infinity?

    hey I was browsing the web a while ago and I found an equation for infinity to equal 1.5 when subjected to a equation. however I can no longer find this. I was wondering if anyone knew what it was and if they could explain how it works. many thanks Ewen
  38. dextercioby

    Insights The Need of Infinity in Physics - Comments

    dextercioby submitted a new PF Insights post The Need of Infinity in Physics Continue reading the Original PF Insights Post.
  39. M

    Finding Residue of Complex Function at Infinity

    Hello everyone, I have a problem with finding a residue of a function: f(z)={\frac{z^3*exp(1/z)}{(1+z)}} in infinity. I tried to present it in Laurent series: \frac{z^3}{1+z} sum_{n=0}^\infty\frac{1}{n!z^n} I know that residue will be equal to coefficient a_{-1}, but i don't know how to find it.
  40. L

    I can't seem to find this limit

    Homework Statement Homework EquationsThe Attempt at a Solution I tried using the rule of multiplying with the "conjugate", for example what's above multiplied by (√n^3+3n)+(√n^3+2n^2+3)/(√n^3+3n)+(√n^3+2n^2+3). But I'm left with a huge mess :( I also tried dividing the top and the bottom by...
  41. G

    MHB Show $\displaystyle \lim_{x \to \infty} \frac{50x^{10}+100}{x^{11}+x^6+1}=0$

    How do you show that $\displaystyle \lim_{x \to \infty} \frac{50x^{10}+100}{x^{11}+x^6+1}=0$ What I tried: $\displaystyle \lim_{x \to \infty} \frac{50x^{10}+100}{x^{11}+x^6+1} =\lim_{x \to \infty} \frac{50+100/x^{11}}{1+1/x^{5}+1/x^{11}} = \frac{50+0}{1+0+0} = 50.$ But this is wrong. (Angry)
  42. G

    MHB Calculating Limit as x Approaches Infinity

    I'm trying to find $\displaystyle \lim_{x \to 20^{+}}\frac{5x^3+1}{20x^3-8000x}$ $\displaystyle \lim_{x \to 20^{+}}\frac{5x^3+1}{20x^3-8000x} =\lim_{x \to 20^{+}}\frac{5+1/x^3}{20-8000/x^2} = \frac{5+\lim_{x \to 20^{+}}1/x^3}{20-\lim_{x \to 20^{+}}8000/x^2} =...
  43. F

    What is the electric field due to hollow sphere at R=z?

    So I derived the E-field of a hollow sphere with a surface charge σ at z and I got: E(r)=\hat{z}\frac{\sigma R^2}{2\varepsilon _{0}z^2}\left ( \frac{R+z}{\left | R+z \right |}-\frac{R-z}{\left | R-z \right |} \right ) at z>R, the equation becomes: E(r)=\hat{z}\frac{\sigma R^2}{\varepsilon...
  44. Grimble

    I  Is Zero x Infinity Really a Real Number?

    This seems a very simple case to me, yet I have heard it said that the answer is some undefined real number. Yet zero times anything means no iterations of whatever the object is; whether that be a real number , an imaginary number or an undefined number. Whatever it is I don't see how one can...
  45. Math Amateur

    MHB Understanding ZFC and the Axiom of Infinity: Simple Explanation and Examples

    I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ... I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ... I am struggling to attain a full understanding of the Axiom of Infinity which reads as shown...
  46. A

    If n is infinity, wavelength is equal to what?

    in the dispersion relation of the surface plasmon the wavelength is proportional to square root of n. according to equation 5 in this paper: https://Newton.ex.ac.uk/research/emag/pubs/pdf/Barnes_JOA_2006.pdf if n goes to infinity, then what will be the value of wavelength. Thank you
  47. I

    Is the sum of all natural numbers equal to -1/12?

    I watched a video where apparently the sum of all natural numbers = -1/12. The video starts by saying S = 1-1+1-1+1-1+1-1... to infinity. He then says this sum does not have an answer, it's constantly between 1 and 0 depending on where you stop it. So he just takes the average and says 1/2. How...
  48. A

    Is (∞ - 1) < ∞ True for Inequalities with Infinity?

    Is this true? (∞ - 1) < ∞
  49. J

    Example of curvature scalar diverging at infinity?

    Reading Geroch's "What is a Singularity in General Relativity?", it seems that polynomial scalar invariants constructed from the Riemann tensor can diverge if we are at infinite distance, and not in a true singularity. Can someone give an example of space-time whose scalar invariant diverges...
  50. DaMeekie

    Do we already measure infinity?

    This idea has been bothering me for a while, it started when I thought that if there was an infinite amount of space inside of an inch. ( or even any measurement in the physical world ) Then I thought that maybe that's not a fair argument on the basis that quantum theory says planks length "h"...
Back
Top