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. K

    Minimum Work Needed to Bring 2 charges from distance infinity

    Two charged particles are brought together from a great distance (r=infinity) to a distance of 2 m. The particle has a positive charge of 3.0 x 10^-5 C and the second has a negative charge of 1.35 x 10^-5 C. What minimum work was accomplished in this process? I can't for the life me figure...
  2. K

    Riemann Sums: Finding the Limit as n Approaches Infinity

    Homework Statement Identify an=the summation from k=1 to n of (2n)/(4k2+1) as a Riemann sum of an appropriate function on an appropriate interval and find the limit as n approaches infinity of an. Homework Equations There is no interval givien so I assume its from 0 to 1. The...
  3. S

    L^p space related question for p= infinity

    Homework Statement We are asked to exhibit a measurable set E such that L^p(E) is separable for p= infinity. Also, we have to show that L^infinity(E) is not separable if E contains a nondegenerate interval.Homework Equations A normed linear space X is separable provided there is a countable...
  4. Y

    Indicate whether sech(x) is invertible for [0, infinity) and explain why

    Homework Statement Indicate whether sech(x) is invertible for [0, infinity) and explain why.The Attempt at a Solution Sooo... I know that: sech(x) = 2 / ( ex + e-x ) I know that to get the inverse equation I'd need to swap the y and the x... but I'm trying to show whether it's invertible...
  5. K

    Analysis(sequences) proof: multiplying infinite limit at infinity by 0

    Homework Statement Let \stackrel{lim}{_{n \rightarrow \infty}}a_{n} = \infty Let c \in R Prove that \stackrel{lim}{_{n \rightarrow \infty}} ca_{n}= \infty for c>0 (i) - \infty for c<0 (ii) 0 for c=0 (iii) Homework Equations Definition of divergence to infinity (infinite limit at...
  6. M

    The Consequences of Making Infinity a Number in Mathematics

    let \infty = 1/0 then 1 =0 * \infty 0 = 0 * 1 then 0= 0 * (0*\infty) then 0 = (0* 0 )* \infty = 0* \infty =1 so 0 = 1 There is nothing called infinity
  7. I

    DEFINITE integral of sinaxsinbx FROM 0 to infinity

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  8. D

    Infinity corrected lens - expanding before tube lens?

    Hello folks, I am on a project that involves combining Raman microscopy + optical tweezers, and I made a setup over the past year. All of this is home built on minimal funds so its not the neatest of all setups. It involves an upright microscope (coupled to trapping laser) and collecting the...
  9. A

    Evaluate trig functions at infinity?

    is it meaningful to evaluate cos and sin at infinity? I ask in relation to Fourier integrals... ie does cos(infinity) have a value
  10. N

    What is the limit of a^x when a tend to infinity and x tend to 0?

    Please teach me this: What is the limit of [a]^{x} when [a]\rightarrow[/infinity] and[x]\rightarrow[/zero].It seem to me that it is divergent as lna,but I can not demontrate.It appears in the renormalization of Quantum Field Theory. Thank you very much in advanced.
  11. S

    F uniformly continuous -> finite slope towards infinity

    f uniformly continuous --> finite slope towards infinity Homework Statement Given f:R \rightarrow R uniformly continuous. Show that \limsup_{x\rightarrow \infty} \displaystyle|f(x)|/x<\infty i.e. \exists C \in R: \, |f(x)|\leq C|x| as x \rightarrow \pm \infty. Homework Equations The...
  12. Z

    Quick question about infinity symbol

    Homework Statement Is it bad to put something like 0 < x < (inf) ? Since infinity is not a number. Homework Equations The Attempt at a Solution
  13. C

    Is the Concept of Infinity Misunderstood in Mathematics and Philosophy?

    I put this in the philosophy section but I guess it could equally go in a maths section being as I suppose it is the philosophy of maths. Infinities are well defined in maths, I doubt anyone could disprove given the set of all natural numbers then the set of all fractions is 1 to 1 and...
  14. Fredrik

    Continuous functions that vanish at infinity

    I'm trying to understand the set C_0(X), defined here as the set of continuous functions f:X\rightarrow\mathbb C such that for each \varepsilon>0, \{x\in X|\,|f(x)|\geq\varepsilon\} is compact. (If you're having trouble viewing page 65, try replacing the .se in the URL with your country domain)...
  15. R

    Proving the lim as n goes to infinity of a function = 2

    I am unsure as to how to prove a that as the limit as n goes to inifinity of a certain function the answer is 2.. I am trying to use the definition of a limit and using \epsilon and N - argument to get a contradiction in order to solve the equation.
  16. F

    Implications of infinity on positional entropy?

    I just read the chapter on entropy in my chemistry text. The book described entropy in terms of possible positions in space that a molecule could take (or even ways in which the atoms within a molecule can shift and rotate). The claim was that gasses had higher entropy than liquids and liquids...
  17. Femme_physics

    Complex numbers - parallel lines meet at infinity ? What does it mean?

    Complex numbers - "parallel lines meet at infinity"? What does it mean? We started learning about complex numbers last week. One of the first things my teacher said was that "We learned that parallel lines never meet. But as it turns out, they meet at infinity." I'm willing to accept it...
  18. L

    Bringing 3 Charges from infinity into a triangle

    Homework Statement How much work is needed to arrange three charges, Q, into an equilateral triangle? The particles are initially infinitely far apart. Take 'a' to be the length of each side of the triangle. Homework Equations U = {(kqq)/L} The Attempt at a Solution I was under...
  19. A

    Work to move charge to infinity?

    Homework Statement The figure below shows three charges at the corners of a rectangle of length x = 0.55 m and height y = 0.35 m. http://www.webassign.net/walker/20-23alt.gif (a) How much work must be done to move the +2.7-µC charge to infinity? Homework Equations W=(\DeltaV)(q)...
  20. A

    Work to move charge to infinity problem

    Homework Statement The figure below shows three charges at the corners of a rectangle of length x = 0.55 m and height y = 0.35 m. http://www.webassign.net/walker/20-23alt.gif (a) How much work must be done to move the +2.7-µC charge to infinity? Homework Equations W=(\DeltaV)(q)...
  21. P

    Solving Path of Particle Subject to Central Force

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  22. P

    Limit approaching infinity with absolute function

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  23. D

    At which points on the x-axis (not at infinity) is the electric potential zero?

    Homework Statement Two point charges -2Q and +3Q are on the x-axis at the origin and at x = L. Find all the points on the x-axis (not at infinity) where the electric potential is zero. Express your answers in terms of L and Q.Homework Equations V= k*Q/rThe Attempt at a Solution I understand...
  24. S

    Limit as x approaches infinity

    Gday, I was contemplating limits today and tried to prove something to myself and I can't figure it out. This is what I am trying to wrap my head around. http://latex.codecogs.com/gif.latex?\lim_{x \to \infty } \sqrt{x^2 + 1} I'm not looking for the "numerical" solution of infinity...
  25. F

    Using operations with infinity

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  26. Y

    Limit of compositions at infinity.

    We often have lim_{x\rightarrow a}f(g(x)) = f(lim_{x\rightarrow a} g(x)) if f(x) is continuous at g(a). But then my question arises where g(x)\rightarrow\infty. I am not sure if there is any meaning to continuity at infinity as it seems that continuity is the property of a particular...
  27. A

    Moving Charges to Infinity: Work Required and Comparison

    Homework Statement The figure below shows three charges at the corners of a rectangle of length x = 0.35 m and height y = 0.22 m. http://www.webassign.net/walker/20-23alt.gif (rectangle image) (a) How much work must be done to move the +2.7 µC charge to infinity? (b) Suppose...
  28. K

    Question about limits at infinity with radicals.

    Homework Statement Find the following limit: \lim_{x \to \infty} \frac{2+\sqrt{(6x)}}{-2+\sqrt{(3x)}} Homework Equations n/aThe Attempt at a Solution I know this shouldn't be that hard, but somehow I keep getting stuck on simplifying the equation. I think the first step is to multiply both...
  29. B

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    'show that the entropy change of the object tends towards deltaQ/T (subcript i) as its mass tends to infinity, in the4 limit where it becomes a heat bath i got: change in entropy =mc ln (Tf/Ti) (mc ln (T subcript f/T subcript i)) but if the mass tends to inifinity...
  30. S

    What is the Correct Notation for Infinity when Taking a Limit?

    I was wondering the following two situations: 1) If x is a real variable, then, is it correct, or is it acceptable rigorously to assign x x = +infinity ? 2) if x is not necessary a real variable, then is it correct, or is it acceptable rigorously to assign x x = +infinity ? I am...
  31. C

    Minimum velocity from infinity

    Homework Statement <In Pic 1> Homework Equations (1/2)mv2 = ΔU The Attempt at a Solution I thought that if i apply the equations i'll have my answer but i got 2.5m/s while answer is 3m/s Please refer the solution given by some book --- Pic2, Pic3 <sorry for bad image, my...
  32. C

    What Is the Correct Approach to Evaluate Limits at Infinity with Square Roots?

    Homework Statement Evaluate the following limits: lim sqrt(x^2-3x+1)-x x->\infty lim sqrt(x^2-3x+1)-x x->-\infty 2. The attempt at a solution http://img816.imageshack.us/img816/9995/limitproblem11.jpg We have to enter in the answers online into a program that tells us if...
  33. L

    Exploring the Mystery of an Expanding, Infinite Universe

    I'm reading a lot of stuff about the expanding universe, and also about it being infinte. So now I was wondering: How can the universe expand while it's infinte? (or am I asking something that nobody really know's the answer to?)
  34. P

    Electric charge and infinity plane

    Homework Statement hello, the problem is that i have charge q which lies in d distance from a plane and I need to find function \sigma which describes density of this charge The Attempt at a Solution I wrote this situation in cylindrical coordinate system where q lies in point A=(0,0,h) and...
  35. S

    Is infinity plus or minus one possible?

    Can you have infinity plus or minus one? Assuming infinity is possible of course.
  36. romsofia

    Is Infinity a Concept or a Number?

    Hey, is infinity a concept or an actual number? This is a discussion with someone, I say it's a concept but then he brought up set theory, and I have no set theory knowledge whatsoever.
  37. D

    Compute the integral of x^a / (1+x^2) for x going from 0 to + infinity

    Compute \int^{\infty}_0 \frac{x^{\alpha}}{1+x^2} dx for some -1<\alpha<1. EDIT: This was slightly wrong. The hint given is that we can integrate from -p to p except for a small semi-circle around 0, and a large semicircle from p to -p, and choose a branch of z^{\alpha}. Wouldn't this in...
  38. A

    Mean tends Infinity in Gaussian Case

    Hi, I am wondering what happens to a Gaussian distribution when its mean tends to infinity. By looking at the equation of a Gaussian one might infer that the limit will go to zero; but does this imply that it reduces to a Dirac-Delta function? More precisely, if we integrate this limit we...
  39. R

    Limit as x approaches infinity, involves sinx and cosx

    y=1/2(sinx-cosx+e(^pi-x)) question: if x approaches infinity, which term or terms will dominate? from my understanding, sinx and cosx will oscillate and the e term will approach zero. so would the answer be sinx-cosx? please and ty
  40. P

    Limits of Infinity: Does f(x) Exist?

    Is it so that for limx->infinity f(x) to exist , limx->+infinity f(x) and limx->-infinity should exist and be equal ? if so then why ?
  41. C

    Help with integral using compelx contour : x/(e^x-1) from 0 to infinity

    Hi, i need to solve this integral : \int _{0}^{\infty }\!{\frac {x}{{{\rm e}^{x}}-1}}{dx} i solved it using series and i got the right answer of Pi^2 / 6 but i need a solution using complex analysis i need help with finding the right contour for this problem. i tried change of...
  42. H

    Define a sequence (fn) from n=1 to infinity of functions

    Homework Statement Define a sequence (fn) from n=1 to infinity of functions on [0,1] by fn(t)=t^n does the sequence converge in (CL^2[0,1],||.||2) Homework Equations The Attempt at a Solution I am struggling on where to start. I am fairly new to the L2 space and so would just...
  43. A

    Simplifying a geometric series with an infinity summation bound

    Homework Statement I am solving some convolutions, and i have come to these solutions. a)\sum2k, summing from -\infty to -1 b)\sum2k, summing from -\infty to n , where n <=-1Homework Equations the geometric series summation formula, from 0 to N \sumak = 1-aN+1 / 1-a , summing from 0 to N The...
  44. M

    Evaluate the following integral from 0 to infinity

    Evaluate the following integral from 0 to infinity. (see attached for better picture) e^(-ax)-e^(bx) ------------------ dx x Remarks: a , b > 0 a < b
  45. J

    Sum to infinity of Heaviside function

    I'm revising my course text for my exam and came across a Fourier series problem finding the Fourier series of the square wave: http://img574.imageshack.us/img574/5862/eq1.png. It is then calculated that the complex Fourier coefficients are...
  46. K

    Exploring Infinity: Is it a Math Concept or Real?

    Is "Infinity" only a mathematical concept or is there anything infinite in reality? I mean ∞ is indeterminate in a way such that any of the following expressions can be constructed: ∞ - ∞ = 1, ∞ - ∞ = 0 ∞ - ∞ = ∞ Is there anything in reality that can actually behave like that?
  47. M

    Limit at Infinity: Solve Without Squeeze Theorem

    Homework Statement \lim_{x \to \infty} \sqrt{x}\sin\frac{1}{x} Homework Equations I don't think you can use the squeeze theorem here...The Attempt at a Solution So I am just studying for an exam that I have tomorrow and I am going through problems that weren't assigned on our homework set...
  48. D

    What does it mean to have an integral with a lower bound of infinity?

    I am working on a proof in which I have an integral with bounds negative infinity to zero, with an even function, i.e., f(y) = f(-y). I took the limit to infinity rather than negative infinity since y is negative (which is OK I think) but now I have an integral that goes from infinity to 0. What...
  49. G

    Does e^(-infinity + i*w*infinity) equal 0 or 1?

    Does e^[(-infinity)+(i*w*infinity)] = 0 or 1? w = omega
  50. G

    Is negative infinity divided by infinity still indeterminate?

    Just as the title states, I'm working on a problem and have come to negative infinity divided by infinity. Is this an indeterminate form? I know that if they are both positive it is indeterminate, but I can't remember if one being negative makes a difference.
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