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

    What does infinity mean to a physicist

    Infinity has always been a problem for me, apart from the concept I do not believe it exists. numbers go on to infinity and even negative infinity but they don't really exist so they don't count (hehe). some people say the universe is infinite in size. well apart from infinite in the same way as...
  2. Y

    Using Fourier Transforms to solve Definite Integrals with Limits 0 to Infinity

    1. Using Fourier Transforms to solve Definite Integrals with Limits 0 to Infinity I'm trying to understand how to use Fourier Transforms to solve Definite Integrals with limits from 0 to Infinity. I understand how to use Fourier Transforms to solve indefinite integrals, but I believe there...
  3. H

    Limit as x approaches negative infinity

    Homework Statement as x approaches negative infinity, what value does this function approach ? limit square root (X^2+X) + X Homework Equations The Attempt at a Solution First, i manipulated the given function to take out absolute (x) from the square root so...
  4. T

    Can Anyone Prove why e^x= (1+x/n)^n as n approaches Infinity

    Can anyone help me ? I am completely lost on this one
  5. Z

    Disputing 1 / 0 = Infinity: Agree or Disagree?

    There are an infinite amount of zero's that can go into 1, therefore we can say 1 / 0 = infinity, but it is useless to say that because infinity isn't a number. That is why we say the answer is undefined. It is also useless to say it "equals" infinity because you cannot get to infinity. We...
  6. pslarsen

    Infinity machines where to buy?

    "Infinity" machines... where to buy? To start out with - they don't exist so let's not discuses that. What does exist are these small machines that can run with extremely little friction which I would love to have here on my desktop - my only problem is that I CANNOT find them and I have...
  7. I

    Electric Field Vanishing at Infinity

    Homework Statement The electric field on the dashed line in the figure vanishes at infinity, but also at two different points a finite distance from the charges. Identify the regions in which you can find E = 0 at a finite distance from the charges. Check all that apply: A)to the...
  8. E

    Limit as x -> infinity of a sine/cosine graph.

    Homework Statement Lim [2 + 3x + sin(x)] / [x + 2cos(x)] (x->infinity) Homework Equations The Attempt at a Solution My roommate asked me to help him solve this homework question, at first glance I noted the derivative to be: [3 + cos(x)] / [1 - 2sin(x)]...
  9. M

    Where is Infinity? Answers to Unsolved Mysteries

    Where is infinity?? Hello , This is an extract from my book about my problem. http://img98.imageshack.us/i/infinityc.jpg/ I was rold before on this forum that this article is wrong so can u define more what's wrong with it and explain it in a better manner?? Thanks in advance
  10. S

    Multiplication of infinity and zero

    What will happen if we multiply infinity with zero? how to describe this situation?
  11. Z

    Find the lim as x approaches infinity

    Homework Statement Find the lim as x approaches infinity of \frac{sin x}{x-\pi} The Attempt at a Solution This was in the section for L'Hopital's Rule, but if you substitute infinity in the functions you don't get an indeterminate form. I don't know what to do next.
  12. M

    Value of this function as n approaches infinity

    Homework Statement f(x) = lim _{n->\infty}(x{n})/(1+x{n}) Homework Equations Suppose that x=1 The Attempt at a Solution Wouldnt f(x) = 1/2? Because 1^n = 1, so the denominator is 2. The solution says that f(x)=1. Why is that?
  13. L

    Limits of Functions at Infinity

    I read that "if f : R -> R is an increasing function, then limit as x tend to infinity of f(x) is either infinity, minus infinity or a real number". f an increasing function means { x < y } => { f(x) < or = f(y) }. How do I prove this (if it is true)? Can I apply this to a function g : R ->...
  14. G

    I believe in infinity, but what does that mean?

    I believe in infinity. I believe the universe is infinite in size and possibilities... I was watching a tv program about this and I fundamentally disagreed with their conclusions. I believe infinity must mean that there can be no repetition and infinite does not mean "impossible". The tv...
  15. B

    Converging to Infinity: Solving the Limit of n!²/(2n)!

    Homework Statement Find the limit of the given sequence as n -> inf. ((n!)^2)/(2n)! Homework Equations We have been told that the squeeze theorem may be helpful. The Attempt at a Solution Using the squeeze theorem, I get stuck. I tried factoring some things out, and seem...
  16. R

    Do Infinity and Statistics Always Go Hand in Hand?

    Hi there, I have a question regarding infinity and statistics. (I hope there aren't too many questions with infinity on these forums) I was wondering if you had some simple procedure, like say rolling a six sided die, and said you did this an infinite amount of times, would it be valid to...
  17. A

    Function, f, with domain (-infinity, + infinity)

    I was working on the following problem from a textbook. The textbook has no answer. I have included my solution - I am not sure whether it is correct Any ideas and or solutions? (guidance) Question: Suppose that f is any function with domain (-infinity, +infinity) a) Does the function g...
  18. M

    How Do You Solve a Limit Involving Infinity in Polynomial Functions?

    Hi, I'm in Engineering Foundation. I'm stuck in one limit question. Find the limit :_ **********________ Lim (3x + V 9x^2 - x ) x-> -infinity by substitution it gives ( inf - inf ) I tried to solve it and get -inf Can anyone help me please ?
  19. N

    Solving Integrals from -∞ to ∞

    Hi all If I have an integral from -∞ to ∞, then is it always true that we can write it as a limit? I.e. if we have a continuous function f, then is it always true that \int_{ - \infty }^\infty {f(x)dx = \mathop {\lim }\limits_{N \to \infty } \int_{ - N}^N {f(x)dx} } ?
  20. T

    Limit as x goes to infinity of e^(-x)* sin(x)

    Homework Statement I am trying to take the following limit lim as x goes to infinity of ( e^-x )*sin(x) Homework Equations The Attempt at a Solution Can I say that it ges to '0' just because the 1/e^x goes to '0'. Or there is a better way to solve it?
  21. K

    Euclid's Proof: Infinity of Primes

    Euclid's proof: 1) Assume there is a finite number of primes. 2) Let Pn be the largest prime. 3) Let X be the P1 * P2 ... * Pn + 1 At this point the statement is that "X cannot be divided by P1 through Pn", but why is that? This is not self-obvious to me. How can I know this? k
  22. I

    Evaluate INT (5+cosx)e^-x from 0 to infinity

    ok so i got (5+cos x)/ex and i compared that with 1/ex (they're both going from 0 to infinity). Turns out that the integral of 1/ex from 0 to infinity converges to 1. But i don't know how to prove that our original function converges as well (which is the answer). Anyone care to help?
  23. I

    Which grows faster as x-> infinity? ln(x^2+4) or x-5?

    which grows faster as x--> infinity? ln(x^2+4) or x-5? so using L'H rule i got lim as x-->infinity of [ln(x2+4)]/(x-5) = [2x/(x2+4)]/1 = 2x/(x2+4) then using L'H rule again i got 2/2x, then again i 0/2 = 0. So, does that mean that x-5 grows faster? And why?
  24. N

    How to calculate residium at infinity?

    i have such a function z^3 \sin \frac{1}{3} i need to calculate its residium at z=infinity if i substitue infinity instead of a"" into the formal formula res(f(x),a)=\lim_{x->a}(f(x)(x-a)) i get infinity am i correct?
  25. D

    Use of infinity in mathematics from the constructivists?

    How are you, if you are, responding to the critique about the use of infinity in mathematics from the constructivists?
  26. R

    Is Matter Infinitely Divisible?

    Hello, This is my first time posting something related to physics. I have never studied physics but I have a question that has been with me since my childhood. When I was about 10 years old, I was sitting alone in my backyard. I had a twig in my hand. I broke the twig in half and was left...
  27. A

    Limit of 10^n/n as n-> infinity

    Hi, Can anyone please suggest a solution to the problem: lim 10n/(n!) n->(infinite)
  28. D

    Diode characteristics/dealing with infinity

    New to the forum, I'll say "Hi," first :) Homework Statement Have a physics assignment to hand in. Very simple experiment - increase the voltage in a circuit and measure the current/voltage across the diode in forward and reverse. The problem comes with dealing with infinity in some of the...
  29. J

    Is the Limit Infinity or Does It Not Exist at a Vertical Asymptote?

    This has been bugging for a while and I haven't found an answer. Say you have a function with a vertical asymptote. This asymptote approaches infinity from both sides. The limit approaching from either side would be infinity. So would you say the limit is infinity or does not exist?
  30. G

    Proving Limit of r/n as n Approaches Infinity

    Homework Statement prove that lim(n\rightarrow\infty)(r1/n) = 1 for r> 0 The Attempt at a Solution let \epsilon > 0 be given we need to find n0 \in N such that \left|r1/n - 1 \left| < \epsilon but not really sure where to go from here?
  31. K

    Square integrable functions blowing up at infinity

    I've been reading Griffths QM recently, and in the book he mentioned a couple of times that though these pathological functions exist, they're not physically realizable. But what's wrong with these functions? What prevents them to be physically realizable ? EDIT：Griffths' statement is wave...
  32. N

    Capacitor voltage at time = infinity

    Homework Statement I am trying to understand how to get the voltages Va and Vc in the following circuit. It is assumed the circuit has been like this for a very long time. Homework Equations Kirchoff's voltage law The Attempt at a Solution So I know that the capacitor acts...
  33. A

    An example of a function that attains the value infinity on R?

    An example of a function that attains the value "infinity" on R? I'm reading a couple of books on introductory measure theory (Royden, Stein-Shakarchi), and both of them talk about functions that can possibly attain the value \infty. But they don't define exactly what this means, or give...
  34. D

    How Do You Calculate the Summation of a Function from n=1 to Infinity?

    Homework Statement what is the summation of a function where n=1 to n=infinity? For example, given a function sin[(pi)nt]. Homework Equations The Attempt at a Solution I asking how I get that I do not know what should I do
  35. J

    Why is it that this integral equals zero as the limits go to infinity?

    x^s/s integrated on the semicircular contour with radius R and center c>0, where x>1, s is the complex variable, and R is meant to go to infinity. please help.
  36. F

    Short question about L infinity

    I want to say that f(x) = |1/x| is in L-infinity(E) when m(E)<infinity becuase the function has and esssup on any measurable set, E. Even if E = (-1, 1) f(0) is not a problem since it is only one point... But wait... what *is* the esssup for this function on (-1, 1)? I think it might not have...
  37. F

    I don't understand this step in the proof about L infinity

    I'm learning the proof that L_{\infty} is complete. I do not understand one of the steps. Let f_n be a cauchy sequence in L_{\infty}(E) then there exists a subset A in E such that f_n is "uniformly cauchy" on E\A. For m,n choose A so that |f_n-f_m| \leq ||f_m - f_n||_{\infty} for all x in...
  38. G

    WHY lim{x-> infinity} f(x)/x = infinity => lim{x-> infinity} f(x) = infinity ?

    Homework Statement Why is it? lim{x-> infinity} f(x)/x = infinity => lim{x-> infinity} f(x) = infinity What is evidence? Thank you very much.
  39. H

    Are Zero and Infinity Singularities in the Physical World?

    Another viewpoint: "Zero and infinity are both symmetry states. Every change (that is arithmetical operation) leaves them essentially unchanged. 50 times zero is zero. Likewise 50 times infinity is still infinity." So if ((like I assume)) mathematics is essential for modeling physical world -...
  40. apeiron

    Zero & Infinity: Equal Symmetry States

    Zero and infinity are both symmetry states. Every change (that is arithmetical operation) leaves them essentially unchanged. 50 times zero is zero. Likewise 50 times infinity is still infinity. Zero represents nothing. Infinity represents everything. Hence - judged on their deep mathematical...
  41. L

    Singularities in C* of f(z) = \frac{{\pi z - \pi {z^3}}}{{\sin (\pi z)}}

    Homework Statement Find and classify the singularities in C* of f(z) = \frac{{\pi z - \pi {z^3}}}{{\sin (\pi z)}}, and give information about Res(f, 0) and Res(f, infinity) The Attempt at a Solution I found that the singularities in C are z = n, with n \in Z, n\neq 0, n\neq 1. These...
  42. D

    Integral evaluated at +/- infinity

    Ok, I'm trying to solve this physics problem and I've come to the following integral (d is taken to be some constant): 1. \int^{+\infty}_{-\infty}{\frac{1}{(x^2 + d^2)^\frac{3}{2}}}dx Now, integrating this I am supposed to get 2. {\frac{x}{d^2\sqrt{x^2 + d^2}}}, evaluated at \pm\infty (Sorry...
  43. I

    Proving Limit at Infinity: n^(1/n) = 1

    Homework Statement How can I prove that: \lim_{n \rightarrow \infty} n^{\frac{1}{n}}=1 Isn't \infty^{0} indeterminate? Thanks! Homework Equations The Attempt at a Solution
  44. M

    Integration with upper limit infinity

    Homework Statement \stackrel{\infty}{0}\int2e^{ky}dy=3/2Homework Equations The Attempt at a Solution I got up to: \stackrel{lim}{x\rightarrow\infty}\[[e^{ky}]^{x}_{0}=3k/4 \stackrel{lim}{x\rightarrow\infty}\[[e^{kx}]=\frac{3k+4}{4} I have no idea how to work that out. Any help will be much...
  45. H

    Exploring the Poincaré Disc: Understanding Infinity

    hi there What is a Poincare' disc and why is the edges of disc represent infinity? thanks
  46. A

    Infinity norm of system matrix

    Homework Statement This problem is related to the system theory, using the H-infinity frame work to determine the maximum gain of a multivarible system. The system is described as G(s) = \begin{pmatrix} \frac{s}{s+1} & \frac{s}{s^2+s+1} \\ \frac{s-1}{s+2} & \frac{s-1}{s+1} \end{pmatrix}...
  47. A

    Determine Infinity Norm of a Transfer Matrix

    I'm trying to understand how the infinity norm of a transfer matrix is calculated. For example, assume a simple transfer matrix G(s) = \begin{pmatrix} \frac{s}{s+1} & \frac{s}{s^2+s+1} \\ \frac{s-1}{s+2} & \frac{s-1}{s+1} \end{pmatrix} Now, I'm trying to compute the \mathcal{H}_{\infty}...
  48. R

    Epsilon-Delta Proof of limit approaching infinity

    **DISCLAIMER - I am super bad at LaTeX** Homework Statement Prove \lim_{x \rightarrow \infty}\frac{1}{1+x^2} = 0 Homework Equations I Think I proved it, but I feel like I'm missing something to make this a proof of ALL \epsilon>0 and not just one case. Maybe I did it right. I...
  49. J

    Limits of functions at infinity

    Suppose that a function f:R to (0,infinity) has the property that f(x) tends 0 as x tends to infinity. Prove that 1/f(x) tends to infinity as x tends to infinity. I don't really know where to start with this problem, I'm assuming it will involve some sort of epsilon proof but that's all I...
  50. J

    Limits of functions at infinity

    Suppose that the two functions f(x) and g(x) both tend to infinity then surely f(x) + g(x) also tends to infinity? How can you prove this though? Similarly f(x)*g(x) would also tend to infinity wouldn't it? f(x) - g(x) and f(x)/g(x) wouldn't tend to anything though surely since infinity minus...
Back
Top