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.
Hi,
I have a quick question about whether or not the infinite series of 1/n converges or diverges. My textbook tells me that it diverges,
but my textbook also says that by the nth term test if we take the limit from n to infinity of a series, if the limit value is not equal to zero the series...
Hello there,
I had another similar post, where asking for proof for Hilbert’s Hotel.
After rethinking this topic, I want to show you a new example. It tries to show why that the sentence, every guest moves into the next room, hides the fact, that we don’t understand what will happen in this...
Hilberts Hotel has infinity numbers of rooms and in every room is exactly one guest.
On Wikipedia Hilberts Hotel gets described as well:
Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest...
From integration by parts, and using y(10) = 0, I get the equation ##2e^{3t-30} = \frac{|y-2|}{|y+1|}.##
Let ##f(t) = 2e^{3t-30}##.
Since it's for t>10, f(10) = 2, and we have ##2=\frac{|y-2|}{|y+1|}##. Depending on the sign I choose to use, I get either that y=-4 or y =0. Since ##t: 10...
In Mathematical Methods in the Physical Sciences by Mary Boas, the author defines the Laplace transform as...
$${L(f)=}\int_0^\infty{f(t)}e^{-pt}{dt=F(p)}$$
The author then states that "...since we integrate from 0 to ##\infty##, ##{L(f)}## is the same no matter how ##{f(t)}## is defined for...
First off, this is just an assumption. My knowledge of the field is extremely limited and I beg you to come and correct my mistakes, so I can learn.
So, I guess we all know how that space-time fabric is bended by gravity. When a star dies, all of the atoms are brought extremely close...
While using L' Hospital's rule in evaluating limits, one comes across limits of the following type: $$\lim_{x \to 0} x \ln x$$ Such limits are generally evaluated by taking ##x## to the denominator and make it ##x^{-1}##. In such a case, an indeterminate form ##\frac{\infty}{\infty}## comes...
Is infinity truly infinite if it has something else in it?
Put differently, say there's an infinite volume of water that has some rocks in it, is the volume of water truly infinite? Though there's a place where there's no water?
I'm writing some notes on set theory, Aleph Null, etc., and was wondering if there's a Notation or Symbol that abbreviates this (inaccessible/strong/uncountable etc. cardinals). I'm not sure if ive seen notation before but it seems like symbols resembling Theta and phi have been used.
Now I expect that most of you on this forum would be familiar with the equality between point nine reoccurring and one:
0.999...=1
If your not familiar please review https://en.wikipedia.org/wiki/0.999...
Now this equality can be used to imply something else, which is rather heterodox...
I have read some of the other posts about this topic but am still left unsatisfied. Could just be me. :cool:
Did the universe, one minute after the big bang, consist of a finite volume of spacetime?
If so, then is it not logically inconsistent that the universe can possibly be infinite now...
Homework Statement
Homework Equations
The Attempt at a Solution
let y = lim x->0+ x^cos(1/x)
lny = cos(1/x)*lnx = (x*cos(1/x)) * (lnx/x)
x*cos(1/x) = 0 (sandwich theorem)
lnx/x = 0 (l'hopital)
so lny = 0
and y = 1
Is this correct?
Homework Statement
$$\lim_{x\to\infty} \left(\frac{n^2+2n+1}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}$$
Homework Equations
3. The Attempt at a Solution [/B]
I tried
##\lim_{x\to\infty} \left(\frac{n^2+2n+3-2}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}=##
##\lim_{x\to\infty}...
Hello!
I'm doing a school project, where I am writing about Infinity in Math and Physics. I've got the math part settled, but it's the physics part that has begun to bother me.
One part of my task is to write about some of the mathematical expressions in physics, where we use infinity - but the...
Hey guys! I have heard of this concept in various places and sort of understands what it attempts to do. Can anybody please explain it to me in more detail like how it works, how to notate it, and how to expand it to infinities and infinitesimals. Thanks in advance!
Aakash Lakshmanan
xphysx.com...
Hello.
First of all, I must say that I'm new to this forum, so I apologize if I'm posting in the wrong section.
I'm a 17 year old with not that much knowledge about physics, so if what I'm talking about makes no sense or is completely stupid, just let me know.
A couple of days ago I asked...
Homework Statement
Limx--> ∞ Ln(x^2-1) -Ln(2x^2+3)
Homework Equations
The 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?
Homework Statement
lim as x tends to -∞ (x)^3/5 - (x)^1/5
Homework Equations
The 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...
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?
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...
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...
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...
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...
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)}##...
Homework Statement
Homework Equations
The 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...