Somewhere beyond the numbers lies the concept of Infinity. But what exactly does "infinity" mean? What rules does it obey? What interesting properties does it have?

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There is no smallest infinity in calculus?

Somewhat of a basic question, but I tried mixing set theory and calculus and the result is a giant mess. From set theory (assume ZFC) we know there is a smallest infinite cardinal, $\aleph_0$, and ...
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3answers
49 views

Calculate exact value of and infinite sum [duplicate]

Im trying to find the exact value of the infinite sum : 3 + 1/3 + 1/27 + 1/243 + 1/2187 + ... I can see that to generate new terms we take the previous term and divide by 9 or multiply by 9. Not ...
2
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0answers
43 views

Is this set countably infinite or not?

"Far away, in the heavenly abode of the great god Indra, there is a wonderful net that has been hung by some artificer in such a manner that it stretches out infinitely in all directions. In ...
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0answers
40 views

Circles and the continuum hypothesis

I was trying to understand the undecidable nature of the continuum hypothesis and came up with the following question: The set of circles with a rational diameter is countably infinite (with ...
3
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3answers
114 views

Summing infinitely many numbers: how to assign a value?

If we take $S = 1-1+1-1+1-1+1-1+...$ we can show (in many different ways) that the result of the sum is $\frac{1}{2}$. One way for example would be to add $S$ to itself but shift it along one place, ...
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2answers
41 views

Limit approaching to negative infinity.

Q. Find $\lim _{x\to -\infty }\left(\frac{x^4\sin\frac{1}{x}+x^2}{1+|x|^3}\right)$ By inserting $x=-\frac{1}{y}$ and as $_{x\to \:-\infty \:}$ then $_{y\to \:0\:\:}$. By applying this my text arrive ...
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1answer
23 views

Help explain the set being constructed in this Cantor-Schroder-Berstein proof

The Cantor-Schroder-Bernstein theorem states that: Suppose $A$ and $B$ are sets. If $|A|\le |B|$ and $|B|\le |A|$, then $|A|=|B|$ Proof: So, $|A|\le|B|$ implies we can choose an injection ...
3
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1answer
39 views

Does there exist a connected 2-regular uncountable graph, or an uncountable path?

Does there exist a connected 2-regular uncountable graph? Can I use the axiom of choice to construct an uncountable path of elements from the reals? The question arose when reading this: Also, ...
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2answers
77 views

Sizes of infinity

I was just thinking about infinity (as you do) and thought the following. "There are infinitely many reals in the interval $x\in[0,1]$ and an 'equal number of reals' $x\in[1,2]$, so there are 'double ...
2
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2answers
39 views

Is there a thing named a “spiral plane” which is a plane but it's spiral?

Hello, I'm wondering if there is such thing like this. Is there a plane which is not flat but spiral and extending for infinity? I have drawn a representation for what I mean but it's not thorough ...
3
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7answers
503 views

Is 0.9 repeating = 1 disproved by asymptotes?

I'm discussing proofs that 0.9 repeating equals 1 with some friends, and they use asymptotes to disprove this. One says if we had the function $y=x/0.000\ldots1$ (and he's only using that impossible ...
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1answer
29 views

Degrees of freedom in each domain in Discrete, Continuous and Mixed Fourier Transforms

I'm having trouble with the different infinities involved in the Discrete and Continuous Fourier Transforms. In the DFT, we have a finite number $N$ time domain samples $x(i), 0\leq i<N$, which ...
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1answer
26 views

Help to prove this inductively defined function is surjective

Suppose that $A$ is a infinite subset of $\mathbb{Z}^+$. We construct a bijection $f:\mathbb{Z}^+ \rightarrow A$ and define $f(n)$ inductively as follows: Base case: Let $f(1)$ be the least element ...
2
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4answers
251 views

Can't find limit tending to infinity of a sequence

I'm stumped by $$\lim_{x \to \infty}\frac{1+3+5+\cdots+(2x-1)}{x+3} - x$$ My obvious first step was to get a lowest common denominator by $x(\frac{x+3}{x+3})$, giving $$\lim_{x \to ...
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0answers
15 views

Check proof of union of denumerable sets is denumerable too

I need to prove: If $A$ and $B$ are denumerable sets then so is their union $A\cup B$. In this case, denumerable is defined as: A set $X$ is said to be denumerable if there is a bijection ...
2
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1answer
63 views

The meaning of infinite series $\sum_{i=0}^\infty 2^{-i}$, its relation to partial sums and Cantor's diagonal argument

Let's define $S(n)$ as $S(n) = \sum_{i=0}^n 2^{-i}$. Obviously, $\lim_{n \to \infty} S(n) = 2$ and also $\forall n \in \mathbb{N}, S(n)<2$. Now my questions are about $Q = \sum_{i=0}^\infty ...
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0answers
51 views

What is the answer of -infinity+infinity? [duplicate]

I was wondering what is the answer of the $-\infty$+$\infty$= _. Kindly help me out.
2
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1answer
74 views

Are $+\infty$ & $-\infty $ elements of the real number line? [duplicate]

Can someone give an explanation/proof of whether these two numbers lie on the real number line?
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3answers
59 views

How to prove that a set is infinite iff it is Dedekind infinite?

I need to prove the following: A set $X$ is infinite if and only if it is equipotent to a proper subset of itself Here, $X$ is defined to be infinite if $|X|$ is not a non-negative integer or ...
3
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2answers
87 views

Is there any infinite quantity small enough to be affected by finite changes?

Hilbert's paradox of the Grand Hotel shows us, among other useful things, that the cardinality of any infinite set is a quantity equal to n more than itself for any finite n. I am interested in ...
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2answers
156 views

Is Infinity Needed in Maths? Does Infinity Actually Exist? [closed]

I'm asking this question as I have been having an on going online debate with a friend of mine. I claimed that Infinity does in fact exist in Maths and in Reality, as there's a whole plethora of ...
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2answers
69 views

Bijection between an infinite set and its union of a countably infinite set

I have $A$ as an infinite set and $S$ as a countably infinite set, (so that means there exists a one-to-one correspondence between $S$ and $\mathbb{N}$). How do I show that there always exists a ...
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1answer
46 views

Why is a complex number plus infinity equal to infinity?

Why is $$2 + 3 i + \infty = \infty$$ according to Mathematica and Wolfram Alpha? Shouldn't it be: $$2 + 3 i + \infty = \infty + 3 i$$ ? After all: $$2 + 3 i + 10 = 12 + 3 i$$ and not: $$2 + 3 i ...
3
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1answer
82 views

how to prove : there are an infinite number of points on the circle

I think the follow problem is equal to the problem set 1.16.(a) in Principles of Mathematical Analysis (walter ruldin), And we take (a, b) in $R^2$, X in $R^i$ how to prove : there are an infinite ...
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2answers
623 views

Can the distance between 2 non-empty sets be infinite?

Intuitively I would immediately assume no, but that's not how things usually work in math and considering there are different kinds of infinities I haven't been able to find the answer. Here's my ...
2
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1answer
62 views

Method for computing limit of a sin function as x tends to zero

I have a question about computing $$ \lim_{x \to 0} \sin\left(\frac{\pi x}{4|x|}\right)$$ I found the limit of $\pi x$ and $4|x|$ seperately and ended with $\sin(\pi/4)$ which is equal to ...
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3answers
60 views

Method for computing limit of a function as $x$ tends to zero

I have a question about computing $$\lim_ {x \to 0} \dfrac{(2/x^3)+(1/x^2)+(1/x)+1}{(1/x^3)+1}.$$ I used a shortcut method of dividing by the highest power but I don't think that I can use this method ...
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4answers
120 views

May seem like a noob question: really, why can't we divide by 0? [duplicate]

Yes, I know, can't be answered, blah, blah, blah.... but here are a few of my theories. I know, plenty of other questions like this, but before marking this as a duplicate, consider this, my ...
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1answer
19 views

One set of functions larger than another set of functions?

This summer I've been slowly working through Halmos's Naive Set Theory. I'm not that far, but I know what lies ahead, which is proving that one infinite set is larger than another (the reals larger ...
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4answers
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Can a set be infinite and bounded?

I don't understand a statement in my math book course, I was restudying the compact sets part of the chapter when at a certain moment there is a corollary saying : 'every infinite and bounded part of ...
2
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1answer
77 views

Teaching the Concept of Infinity to Children.

I was recently out with the family and we left it up to the children where we ate lunch (11 and 9 years old). They couldn't agree and were going back and forth calling each other names. This ...
2
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1answer
77 views

Is there a mathematical concept of fractions using transfinite numbers as numerators and denominators?

http://de.wikipedia.org/wiki/Cantors_erstes_Diagonalargument (German) http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument (English) While looking at Cantors method of proof, which he used to ...
7
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3answers
137 views

The nature of infinities

I have been thinking about the nature of infinity lately. I have no experience with higher mathematics or theorems regarding infinity, so please forgive me if my ideas on this topic are extremely ...
0
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1answer
11 views

Deffered annuity with perpetuity

An annuity immediate has $40$ initial quarterly payments of $20$ followed by perpetuity of quarterly payments of $25$ starting in the eleventh year. Find the present value at $4\% $ convertible ...
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1answer
39 views

Comparing density of countable infinite sets by examining the association

The two questions that i am asking are in bold. To be clear, i am talking about whole number here. Having seen 3 is everywhere by Numberphile that shows that almost 100% of the whole the numbers ...
0
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2answers
61 views

The limit of $(x^3+\cos x+e^{-2x})/(x^2 \sqrt{x^2+1})$ as $x\to\infty$

I have this infinity problem which I do not know the answer to: $$\lim_{x\to\infty}\frac{x^3+\cos x+e^{-2x}}{x^2 \sqrt{x^2+1}}$$ I thaught that because $x^3$ is the fastest growing part, this would ...
8
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3answers
482 views

Does 0% chance mean impossible? [duplicate]

Suppose we pick a random real number between 0 and 1 and call it $x$. There are $2^{\aleph_0}$ possible values, so the chance of picking any specific number (such as $x$) in that range is 0. But in ...
2
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1answer
54 views

Question about the cardinality of sets and infinity

Let's say we have $\mathbb{N}$, the set of natural numbers: $\{1, 2, 3, 4, 5...\}$ ...which has a cardinality of infinity, and the set $A_x$ which consists of the variable "$x$" (so $\{x\}$). If I ...
0
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0answers
43 views

Cardinality of infinite between the set of rationals and set of reals

I remember learning that whether or not there is a cardinality of infinity between the set of rational numbers and the set of real numbers is unprovable. Is this true, and if so, how do we know it to ...
3
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0answers
69 views

Is it possible for infinite sets to exist in ZFC with the negation of the Axiom of Infinity? [duplicate]

The Axiom of Infinity states that at least one inductive set exists. Inductive sets are infinite, but not all infinite sets are inductive. Suppose that we take ZFC with the negation of the Axiom of ...
29
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2answers
4k views

Are weird numbers more rare than prime numbers?

By taking a look at the first few weird numbers: $$(70, 836, 4030, 5830, 7192, 7912, 9272, 10430)$$ It is certain that prime numbers occurs more often within this range of numbers. But are weird ...
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4answers
99 views

Evaluate the limit: $\lim_{x\to \infty}$

Evaluate the limit: $$\lim_{x\to\infty} \frac{(2x^2 +1)^2}{(x-1)^2(x^2+x)}$$ The answer is 4 and I don't understand why, but why can't I just do something like:$$\frac{(\infty)}{(\infty)(\infty)} = ...
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1answer
50 views

Are distance-related paradoxes limited by the size of an atom?

See these 2 paradoxes: Coastline paradox The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. ...
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3answers
65 views

Infinitesimal Unit of Measurement

This is just a question that popped into my head which I lack the knowledge to answer (or even to know whether there is an answer, honestly). Does the idea of an infinitesimal unit of measurement even ...
3
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2answers
113 views

I need to understand why the limit of $x\cdot \sin (1/x)$ as $x$ tends to infinity is 1

here's the question, how can I solve this: $$\lim_{x \rightarrow \infty} x\sin (1/x) $$ Now, from textbooks I know it is possible to use the following substitution $x=1/t$, then, the ecuation is ...
1
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6answers
104 views

Help me understand infinity [duplicate]

I asked a math professor once about infinity and his answer puzzled me. I asked if i had two sets of numbers: A = all the whole numbers in infinity B = all the whole and half numbers (1, 1.5, 2, ...
0
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1answer
54 views

How is this definition of a constant divided by zero called?

I divide a constant by zero. One example is the following: 2/0 My father told me he learned at school earlier that the result is "not defined". If I enter this arithmetic problem in Wolfram Alpha, I ...
13
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6answers
2k views

A strange puzzle having two possible solutions

A friend of mine asked me the following question: Suppose you have a basket in which there is a coin. The coin is marked with the number one. At noon less one minute, someone takes the coin ...
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4answers
95 views

Which of these sets is bigger?

I am a fourth year computer science student and I am taking second year level maths because they are very useful for computer stuff. At the end of the linear algebra lecture the Prof left us with a ...
0
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1answer
130 views

Does negative infinity squared = positive infinity?

I googled this question and saw this answer but I wasn't satisfied.