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What is $\infty - \infty$?

Is it $\infty$ or $0$ or what?

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Very relevant: en.wikipedia.org/wiki/Indeterminate_form. Hopefully, someone will write a nice answer. –  Srivatsan Aug 30 '11 at 18:19
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@Sri I see no "indeterminate forms" in the question. –  Bill Dubuque Aug 30 '11 at 18:34
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@Sri "Infinity" has many different meanings in mathematics. As such, the question is ill-posed as it stands. –  Bill Dubuque Aug 30 '11 at 18:48
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With natural numbers the result of taking $m$ items away from a set of $n$ items, $m\le n$, will result in a set with a unique number of items in it (no matter which subset of $m$ items were taken away), hence it makes sense to label the result as a specific number, namely $n-m$. With an infinite ($\aleph_0$) set of items, taking away an infinite subset doesn't uniquely determine the cardinality of the resulting set, hence $\infty-\infty$ is ill-defined in this context. –  anon Aug 30 '11 at 19:28
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11 Answers 11

In any number system where one class of numbers (the infinite) is bigger than all the numbers in another (the finite) there will be should usually be more than 1 "infinity". For example, in the hyper-reals, if $H$ is infinite, $2H, \frac H2, H+1,$ and $H-1$ are all infinite.

$H-2H=-H$ which is negative infinite.

$H-\frac 2H=H/2$ which is positive infinite.

$H-(H+1)=-1$ which is negative finite.

$H-(H-1)=1$ which is positive finite.

$H-H=0$ which is zero.

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From a layman's perspective, imagine that I have an infinite number of hotel rooms, each numbered 1, 2, 3, 4, ...

Then I give you all of them. I would have none left, so $\infty - \infty = 0$

On the other hand, if I give you all of the odd-numbered ones, then I still have an infinite number left. So $\infty - \infty = \infty$.

Now suppose that I give you all of them except for the first seven. Then $\infty - \infty = 7$.
While this doesn't explain why this is indeterminate, hopefully you can agree that it is indeterminate!

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wow i actually understand this ! –  Pacerier Aug 30 '11 at 19:16
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I don't quite understand that last comment, but I'm almost certain that the answer is "no". –  Altar Ego Aug 30 '11 at 19:19
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@Altar: This is a well-known paradox. Google "Hilbert's hotel" for more information. –  Fredrik Meyer Aug 30 '11 at 19:27
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@Altar Ego: Congratulations on getting away from the usual limit examples. It would spoil the clarity of what you say, but let $A$ is a (finite) set of hotel rooms, say $a$ of them, and if $B$ is the set of occupied hotel rooms, and there are $b$ such, then the number of unoccupied rooms is $a-b$. So subtraction does give the answer in the finite case. –  André Nicolas Aug 30 '11 at 19:42
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@trony: I had "there are as many even numbers as there are natural numbers" in mind... –  J. M. Aug 30 '11 at 23:50

Perhaps an uninteresting way of speaking about infinity, but one you surely will understand, the first one I was taught and which is at the level of Introductory Calculus. If you consider the sequence

$$1,4,9,16,25,36,\ldots ,n^{2},\ldots \tag{1}$$

and the sequence

$$1,2,3,4,5,6,\ldots ,n,\ldots \tag{2}$$

both go to infinity as $n$ tends to infinity (I will add if you deem necessary the meaning of "going to infinity"). The sequence obtained by subtracting $(2)$ from $(1)$ goes to infinity too

$$0,2,6,12,20,30,\ldots ,n^{2}-n,\ldots \tag{3}$$

If you consider now the sequence

$$\frac{1}{1},\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{24}{5},\frac{35}{6}% ,\ldots ,\frac{n^{2}-1}{n},\ldots \tag{4}$$

which goes to infinity too and subtract it from $(2)$ you get a sequence which tends to $0$

$$0,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},\ldots ,\frac{% 1}{n},\ldots \tag{5}$$

If you take sequence $(2)$ and subtract from it the sequence

$$0,1,2,3,4,5,\ldots ,n-1,\ldots \tag{6}$$

you get the constant sequence

$$1,1,1,1,1,1,\ldots \tag{7}$$

So the difference of two sequences both going to infinity may be a sequence which tends to $\pm\infty $, $0$ or a finite number.

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Or negative infinity! –  nightcracker Oct 18 '11 at 10:08
    
@nightcracker: That's right. –  Américo Tavares Oct 18 '11 at 11:11

There is a simple way to show how we can define such problems. .Simply write the equation and try to find $x$.

$\infty - \infty =x$

$\infty =x+ \infty$

If you try to put some values of $x$, such as 1,2,3,.... you will notice that all numbers will be correct because if you add any number to infinity,it will be again infinity or if you select any negative number , infinity will not loss anything from infinity, it will be again infinity. If so what is $x$? x can be any number, therefore we say such x as indeterminate.

If you try the same method to other indeterminate expression such as $$\frac{0}{0}$$

$$\frac{0}{0}=x$$ what is $x$?

$$\frac{0}{0}=\frac{x}{1}$$

$$x.0=0.1$$

$$x.0=0$$ $x$ can be any number because if we multiple any number with zero result will be zero. Again similiar result as we got for $\infty - \infty$.

Another important property that all indeterminate expressions can transform to another form. please see examples below.

Example 1: $$0.\infty=$$

$$=0.\frac{1}{0}=\frac{0}{0}$$


Example 2: $$\frac{\infty}{\infty}=$$

$$ =\frac{\frac{1}{0}} {\frac{1}{0}} =\frac{0}{0}$$


Example 3: $$\infty - \infty=$$

$$ =\frac{1}{0}-\frac{1}{0}= \frac{1-1}{0}=\frac{0}{0}$$


Example 4: $$1^\infty=x$$

$$ln(x) = ln(1^\infty)= \infty.ln(1)=\infty.0=\frac{1}{0}.0= \frac{0}{0}$$

If $ln(x)$ is indeterminate , thus $x$ is also indeterminate

Similiar way can be shown for $\infty ^ 0$ as shown in example 4. These examples show that indeterminate expressions can convert to other type. It is very important to find limit values in mathematics.

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I can't comment on the merit of the bottom half of this answer, but I can say that the top is respectable. As such, I'll act on good faith and +1. Feel free to modify the bottom--I do not understand it enough to fully help you clarify it. –  000 Mar 17 '12 at 21:00

If we interpret $\infty$ to mean some infinite surreal number such as $\omega = \{ \mathbb{N} \mathrel{|} \emptyset \}$, then yes, $\infty - \infty = 0$, because all surreal numbers have additive inverses.

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The quantity $\infty - \infty$ is called indeterminant because $\infty - \infty$ could be anything in the set $[-\infty, \infty] = \mathbb{R} \cup \{\pm \infty\}$. Consider the limit

$$ \lim_{x \to \infty} (f(x) - g(x)). $$ If $f(x)$ and $g(x)$ are polynomials (or arbitrary functions which tend to infinity), the limit is of the form $\infty - \infty$, but we can concoct examples where the limit can be any number. For example, let $\alpha$ be an arbitrary real number, and define $f(x) = x+ \alpha$ and $g(x) = x$. Then the limit is $\alpha$. Furthermore, if $f(x) = x^2$ and $g(x) = x$ (or vice versa), then the limit becomes $+ \infty$ (or $-\infty$). Therefore there is no reasonable way to define $\infty - \infty$.

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I will take $\infty$ to mean a quantity which grows without bound. There are many such quantities, but the distinction is usually made after one is introduced to calculus, and since my answer is based in elementary calculus, I will not worry about the distinction here. –  JavaMan Aug 30 '11 at 19:07
    
It is "indeterminate", not "indeterminant". Also I would say indeterminate form or expression, because being indeterminate here means exactly that no definite value (quantity) can be unambiguously associated to the expression. –  Marc van Leeuwen Aug 21 '13 at 10:25

infinity minus infinity is not defined since infinity itself is not defined

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Welcome to math.SE! I'm afraid this answer is incorrect; there are many valid ways of defining infinity. Look at any of the answers above for a correct explanation of the issue. Please improve your answer, or it may be deleted. –  Zev Chonoles Aug 30 '11 at 21:51
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Will it really be deleted just for being incorrect? –  Bonanza Aug 30 '11 at 23:37
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It's not even wrong.... –  The Chaz 2.0 Aug 30 '11 at 23:53
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"Infinite" as a modifier is defined in various contexts in various ways. I would be surprised to see bare "infinity" defined in a mathematical text. Maybe in a book on mathematical theology. One could argue that the above answer is the only right one. –  André Nicolas Aug 30 '11 at 23:59
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This should not be deleted. André has a very good point. –  Jonas Meyer Aug 31 '11 at 2:14

$$ \int_0^1 \int_0^1 \frac{x^2-y^2}{(x^2 + y^2)^2} \, dy\,dx \text{ is actually different from }\int_0^1 \int_0^1 \frac{x^2-y^2}{(x^2 + y^2)^2} \, dx\,dy. $$ One of these is $\pi/4$; the other is $-\pi/4$.

That's infinity minus infinity. I.e. if you integrate over the part of the square where $x^2-y^2$ is positive, you get $\infty$, and if you integrate over the part that's negative, you get $-\infty$.

If you have $\infty$ minus a finite positive number, then the integral would be $\infty$ either way; if you have $-\infty$ plus a finite positive number, then it's $-\infty$ either way. And if both the positive and negative parts are finite, then you get the same number either way. This sort of bad behavior of integrals, where changing the order of integration can change the number you get as the bottom line, can happen only when the positive and negative parts are both infinite.

The same thing happens with infinite series, as you'll see if you google: conditional convergence Riemann rearrange

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One reason the answer is indeterminate is because you can find sequences $x_n,y_n$ of real numbers such that $x_n,y_n \to \infty$ and $(x_n-y_n)$ can converge to any real value or $\pm \infty$.

For example pick $a \in \Bbb{R}$ and $x_n=n+a,y_n=n$. Then $x_n-y_n=a,\ \forall n$ and therefore $x_n-y_n \to a$.

For $x_n=2n,y_n=n$ you get $\infty$. Switch the order and get $-\infty$.

Indeterminations are usually taught in analysis courses, and the main reason they are called this way is that you cannot say what is the value of the limit from the start. Other examples are $\infty \cdot 0, 1^\infty, \infty^0, \frac{\infty}{\infty},\frac{0}{0}$...

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When trying to invent new number systems with numbers we name, for example, "infinity", we must define the rules of operation. If you decide to adjoin the symbol $\infty$ to, say, the real numbers, then you must decide which properties you want the symbol to have.

For example, do you want (which is reasonable) $x+\infty=\infty$ for every real number $x$? If so, your new system, $\mathbb{R} \cup \{\infty\}$ can no longer be a ring, so you lose some of the important properties of the system.

Usually, the symbol $\infty$ is only used to indicate that limits "grow beyond any number", for example. In this case, $\infty-\infty$ depends on what limits are in question.

In other situations, $\infty$ is used to formally "complete" a topological space, say $\mathbb{C}$. All new students of topology learn that the sphere and $\mathbb{C} \cup \{\infty \}$ are homeomorphic spaces, that is, essentially the same.

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There has been an interesting paper on this issue in the first issue of Rejecta Mathematica: math.rejecta.org/vol2-num1/44-48 –  Michael Greinecker Dec 29 '11 at 23:15

The result is Indeterminate.

The reason being because you could never come to a concise answer:

Is the 1st infinity larger? Then answer would be +infinity.

Is the 2nd infinity larger? Then answer would be -infinity.

Are the same size? Then answer could be 0.

Since the size of infinity is unknown, we cannot determine any of these situations and therefore the answer is Indeterminate.

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Well: Take this example $\lim_{x \to \infty} (x+2)-(x)=2$. This is an expression of the form $\infty-\infty$, but "the result" is 2, even though one "infinity is larger" than the other. –  Fredrik Meyer Aug 30 '11 at 18:55
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But you've given an answer to a different thing from to the question: you determined the expression, but in the question it is indetermined (underdetermined,indeterminate) –  Gottfried Helms Aug 31 '11 at 2:57

protected by Zev Chonoles Aug 30 '11 at 23:16

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