Using operations with infinity

[f25274]

I've gotten claims that infinity is not a number but an idea.
How do infinities work in operations?
Are there "smaller" and "bigger" infinities?
If ∞+1=∞, is ∞-∞=1?

 

[klackity]

Infinity is sometimes thought of as a number in complex analysis. The Riemann Sphere is a way of representing the complex numbers where a single point is added at infinity: http://en.wikipedia.org/wiki/Riemann_sphere

Even then, however, we still can't define notions like "∞-∞". However, ∞+1 = ∞ makes perfect sense.

We can formalize it as follows:

x*∞ = ∞ for any x not equal to 0
∞+x = ∞ for any x not equal to ∞
∞-x = ∞ for any x not equal to ∞
x/0 = ∞ for any x not equal to 0

But notice that we can't apply the normal rules of algebra to equations anymore. For example, we cannot multiply both sides of the equation "x/0 = ∞" by 0, because we are not allowed to multiply ∞ by 0 (in much the same way we are not normally allowed to divide by zero!).

 

[abhishek ghos]

think of a large number and you can think of an even larger number, this doesn't ends anywhere. infinity is that number which is larger than every number you can think of and hence acts as a limit to such numbers which are still open at the top. you can see it like all numbers bigger than infinity can also be defined as infinity and all numbers less than infinity by a specific amount are also tending to infinity.

 

[klackity]

And as to whether there are "smaller" or "bigger" infinities, it turns out there are!

In this case, however, "infinity" is not considered a number, but a size. We use different notions of infinity to describe a notion of the size of a collection (or set) of objects. For example, there are infinitely many natural numbers.

When using infinity to describe the size of a collection of things, we call it "cardinality". Instead of saying "There are infinitely many rational numbers" we say, "The rational numbers are countable". "Countable" is a cardinality (i.e., a type of (infinite) size).

We say a collection of objects is countable when we can assign a unique natural number to each of those objects, such that every natural number is used exactly once.

Now, we can also say that a collection of objects has a specific finite cardinality. For example, the set of letters {r,b,g,f,m} has cardinality 5, because I can use ALL the numbers 1 through 5 to label them and do so uniquely. i.e.:

1. f
2. m
3. g
4. r
5. b

The rationals are countable, because we can make a similar list of all fractions:

1. 1/2
2. 35/29
3. -4/5
...

I won't go into how to prove that you can do that!

Anyway, it turns out that the cardinality of natural numbers is the "smallest" infinity there is. The next biggest is the cardinality of the real numbers.

 

[f25274]

I think I get it, so if a sequence includes all positive integers you have 0,1,2,3,4,5,...
an infinite number of numbers but if it includes all integers you have ...,-3,-2,-1,0,1,2,3,... which includes infinite numbers but more than the first sequence. Am I right?

 

[Hurkyl]

I've gotten claims that infinity is not a number but an idea.
How do infinities work in operations?
Are there "smaller" and "bigger" infinities?
If ∞+1=∞, is ∞-∞=1?

There are a variety of algebraic and geometric structures that have infinite elements.

The progression of numbers you learned in school is quite misleading, because those number systems have unusually wide application, and they give the impression there's a "one true system" of numbers and we are only considering which ones to use.

The infinite numbers you use to measure cardinality, for example, have essentially nothing to do with the points $+\infty$ and $-\infty$ on the extended real number line that you frequently use in calculus.



If you want to measure sequences in a way that ...,-3,-2,-1,0,1,2,3,... is bigger than 0,1,2,3,4,5,..., then cardinality is not appropriate for that purpose -- the sets of numbers in those sequences have the same cardinality ($\aleph_0$, the cardinality of the natural numbers), as can easily be seen by reordering them:

0,  1, 2,  3, 4,  5, 6, ...
0, -1, 1, -2, 2, -3, 3, ...

The two sequences you stated have different order types, though.

Order type can't distinguish between

0, 1, 2, 3, 4, 5, 6, ...
1, 2, 3, 4, 5, 6, 7, ...

these two sequences have the same order type.

You could, of course, compare them by instead trying to measure the difference of the two sets.

 

[f25274]

Let me clarify this:
I know there are more terms between 0 and 2 (counting fractions) than between 0 and 1.
They both have infinite terms between them but is the cardinality between 0 and 2 but is it twice the cardinality between 0 and 1?

 

[spamiam]

They both have infinite terms between them but is the cardinality between 0 and 2 but is it twice the cardinality between 0 and 1?

Surprisingly, no. When comparing cardinalities of two sets A and B, we look for one-to-one correspondences between them, which is a way of identifying each element of A with a unique element in B. We say that two sets have the same cardinality if there is such a correspondence between them. I can easily make such a map between [0,1] and [0,2]: just multiply by two. It takes a little proving to show that this function is really one-to-one (that is, each element of [0,1] gets identified with a unique element of [0,2]) and onto (every element of [0,2] gets "hit" by something from [0,1]).

So even though intuitively, there are "more" numbers in [0,1] than in [0,2], these sets have the same cardinality. Infinite sets can lead to some very counter-intuitive results!

 

[jambaugh]

...They both have infinite terms between them but is the cardinality between 0 and 2 but is it twice the cardinality between 0 and 1?

Surprisingly, no. [...]
So even though intuitively, there are "more" numbers in [0,1] than in [0,2], these sets have the same cardinality. Infinite sets can lead to some very counter-intuitive results!

Technically the answer to the question as posed is "Yes", because twice an infinite cardinal is equal to that cardinal value. This isn't so strange, after all, the cardinality of the empty set is also equal to twice its value as well.

You can define addition and multiplication (and powers) of cardinal numbers consistently as long as you avoid subtraction and division (which is simply the absence of negative or reciprocal cardinal numbers). There are corresponding set operations e.g. disjoint unions for +, cartesian products for *, and "sets of all mappings" for ^.

BTW The OP mentioned "...that infinity is not a number but an idea." but numbers are also ideas. The real question is how are they used. We have e.g. infinite cardinal numbers (counts of things) and infinite ordinal numbers (positions in sequences of things). We have multiple orders of infinity in each case as well.

Check out the wikipedia article: http://en.wikipedia.org/wiki/Transfinite_number" for more details.

 

[spamiam]

Technically the answer to the question as posed is "Yes", because twice an infinite cardinal is equal to that cardinal value. This isn't so strange, after all, the cardinality of the empty set is also equal to twice its value as well.

Oops, my mistake!

 

[A.M.L.]

I have to agree with the assessment that infinity is an idea, not a number. I think that's rather intuitive as well, but that's a personal sentiment.

For example:

Taking the integral of 1 to infinity of 1/x^2 does not exist as a Riemann Integral.

It has a domain that is unbounded.

If one tries to invoke the Second Fundamental Theorem and plug and chug, it works.

By conventional wisdom,

(-1/infinity) - (-1/1) = 1

However, if you were to write this on a test, it would not be given credit. -1 CANNOT be divided by infinity. Afterall, infinity contains many, many, many numbers. Is it 1/2 or 1/100000?

Instead, we take the integral from 1 to n. We then take the indefinite integral and plug in n.

This gives us -1/n - (-1/1).

To find your integral, we find the limit of the above as n approaches 0.

limit as n approaches 0 of (-1/n) - (-1/1) = 1.

Giving us the correct answer.

Hope that helps. I tried to keep my explanation limited to someone who has taken basic Calculus courses.

 

[TylerH]

That's a really long way of saying that infinity is not in the set of real numbers.