# can there be a negative infinity?

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if there's a positive infinity, can there be a negative infinity?

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Hi BT,

In short, yes.

So here's a little homework for you: can you think of an example...?

Cheers

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an example of negative infinity? well if you subtract infinity from infinity, instead of being zero, it would be negative infinity.

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an example of negative infinity? well if you subtract infinity from infinity, instead of being zero, it would be negative infinity.

It doesn't quite work that way...

Try to think of something that can be infinite, and then think if that (whatever it is) can take on negative values. Keep trying until you come up with something.

cheers

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no, infinity subtracted from infinity would still be infinity.

negative infinities can exist when you take limits of certain curves... this topic is covered calculus. How much maths have you taken?

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no, infinity subtracted from infinity would still be infinity.

Hi ecoli,

It doesn't quite work that way either...

Infinity - infinity is indeterminate.

It is interesting to note that infinity doesn't always equal infinity. There are different degrees of infinity (or so it is thought...). This has to do with something called cardinality.

The cardinality of a set is just the number of elements in the set. For example {1,4,d,y,&} has a cardinality of 5.

The cardinality of the positive integers {1,2,3,...,infinity} is called aleph-null. It is also the cardinality of all the integers including zero {-infinity,...,-3,-2,-1,0,1,2,3,...,infinity}. This set is what is called countably infinite.

Now the reals on the other hand, are also infinite, but that set is uncountably infinite (try it!). The cardinality of the reals is called aleph-one.

If you can find a one-to-one (injective) and onto (surjective) mapping [i.e., a bijection] from one set to another, then you can show that the cardinality of the two sets are equal. This cannot be done between the integers and the reals.

If this stuff interests you, read up on Georg Cantor and the continuum hypothesis.

Cheers

w=f[z]

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You know, even the title of threads like this make my head hurt. I get some sinking feeling I'm going to be sucked into some sort of Star Trek like paradox.

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no, infinity subtracted from infinity would still be infinity.

negative infinities can exist when you take limits of certain curves... this topic is covered calculus. How much maths have you taken?

Many of us may have taken more than you may have...but in my case it was 30 years ago. Pre calculaters when a science student's best friend was the slide rule.

However, you are correct if I stretch my few remaining brain cells back to university days. We did a problem once with the proof that negative infinity and infinity were one and the same BUT that infinity minus negative infinity did not equal zero. this was proven in linear algebra and also in what we used to call projectile curving planes. The word Reeman Sphere (sp?) pops into my brain but if I think about it any longer a fuse is going to blow. Jeopardy is coming on and I need to save some cells so my wife doesn't humiliate me.

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However, you are correct if I stretch my few remaining brain cells back to university days. We did a problem once with the proof that negative infinity and infinity were one and the same BUT that infinity minus negative infinity did not equal zero.

It all depends on what setting you pick. If you take the real number line, then you can project this onto $S^1 \backslash \{a\}$, and then a is called 'the point at infinity', giving the projective line. So in this sense, "$\infty = -\infty$". Conversely, since $\mathbb{R}$ is a totally ordered set it might be more convenient to define the extended reals: $\overline{\mathbb{R}} = \mathbb{R}\cup\{\infty\}\cup\{-\infty\}$.

this was proven in linear algebra and also in what we used to call projectile curving planes. The word Reeman Sphere (sp?) pops into my brain but if I think about it any longer a fuse is going to blow. Jeopardy is coming on and I need to save some cells so my wife doesn't humiliate me.

The Riemann sphere is exactly the same thing as identifying a point at infinity by using stereographic projection onto the sphere.

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I think it should be classified that most of these posts are all dealing with different things referred to as infinity.

Ecoli is talking about the infinity of calculus which technically exists only as a limit.

w=f[z] is talking about the infinities which are transfinite cardinalities.

KLB is talking about To Infinity, and Beyond!

And geoguy and dave are talking about "the point at infinity" which exists in some non-euclidean geometries, most notably (as was said) Reimann geometry, which is a geometry on a sphere.

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The question then, is what is bored_teen talking about?

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of course there can be negative infinity, same way there is positive;

what's wrong with that explanation?

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of course there can be negative infinity, same way there is positive;

what's wrong with that explanation?

That $\sum^{\infty}-1$ only tends to negative infinity, there is no infinity (negative or otherwise) in that context except as a limit?
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hmm if we take the concept of infinity as a whole, then it exists of course, otherwise we wouldn't know about it would we, it's not imaginery? infinity in itself maybe doesn't exist, but the concept of it does, and that is one of the ways how we can express that concept, isn't it?

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Most of maths we only know about as a result of us making it up, of course it's imaginary. Just like numbers.

Anyways, the example you gave merely showed that negative infinity can be a limit, which whilst true, is far from the whole story.

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hmm if we take the concept of infinity as a whole, then it exists of course, otherwise we wouldn't know about it would we, it's not imaginery? infinity in itself maybe doesn't exist, but the concept of it does, and that is one of the ways how we can express that concept, isn't it?

Can you restate what you mean?

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note to self: create no more math threads!

no, infinity subtracted from infinity would still be infinity.

negative infinities can exist when you take limits of certain curves... this topic is covered calculus. How much maths have you taken?

high school algebra, high school geometry. next year is high school algebra 2. so not a lot.

so there's lot's of infinities? or infiniti? either way, i'll be gone for a week or so. bye!

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Well, that's just my idea of what I understand about infinity, and so far, I have not had any problems based on that understanding and if you could make me think otherwise I'd be thankful for you would be pointing out something I did not previously know (which is very likely since I only have 1 semester worth of maths). Otherwise it fits into my reasoning well. When I think about infinity I take a point 0 on 1 dimensional axis, and the evermoving movement in the negative direciton, say left, can be represented as $-\infty$. Likewise, I think of positive infinity as the evermoving movement (excuse the expression) in the positive direction. If you apply this to some problem in physics or math then it would depend on what kind of problem it is, wouldn't it? An example of my own would be movement in 1 dimension (having 2 directions, forward and backward) such as time. Time running forward forever (we don't know that of course, but say if it was running forward forever) could be represented as positive infinity, likewise, if it was running backwards forever it could be represented as negative infinity. (but bear in mind that point of reference t= big bang can be used if we make an assumption that time existed before the big bang, that is, that time did not start with big bang (otherwise negative infinity with time would not exist), now how true that is I don't know and at this point I don't care, and is also not to be used as a counterattack on my example, feel free to come up with a better example if you don't like concept of time)

As you say the tree (couldn't you have thought of a better name? jks), you could think of it as imaginary, of course, but if you write it down on paper "infinity" the word itself is not imaginary. However, if you think of it objectively, you could say it's imaginery as of course, you can never reach infinity by definition. (this is really our difference in understanding of the word "imaginary")

cosine; That's what I meant (above), infinity exists as a concept, you can look it up in the dictionary can't you? But infinity by definition doesn't exist, simply because you can't see the end.

To our highschool friend, wouldn't it be the simplest to explain it that way too? Take a line with center zero, the unlimited movement in the negative direction (define negative as to the left of point 0) would be negative infinity, and to the right, positive.

$-\infty$..................... 0 .....................$+\infty$

Don't let the dotted lines trick you, they continue forever in both directions. And point of reference 0 can be started at any point in time. For example, if you started point 0 (reference point) in 1944, you could write 1944 instead of zero, but then again 1944 is human-defined point in time (1944 Anno Domini).

Example when working with that explanation: can you predict what's going to happen to the following function as x approaches into +infinity (becomes unstoppably large in positive direction):

$f(x) = \frac{1}{x}$ ?

Disagree with that?

As a sidenote, I don't understand this bit about limits... If infinity is infinite how can it have a limit? There is noone that can prove that, it's against the definition isn't it? I remember finding limits at infinity in my math unit, but they referred to limits of the function and not to the limits of infinity.. right?

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note to self: create no more math threads!
Nothing wrong with lots of things to talk about.

so there's lot's of infinities? or infiniti?
Think about how many numbers there are in the set of real numbers, now how many in the natural numbers. Both infinite but not the same.
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As a sidenote, I don't understand this bit about limits... If infinity is infinite how can it have a limit? There is noone that can prove that, it's against the definition isn't it? I remember finding limits at infinity in my math unit, but they referred to limits of the function and not to the limits of infinity.. right?

Well there are several types of infinities we talked about. (cf: http://www.scienceforums.net/forum/showpost.php?p=343622&postcount=10)

But the sense in which you are talking about is infinity as a limiting concept. (cf: http://en.wikipedia.org/wiki/Limit_%28mathematics%29)

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"Infinity" can mean several different things, depending on context. Most of the time, "negative infinity" is just as meaningful as "infinity." Sometimes it isn't.

How many even numbers are there? Now, how many integers are there? Twice as many, right? Since every even number is also an integer, and to every even number you can just add one and get an integer that is not an even number? But wait. Look at your list of integers and double every one. Now look at your list of answers: the even numbers, one for each integer, and no repeats! Now it looks like there is the same number of integers and of even numbers, because you can match them up, one to one.

How can that be? Well, that's one of the reasons we say that infinity is not a number. We can't have numbers be both equal to themselves and not equal to themselves. That's why we say that infinity does not equal infinity. It also does not NOT equal infinity, because "equal" is something which just doesn't apply to the concept of infinity.

BTW, there are different "kinds" of infinity. The two "infinities" that I used, the set of integers and the set of even numbers, are considered the same kind because you can match them up one to one (you can also do this with all rational numbers). However, there are "infinities" which you can't do that with. You can't do it with irrational numbers, for example. You can give a different irrational number for each rational number, but you can't do it the other way around, and so the set of irrationals is a "higher" infinity than the set of rationals. There are, actually, an infinite number of different kinds of infinities, each "more" infinite than the last.

If that doesn't freak you out, you're probably doing something wrong...

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• 1 month later...

There is no such thing as infinity just forever. So there is no infinite numbers because infinite numbers are not real numbers nor are negative real numbers. You cannot mathamatisize any known value against infinity. Forever means to go on without end. Infinite means an unknown quantity given it's length. All quantites are knowable but not postustable. Given an unknown quantity are question mark would be prefered by myself and some others.

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Perhaps there is no such thing as zero either? [/TongueInCheek]

Try quantizing love. When you fail, it does not indicate it's nonexistence, only your own inability to quantize it.

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There is no such thing as infinity just forever. So there is no infinite numbers because infinite numbers are not real numbers nor are negative real numbers.
So numbers have to be real to be useful do they?

And did you just say that negative reals aren't real?

You cannot mathamatisize any known value against infinity.
If you're going to be making up words you can do pretty much whatever you like.
Forever means to go on without end.
Yes...
Infinite means an unknown quantity given it's length.
No. Infinite means without end (it's different to forever, forever implies something happening whereas infinity is quite static).
All quantites are knowable but not postustable. [sic]
What type of quantity is it not possible to postulate?
Given an unknown quantity are question mark would be prefered by myself and some others.
The modern convention for unknown quantities is $n$ or $x$. A question mark may refer to the Minkowski question mark function, but what use has it for an unknown quantity? Who are these others? And what have unknown quantities got to do with infinity?
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You are correct. Forever means a beginning. I think I had it backwards. Good call. I do not belive in infinite numbers because I feel they have no useful quanitities. Since they have no quantity at all. Postulating the size of the universe is one since a person cannot truely know the it's size or even the theories behind it's conception.

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