Question

What is the difference between +∞ and ∞ ?

Answer 1

I don't know that there is one.

Answer 2

0

Answer 3

"+" if we are comparing and contrasting.

Answer 4

I thought \(\infty\) is \(+\infty\), just as \(7\) is \(+7\).

Answer 5

\[+\infty-\infty= ?\]

Answer 6

I think of it as: \[
+\infty \equiv 0+\infty = \infty
\]

Answer 7

Just as \[
-\infty \equiv 0-\infty
\]

Answer 8

no

Answer 9

infinity is a concept, not a number that can be subtracted...
or did you ask the difference in concepts :P

Answer 10

@hartnn I disagree.

Answer 11

+∞ means above 0? I don't know...

Answer 12

@UnkleRhaukus What are you talking about?

Answer 13

infinity is a number ?

Answer 14

It's both

Answer 15

It is not technically a number, but that doesn't mean much. The 'number' is slightly arbitrary classification anyway, just like how prime numbers exclude.

Answer 16

The only point of 'infinity is not a number' is to basically to exclude infinity from most of our algebraic properties and such.

Answer 17

It's actually just a concept, maybe because we can't comprehend it. But then again googolplex is also a number but we can't really comprehend that either.

Answer 18

infinity breaks a lot of algebra, so we exclude it to avoid having to put an asterisk next to everything.

Answer 19

Maybe the plus sign gives infinity direction or value

Answer 20

Saying "infinity is not a number" is like saying "\(i\) is not a real number". The only point is that it doesn't apply in formulas that apply only to numbers. Same way certain formulas about real numbers don't apply to complex numbers.

Answer 21

+∞ is above the x axis.....
And - ∞ is below the x axis.....

Answer 22

\[\infty = scalar~~~ + \infty = vector ? \]

Answer 23

We can add, subtract, and multiply infinity as if it were a number, it's just that it often leads to an indeterminant form. \(0-\infty \) and \(0+\infty\) are prefectly valid.

Answer 24

Wait... did you mean difference as in ... \(+\infty - \infty\)? I thought you meant the difference as in like, are they fundamentally different concepts...

Answer 25

My professor was insisting that +∞ is more precise than ∞, and that ∞ was abuse of notation; I really didn't understand

Answer 26

But \(+7\) isn't an abuse of notation? I'm not sure I agree with him.

Answer 27

whoops, I mean \(7\) instead of \(+7\).

Answer 28

Always increasing to an undetermined numerical value?

Answer 29

Well when using limits it doesn't really effect it?

Answer 30

Unless you view left and right side limits

Answer 31

Or piecewise functions

Answer 32

I think your prof is just being picky, Unkle.

Answer 33

Either notation is fine

\[\lim_{x \rightarrow \infty} ~~~ or ~~~ \lim_{x \rightarrow + \infty}\] unless you look at both + and - and see if it exists or not, other wise I don't see what the big deal is?

Answer 34

Unless \(\infty\) without a \(+\) somehow adds ambiguity... but I don't see how it would?

Answer 35

Abuse of notation!

Answer 36

I vaguely remember some set of numbers that included one infinity that stood for both positive and negative infinity, because it wrapped around on itself. I doubt that's the real reason for distinguishing the two and there probably isn't really one, but just a thought I had. lol

Answer 37

There are various types of complex infinities.

Answer 38

Like \[
\lim_{n\to \infty} an+bni
\]

Answer 39

go on, , ,

Answer 40

What else is there to say? Basically you have \[
\theta = \arctan\left(\frac ba\right)
\]Since \(\theta \in [0,2\pi)\), the set of complex infinities is an infinite, uncountable set.

Answer 41

For \(\theta = 0\), you have \(+\infty\). For \(\theta = \pi\), you have \(-\infty\). You can multiply these infinity and get \((+\infty)(-\infty) = -\infty\). You can probably do something similar with complex infinity.

Answer 42

If you do \[
\lim_{n\to \infty}n + i
\]Then it would seem to go to \(\infty + i\). What would happen then? The angle would appear to be \(0\).

Answer 43

So does \(\infty+i = \infty\) then?

Answer 44

hmmm

Answer 45

Well, it's not too far fetched. \(\infty + 1 = \infty\).

Answer 46

One thing to keep in mind about all of this... and that is that \(\infty \) is shorthand for \(\lim_{n\to \infty} n\) when it is used outside of a limit.

Answer 47

So what I really mean by \(\infty + 1 = \infty\) is \[
\lim_{n\to \infty}n+1 = \lim_{n\to\infty}n
\]

Answer 48

isn't that a recursive definition?

Answer 49

No, I said "when it is outside of a limit".

Answer 50

Both \[
\lim_{x\to \infty}f(x) = L
\]and \[
\lim_{x\to a}f(x) = \infty
\]and all the other combinations have their own definitions. And those definitions do not include infinity.

Answer 51

Then what does ∞ mean when it is inside the limit

Answer 52

Outside of the context of limts, I don't think \(\infty\) has a meaning as far as I know.

Answer 53

\[
\lim_{x\to \infty}f(x) = L
\]This means \[\forall \epsilon \exists \delta \forall x\quad
x>N\implies |f(x)-L|<\epsilon
\]

Answer 54

Whoops, change \(\exists \delta\) to \(\exists N\).

Answer 55

Or just put \(\delta\) in there.

Answer 56

\[
\lim_{x\to -\infty}f(x) = L
\]This means \[\forall \epsilon \exists \delta \forall x\quad
x<\delta \implies |f(x)-L|<\epsilon
\]

Answer 57

\[
\lim_{x\to a}f(x) = \infty
\]This means \[\forall \epsilon \exists \delta \forall x\quad
|x-a|<\delta \implies f(x)>\epsilon
\]
\[
\lim_{x\to a}f(x) =- \infty
\]This means \[\forall \epsilon \exists \delta \forall x\quad
|x-a|<\delta \implies f(x)<\epsilon
\]

Answer 58

Notice how the definitions for limits where you have \(x\to \pm \infty\), they resemble one-sided limits. Technically they are one-sided limits since \(\infty\) is like a wall and you can't approach from the other side.

Answer 59

In my opinion, these definitions of limits also define infinity, and infinity doesn't have any formal meaning outside of them.

Answer 60

Even \[
\int\limits_{-\infty}^{\infty}
\]is just \[
\lim_{t\to-\infty}\lim_{s\to \infty} \int\limits_t^s
\]

Answer 61

Limits are unwieldy and don't always play by the rules of algebra, thus infinity is unwieldy as well.

Answer 62

this is a diffrence :D