Infinity (2 Viewers)

[XcarvengerX]

Which one of these are correct. Why or why not?
infinity + 1 = infinity
infinity - infinity = 0
infinity x infinity = infinity
infinity - 1 = infinity
infinity : infinity = 1
0 : infinity = 0
1 : infinity = 0
infinity : 0 = undefine
x to the power of infinity = infinity
derivative of x to the power of infinity = infinity
integrate x to the power of infinity = 0

I don't know the anwers too as this is open for discussion.
If you have more, just add it here.

 

[darkliight]

Just my thoughts and I'm not sure if I'm right, but, I don't think many of them are correct strictly speaking. Infinity is not a number so you can not do normal operations with it. We can however take ...

lim_{n->oo} (n + 1); tends to infinity (ie. it diverges)
lim_{n->oo} (n - n) = lim_{n->oo} 0 = 0
lim_{n->oo} n^2; tends to infinity
lim_{n->oo} n - 1; tends to infinity
lim_{n->oo} n/n = lim_{n->oo} 1 = 1
lim_{n->oo} 0/n = lim_{n->oo} 0 = 0
lim_{n->oo} 1/n = 0
lim_{n->oo} n/0; undefined for all n
lim_{n->oo} x^n would depend on x, |x| < 1 tends to 0, |x| > 1 tends to oo
The calculus stuff I don't think would be defined, as oo isn't a number, think back to first principles and the definition of the derivative.

Just my two cents.

 

[Raginsheep]

Agree with above. Infinity is not a number but rather an abstract concept.

BTW, infinity-infinity=0 may not be entirely correct or some of the others.
eg. lim (x->oo) x - lnx =/= 0.

 

[XcarvengerX]

Infinity is the largest number, isn't it? It is the limit of number or everything in general.

Also just notice that anything over infinity is 0, so infinity times 0 would be undefined as it can be anything, like the case of anything over 0.

 

[Templar]

For x>=2, x^inf is a set of infinity with more members than the original infinity.

Or, x^N₀>N₀

 

[Raginsheep]

Infinity is the largest number, isn't it? It is the limit of number or everything in general.

Also just notice that anything over infinity is 0, so infinity times 0 would be undefined as it can be anything, like the case of anything over 0.

Like I said, infinity is not a number. It's pretty much just a technical term of saying "something really big". Of course, something really big is different to something even bigger.

 

[XcarvengerX]

If infinity is not something that is the biggest (or the limit), then what is that something even bigger than this something really big (infinity)?

For 1 to the power of infinity, the answer is still 1.

 

[Raginsheep]

Like I said, infinity is a concept which that does not have absolute definitions.
I gave you the example before where y=x for very large x becomes y=00. y=lnx for very large x also becomes y=00. BUT, y=x > y=lnx for very large x and thus one infinity is bigger than another infinity.

 

[XcarvengerX]

Like I said, infinity is a concept which that does not have absolute definitions.
I gave you the example before where y=x for very large x becomes y=00. y=lnx for very large x also becomes y=00. BUT, y=x > y=lnx for very large x and thus one infinity is bigger than another infinity.

But at infinity, it will equal even though it seems like y=lnx is smaller than y=x. I think it's called divergence or something, like in the case of infinity² is equal to infinity even though it seems like infinity is smaller than infinity².

They shouldn't removed "Convergence and divergence of infinite series" topic from 4 unit Math syllabus.

 

[Raginsheep]

http://en.wikipedia.org/wiki/Infinity#Mathematical_infinity

 

[Templar]

Too bad the Continuum Hypothesis is undecidable. Then we would have all the sets of infinite nicely constructed and following each other.

 

[XcarvengerX]

Is this article reliable? Looks like it is.

An extract from the article above:

Infinities as part of the extended real number line

Infinity is not a real number but may be considered part of the extended real number line, in which arithmetic operations involving infinity may be performed. In this system, infinity has the following arithmetic properties:

Infinity with itself

1. infinity + infinity = infinity times infinity = (-infinity) times (-infinity) = infinity
2. (-infinity) + (-infinity) = infinity times (-infinity) = (-infinity) times infinity = -infinity

Operations involving infinity and real numbers

1. -infinity < x < infinity
2. x + infinity = infinity and x + (-infinity) = -infinity
3. x - infinity = -infinity
4. x - (-infinity) = infinity
5. (x over infinity) = 0 and (x over -infinity) = 0
6. If 0 < x < infinity then x times infinity = infinity and x times (-infinity) = (-infinity).
7. If -infinity < x < 0 then x times infinity = -infinity and x times (-infinity) = infinity.

Undefined operations

1. 0 times infinity and 0 times (-infinity)
2. infinity + (-infinity) and (-infinity) + infinity
3. (±infinity over ±infinity)
4. (±infinity)⁰
5. 1^(±infinity)

Notice that [(x over infinity) = 0] is not equivalent to [0 times infinity = x]. This is because zero times infinity is undefined.

Infinity may be considered as number, so arithmetic operations involving infinity may be performed. However, I don't think derivative and integration of infinity could be performed. Same as imaginary number i...

 

[SeDaTeD]

Oh, you can do differentiation and integration using i. It's just a constant. But be careful if you are dealing with functions of complex variables, but you won't need to worry about that yet.

 

[Templar]

Oh, you can do differentiation and integration using i. It's just a constant. But be careful if you are dealing with functions of complex variables, but you won't need to worry about that yet.

Integration in the complex field gets messy though.

Infinity is all things greater and smaller than 0.. So inf - inf = 0 is based on the assumption that it is a positive number that is relative to 0.. It isn't relative to anything and breaks all boundaries.

Do you know what you're talking about?

 

[SeDaTeD]

We best ignore him.

 

[XcarvengerX]

Oh, you can do differentiation and integration using i. It's just a constant. But be careful if you are dealing with functions of complex variables, but you won't need to worry about that yet.

Do you mean I can differentiate z=x+iy? Do I use dy/dx or dz/di? If you don't mind, please tell me how to do it. Thanks.

 

[Templar]

You can differentiate and integrate functions containing complex variables. However they are different to real functions and can get messy, so it's best if you read it from a book than trying to be told here.

 

[dom001]

The most obvious thing to consider is that infinite is not finite. it is not a number, it has no value; rather, it is a concept.
Therefore,

Which one of these are correct. Why or why not?

infinity + 1 = infinity
true

infinity - infinity = 0
to be honest, i dont think thats possible. unsure about it.

infinity x infinity = infinity
true. infinity^2 ~ infinity + infinity ~ infinity

infinity - 1 = infinity
true

infinity : infinity = 1
same as the infinity - infinity. not sure.

0 : infinity = 0
true. though infinity is not a number, 0 divided by any value remains zero, regardless.

1 : infinity = 0
not true. it would be something like 1/infinity = 1x10^-oo.

infinity : 0 = undefine
true. same as 0/infinity

x to the power of infinity = infinity
true (x in R, x>1, x<o. for 0<=x<=1, it is 10^-oo)

derivative of x to the power of infinity = infinity
d(x^oo)dx = oo(x^oo-1) ~oo(x^oo). true for x in R, x=/ 1 or 0. You could also argue the interval 0<x<1.

integrate x to the power of infinity = 0
S(x^oo)dx = (1/oo)x^oo-1 ~ (10^-oo)(x^oo). well this is a bit weird; any comments on this one would have to include intervals for x, and even so, im not sure it would be at all valid.

 

[Templar]

Actually, any number divided by infinity is 0.

inf-inf is undefined, so is inf/inf.

 

[XcarvengerX]

The most obvious thing to consider is that infinite is not finite. it is not a number, it has no value; rather, it is a concept.

Ok. Imagine this:
How many numbers lie between 0 and 1? Infinite.
How many numbers lie between 0 and 2? Infinite.
It is a common sense that the second infinite must be twice as big as the first infinite, but we still called it as infinite anyway. As Raginsheep posted previously:

Like I said, infinity is a concept which that does not have absolute definitions.
I gave you the example before where y=x for very large x becomes y=00. y=lnx for very large x also becomes y=00. BUT, y=x > y=lnx for very large x and thus one infinity is bigger than another infinity.