Is 1 divided by infinite zero? (1/inf.=0)?

Question

So it's wrong to say

1. 1/∞=0
2. 1/∞*∞/1=1
3. 0*∞=1

?

Thanks a lot for the many good answers - think I got it now! I think I once fell over this calculus stuff like limes and so on in school (integral calculus), but I can't really remenber.

Anyway, I couldn't decide what's the best answer, so I put it to vote. I tend between Mindscape's, AnyMouse's, DarkNight's and Jim's answer.... But let the people decide ;-) !

Answer 1

You're close, but no (if you are working in the Real number system).

Here is what you are talking about when you say 1/inf.=0

You are talking about 1/something and the something is TENDING towards infinity. Infinity isn't a Real number though so you can't actually define 1/inf. using real numbers. (there are other ways to define 1/inf. dealing with various infinitesimals and transfinite numbers but that will take us pretty far a field).

Anyhoo, I saw someone said 1/0 = inf. You run into the same issue. 1/(something) = (something else). As (something) TENDS to 0, (something else) TENDS to infinity. But we do NOT talk about 1/0 when working in the Real number system, because it leads to inconsistencies.

Here's something that will help you see why

1) a = b
2) a*b = b^2
3) a*b-a^2 = b^2 - a^2
4) a(b-a) = (b+a) * (b-a)
5) a = b+a
6) a = a+a
7) 1 = 2

We don't like things like that in math. We got there because in going from 4 to 5 we divided by zero.

Similarly if you treat infinity like a real number and say 1/inf = 0 you can get

1) 1/inf. = 0
2) 2*(1/inf) = 2*0 = 0
3) 1/inf. = 2*(1/inf).
4) 1 = 2

Also something we don't like. There ARE ways to get around things like this, but they require extending your number system beyond the Reals (Real numbers are basically any number that you've probably seen except for maybe the imaginary number Sqrt[-1]).

Answer 2

It's close to 0 but not exactly zero as people before me said.

See 1/inf goes to 0 , but so does 1/(- inf) go to 0.
One is written 0 with a small +, the other with a small minus (as a superscript) That is a little over 0, or a little under 0.
And there are problems where it matters which way you are approaching the zero, from the positive side or from the negative.

For instance you are solving a limit and finally you reduce it to: 1/0- then you know your answer is - infinity.

2 is wrong, you can't simplify infinity.
3 is wrong (because of 2).

Answer 3

Well, infinity is not a number. It is a limit. No matter how big a number you think of, there is always a bigger number than that.. and so on. So this process leads us to the concept of infinity. Since infinity is not a defined number (it's a limit) therefore the ordinary rules of algebra don't really apply to it.
In limiting situations you do the algebra (if any possible first, then apply the limiting condition)

So your equation 1 / ∞ really should be written as
lim 1/x
x→∞

And yes the answer to that is indeed Zero.

For the second one your equation should be
lim (1/x)*(x/1)
x→∞

And yes for that the answer is 1.

For the third one your equation should read:
lim (0.x)
x→∞

Unfortunately since you do the algebra before taking limits, the part inside the parenthesis is already 0. So the limit has no effect on it. And the answer is Zero.

Answer 4

Divsion by zero is always undefined.

Undefinded means that the result of the calcullation is not a specific number.

If we were to suppose that x/0 were a specific number, say z, we'd have the equation
x/0=z or x=0 times z.

There are no real valued numbers that are solutions to this equation so that is one reason why mathematicians say that divisiion by zero is undefined, _always_.

Even in your case of 1,

1/x=# shows us that x and # are not uniquely determined by the equation.

Indeterminate forms are studied in calculus and analysis classes. The basic iidea is that if there isn't a way to write the number down, it is an indeterminate form: the answer is not unique. So, infinity+infinity=infinity; but infinity-infinity is indeterminate.

Take more calculus classes! :)

You have asked a very nice question!

Answer 5

by using limits in calculus, 1/infinity is equal to zero. The reason being that infinity is so large, and it is on the bottom of a fraction. The larger the number on the bottom, the smaller the fraction becomes. Since infinity is never ending, it is easier to just call 1/inf=zero because the human mind has no other way to percieve the infinitessimally small number.

Answer 6

Yes that is correct. Don't let someone tell you it is a small number, or something like that. It is a common mistake by students to assume it is a really small number.

The answer is 0, and conversely 1/0=inf. Also anything divided by infinity is 0, not just 1. And infinity divided by anything is always infinity.

Answer 7

(1) If 1/ ∞ = 0
then 1 = (0)(∞)
=> 1/0 = ∞
But when you divide a number by zero, the answer is undefined.

1/∞ * ∞/1 = (almost zero)(∞) = 1
The answer is one in this case.

(3) If 1/ ∞ = 0
then 1 = (0)(∞)
but (0)(∞) = 0
=> 1 = 0 This cannot be the case.

Answer 8

Using infinity as a real number leads people to believe that they can do strange stuff. You can see from the above responses that many people who consider themselves versed enough to answer the question do not understand the subtlety themselves.

Infinity is not a real number. What does 1/infinity mean? You mean that you are considering the limit of 1/x, as x approaches infinity. This is zero. It is not a "small number"... Those people who tell you that are confusing a limiting process with the limit itself. For any x very large, 1/x itself is a small but positive number, but the limit IS 0. 1/infinity is the limit. It is zero.

Notice that this does NOT mean that 1 = infinity * 0, as the properties of real numbers do not apply to infinity. In fact, there are lots of ways to go to infinity ( in terms of rates ). In the limit process, we can see "infinity * 0" approaching any real number, or infinity itself. ( 2x approaches infinity and 1/x approaches zero as x approaches infinity, but their product is always 2 )

Nor does 1/0 = infinity. Consider the limit of 1/x as x tends to 0. There are two directions in which to approach 0. One side tends to infinity, but the other direction tends to -infinity. The limit does not exist.

Answer 9

No. It would be an infinitely small number, but not zero. Any number divided by 0 is 0. 0 can not be divided by any number. It's zero. 1/x when x is not equal to zero has some value, but never zero. Infinity can not really be expressed as a number, so 1/infinity = 1/infinity, not zero.

Answer 10

I agree with Hmid inf.=0/1 inf.=0