Is 1 divided by infinity equal to zero?

[Nick666]

Well? Is it equal to zero ? If there are threads with this subject, redirect me to them please.

 

[Kummer]

Well? Is it equal to zero ? If there are threads with this subject, redirect me to them please.

Please look in the philosophy forum.
(Because this is not a topic mathematicians discuss, just philosophers.)

 

[mgb_phys]

To an engineer or physicist yes.
We aren't as squeamish as mathematicians when it comes to needing an answer.

 

[Kummer]

To an engineer or physicist yes.
We aren't as squeamish as mathematicians when it comes to needing an answer.

In that case we can read about the extended real line.

 

[homology]

Or, heck, check out nonstandard analysis... But I assume you're (Nick666) talking about calculus and perhaps a limit that comes up? The idea is that 1/big is small, and 1/bigger is smaller, and so I can always choose an x to make 1/x as small as you'd like it (or as close to zero).

Cheers,

Kevin

 

[Moridin]

Here is the correct mathematical notation:

$$\lim_{x\to\infty} \frac{1}{x} = 0$$

1 over infinity is not a valid computation. Actually, it should be that "the limit of 1 over x as x goes to infinity is equal to zero".

 

[matt grime]

Oh dear, so many misconceptions here.

1/oo is a perfectly good symbol. In the extended complex plane it is 0. As it would be in the extended reals - you do not need limits at all to answer that. However, the symbol 1/oo does not have a canonical meaning - I can think of no symbol in mathematics that has a canonical meaning. It's not even true that there is a unique meaning for the symbol 1, or 0 for that matter, is there, so why should there be such a meaning here?

 

[jostpuur]

Well? Is it equal to zero ? If there are threads with this subject, redirect me to them please.

Nick666, do you know yourself what you mean with the infinity? Is there a definition you are using?

 

[Nick666]

Let oo be 999... :) . ( oh, can 999... be infinity ?)

 

[HallsofIvy]

Now you're just pulling our chain!

 

[jostpuur]

Let oo be 999... :) . ( oh, can 999... be infinity ?)

If you write 999..., I'm afraid I'll have to ask again, that do you know yourself what you mean by that?

For example, a number 123 is $1\cdot 10^2 + 2\cdot 10^1 + 3\cdot 10^0$. In general natural numbers can be written as $\sum_{k=0}^N a_k 10^k$, where for all k $a_k\in\{0,1,2,\ldots,9\}$. Your number starts like $9\cdot 10^{?} + \cdots$, and what do you have up there in the exponent?

Writing ...999 would make more sense, because it would be $\sum_{k=0}^{\infty} 9\cdot 10^k$, but I don't know what this means either, because the sum doesn't converge towards any natural number.

It seems your problem is, that you don't know what you mean with the infinity. If you are interested in the basics of analysis, I think Moridin's answer has the point. $\infty$ is a symbol, that usually means that there is some kind of limiting process. The symbol doesn't have an independent meaning there, but it gets meaning in expressions like $\lim_{n\to\infty}$ and $\sum_{k=0}^{\infty}$.

 

[Nick666]

999... As in infinitely many 9`s .

And 1/"that sum you wrote" = ?

 

[Nick666]

And another question about the sum you wrote. Isnt every element of that sum a natural number? (9, 90, 900, 9000 etc) Or let me put it another way. 10^k, when k ->oo , isn't that a natural number ? I mean, if we multiply 10 by 10 by 10... and so on, shouldn't we get a natural number?

 

[CompuChip]

And another question about the sum you wrote. Isnt every element of that sum a natural number? (9, 90, 900, 9000 etc)

Yes it is.
So is every partial sum (cutting off the summation after a finite number of terms).
But the sum itself isn't.

 

[Nick666]

See my above edited post.

But if we add a bunch of natural numbers, no matter how many, isn't it logic that we should also get a natural number ? (or maybe this is why I got low grades at math haha)

 

[jostpuur]

And another question about the sum you wrote. Isnt every element of that sum a natural number? (9, 90, 900, 9000 etc) Or let me put it another way. 10^k, when k ->oo , isn't that a natural number ? I mean, if we multiply 10 by 10 by 10... and so on, shouldn't we get a natural number?

No at all! $\lim_{k\to\infty} 10^k$ is not a natural number.

 

[Moridin]

Oh dear, so many misconceptions here.

1/oo is a perfectly good symbol. In the extended complex plane it is 0. As it would be in the extended reals - you do not need limits at all to answer that. However, the symbol 1/oo does not have a canonical meaning - I can think of no symbol in mathematics that has a canonical meaning. It's not even true that there is a unique meaning for the symbol 1, or 0 for that matter, is there, so why should there be such a meaning here?

As you cannot compute [itex]\infty[/tex] (division by zero is undefined), how would it be possible to compute something that involves it without using limits? I'm not sure I understand.[/itex]

 

[matt grime]

It's just a symbol. One that is used in the context of limits in analysis, and one that is not "computed' (whatever that means) in terms of limits in other contexts.

 

[Moridin]

http://www.m-w.com/dictionary/compute

I think I understand your point now. Thanks.

 

[Nick666]

I still don't understand how, if you add a natural number to a natural number and another natural number and so on,you don't get a natural number. If you add 1 apple and 1 apple and 1 apple and so on, don't you get an infinite number... of apples ?

 

[daveb]

A natural number is an integer. Infinity is not an integer.

 

[CRGreathouse]

I still don't understand how, if you add a natural number to a natural number and another natural number and so on,you don't get a natural number. If you add 1 apple and 1 apple and 1 apple and so on, don't you get an infinite number... of apples ?

If you add any finite number of integers, you get an integer. If you add an infinite number of integers you could get a natural number or an undefined result. In the 'extended integers' you could get infinity or -infinity in addition to those two.

 

[Kummer]

If you add any finite number of integers, you get an integer. If you add an infinite number of integers you could get a natural number or an undefined result. In the 'extended integers' you could get infinity or -infinity in addition to those two.

I really hope the moderators move it to the Philosophy forum.


My comments are not to address this "question" but to show you a beautiful phenomona that happens with infinite things.

Now,
$$1$$ is rational.
$$1+\frac{1}{2^2}$$ is rational.
$$1+\frac{1}{2^2}+\frac{1}{3^2}$$ is rational.
...
And so one.

You are assuming that since the finite sums are rational, then infinitely many of them are rational. But that is not true since,
$$1+\frac{1}{2^2}+\frac{1}{3^3}+... = \frac{\pi ^2}{6}$$.
Not a rational.

~~~~~
Here is another example,
Consider the numbers,
$$\{ 1 , 2 , 3, \}$$
This set has a maximum value , namely, 3.

In fact given any finite set (non-empty) it has a maximum number. And if this set happens to be a set of rational numbers then its maximum is also a rational number.

But look what happens with infinite sets. Say we have,
$$\{ \mbox{ all rational numbers } < \sqrt{2} \}$$.
What is the maximum for that set?
It is not a rational number because we can always choose a larger one getting closer to $$\sqrt{2}$$. In fact its maximum is $$\sqrt{2}$$ it is not longer a rational number! Eventhough the set was filled with only rational numbers.

So part of mathematics studies infinite things. Whether infinitely many real numbers (that is called real analysis) or infinite sets (that is called set theory) and so one. Because the finite things cause no problem. It is the infinite concepts that are fun and sometimes strange.

Note: Have you ever wondered why professional mathematicians never argue (or even discuss) these topics? Because as I said it has little to do with math (perhaps even nothing).

 

[morphism]

1, 2, 3, ..., $\omega$!

 

[Gib Z]

Note: Have you ever wondered why professional mathematicians never argue (or even discuss) these topics? Because as I said it has little to do with math (perhaps even nothing).

How do you expect him to know what professional mathematicians discuss lol

 

[Nick666]

Look, this is the math HELP forum, it doesn't say that this is the professional mathematicians forum.

I`m 22 years old, math in high school was inexistent for me, and a few months ago I stumbled upon 0.9...=1 and I was like what ? And that's how all these question came into my mind.

I`m just sorry that when we started analysis in high school, the boring and also frightening teacher didnt ask "silly" questions like these to us. It would have driven me to study it maybe.

 

[CompuChip]

You should. It's a pity you stopped because of the teacher, but it's really a very interesting field. And it's very rigorous, if you need practice in math :)

 

[Palindrom]

I think the problem here is a non existing concept of the limit.

Nick666 - If I ask you to compute 23 + 42, you'll know exactly how to do it. Even if you don't, your calculator can give you the answer.
How about 1 + 1/2 + 1/4 + 1/8 +...? How would you compute it?

And don't go looking for the answer. If you can't answer it yourself, then you need to learn a little subject before attacking the question "is 1 + 1 +... a natural number?".

 

[Winzer]

I think $$\frac{1}{\infty}$$ makes sense because $$\infty$$ is some unattainable value that can't be reached becasue we are now approaching it fast enough. So : like the function $$f(x)=\frac{1}{x}$$ we take n to infinity we approaches $$\frac{1}{enormous number}$$ and can say it approaches 0. wheather it really gets there or not I can't answer

 

[matt grime]

I want to cry.