Infinitely small number added infinitely?

I didn't ask you for the sum of infinitely many zeros. I asked you for the sum of an infinite number of the infinitesimally small number. You are simply equating the infinitesimally small number with zero. Not good enough. I am asking you to cancel out the infinitely regressive nature of the infinitesimally small number with an infinitely progressive summation of that number added to itself. You are not explaining why a cancellation effect between the two competing infinities would not occur. You are simply ignoring the progressive infinity of the question.

 

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Yes, it will never get to zero. It just approaches zero..

 

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A number reached upon an indefinite descent toward zero.

 

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Dude.. the sum would never end. Never..

 

If it never gets to zero, than how can it be the same as zero?

 

I don't agree with that. I'm saying it isn't.

 

The sums would cancel out the descending infinity into zero, and the result would be the number closest to zero without actually being zero.

 

You still don't understand what an infinite sum is. If you take the series in Zeno's paradox, none of the components in the sum are infinitesimally small. Every term in the series is finite, not infinitesimal. You never reach a fraction which is infinitesimally small, no matter how far you go. Adding up more and more of the fractions in this series gets you closer and closer to 1, and you can get as close as you want to 1 by taking more and more terms in the series.

 

Again, your problem is you want to jump the gun by discussing infinity without first learning the mathematical definition of infinity, or infinite sums, etc.

 

So no, the infinite series in Zeno's paradox does not say that you get something from nothing.

 

An infinitely small number is not finite but infinite. Nonethless, an infinitely small number can be contained in a finite summation of an infinite set. This is Zeno's paradox. Zeno cannot understand how it is possible to progress through an infinity of smaller and smaller fractional mid-points, and still go from point A to point B. You have not solved Zeno's paradox. You are simply restating the obvious, that an infinite series of numbers can sum up to a finite number. Your observation still does not get Zeno from point A to point B. You have not dealt with the fundamental issue of infinities. If I am mistaken, then please solve Zeno's paradox for me.

 

Yeah you completely misunderstand. Your phrasing isn't even gramatically correct, just look at your first sentence where you contradict yourself by saying that an infinitesimally small number is infinite. And your second sentence as related to Zeno's paradox is wrong, because 1 is not an infinitesimally small number. As I say, you make false assumptions, you get a false result back. They have a saying in logic, "you put garbage in, you get garbage out".

Zeno's argument was that if a tortoise travels at say 1mm per second, and has to cross a distance of, say, 1mm, then he thought it should take infinitely long for the tortoise to reach its destination. He said it would take 1/2 a second to cross half the initial distance, then 1/4 of a second to cross half of the remaining distance, then 1/8 to cross half of what remains after that, and so on. But what he didn't realize is that no matter how many terms you take in this series, the tortoise never travels for more than 1 second before it reaches its destination.

 

Please explain to me why you're asking questions about infinity before you've learned what infinity is as a mathematical construct?

 

No, you seem to be operating under a misunderstanding of Zeno's paradox. Zeno (mistakenly) assumed that if you sum an infinite series of non-zero numbers, the result must be infinity. Since the distances between two points can be broken into an infinite set of non-zero distances and each distance takes a certain time to traverse, Zeno assumed (based on his mistaken belief that the sum of an infinite series of finite numbers must be infinite) that it should take an infinitely long time to move between two points.

 

There is no such number. By definition, two numbers are equal to each other if there is no number between them. If there is no number that can exist between zero and your "infinitely small number," then they are both exactly equal, by definition. That's part of the formal definition of "equal" in mathematics.

 

No, I am referring to Zeno's Dichotomy Paradox. See http://en.wikipedia.org/wiki/Zeno's_paradoxes And if you read that site, you will realize that some disagree with you that Zeno's paradox has been solved.

 

Its your understanding of infinity that is at fault.

Consider:

0-------------------------------------1

Divide by 2

0......................0.5...........................1

Divide by 2

0...........0.25.......0.5...........0.75...............1


So, you can get as close to 0 or 1 as you wish without actually reaching either number. That is how infinity is defined. If you disagree with this you are, in fact, at odds with mathematics as we understand it.

 

Please cite your authority for that definition.

 

Well I can't speak for Nasor, but his/her definition follows directly from the axioms which are used to define the set of real numbers.

 

Myles, you didn't answer the question I posted. Also, you are not quite correct in stating that the sum of any two numbers will be greater than either of the two number. For example, 0+0=0. Also, 5+-3=2.