Zeno's Paradox Question

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JoshuaSass

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Hey people!

So I have a question... But I think you guys already realized that

So Zeno's Paradox, Infinite half distances to point A will never fully reach point A, works like an asymptote, but what are some things that can prove that moving at half distances, you will eventually reach point A? I see a lot of people showing proofs, and I'm genuinely looking for people to show me what they are, even if they are proven wrong, I would love to see them!

Thanks people!

-Numbers are the Music of the Brain- Favorite quote from a teacher, 2018
 
So I know if a different forum, I found someone talking about repeatedly folding a paper in half is the same, but I was confused, because the paradox is that you will never reach 1, but your paper is always going to reach 1 because it starts at 1. Like 1+1=2 same as 2=2, it does not matter as long as the number is still equal, so is this an example of Zeno's Paradox or...
 
As you were asked, please SHOW what you are talking about, rather than referring indirectly to it. It will be the details that matter. I have no idea what you are saying.
 
Okay, let me try to rephrase, sorry I am bad with words, English is not my first language.

So If we take a 1 inch by 1 inch square peice of paper, and fold it multiple times, is that not the same as Zeno's Paradox of Infinite half distances? Or how a limit works, sure the number can approach the limit, but will never touch it.
 
JoshuaSass said:
Okay, let me try to rephrase, sorry I am bad with words, English is not my first language.
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You could just quote what somebody said about this, which is what I was asking for.

JoshuaSass said:
So If we take a 1 inch by 1 inch square peice of paper, and fold it multiple times, is that not the same as Zeno's Paradox of Infinite half distances? Or how a limit works, sure the number can approach the limit, but will never touch it.
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But why did you say,
JoshuaSass said:
but your paper is always going to reach 1 because it starts at 1.
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You seem to be saying this is contrary to Zeno's paradox; how??
 
JoshuaSass said:
Okay, let me try to rephrase, sorry I am bad with words, English is not my first language.

So If we take a 1 inch by 1 inch square peice of paper, and fold it multiple times, is that not the same as Zeno's Paradox of Infinite half distances? Or how a limit works, sure the number can approach the limit, but will never touch it.
Click to expand...
Unfortunately I hit "Like" instead of "Reply" to your post above; I'm afraid I didn't really "like" it at all though I am happy to grant you some "leeway" considering English is not your native tongue.

Reading through the thread (up to @Dr.Peterson's last post which arrived while I was composing this) prompts me to state the following....

\(\displaystyle \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdot\cdot\cdot = 1\)​

because that is a convergent, geometric series, ie:-

\(\displaystyle \sum_{n=1}^{\infty }\left( \frac{1}{2} \right)^n=1\quad\) not \(\displaystyle \quad\sum_{n=1}^{\infty }\left( \frac{1}{2} \right)^n \approx 1\)​

Now Zeno and his ancient Greek buddies had no idea about such things (or the concept of a limit) which is why they were all so confused by the (apparent) paradoxes that he postulated. Do you really think that if you are running, head down, towards a wall you will never reach it because you are having to cover an infinite number of decreasingly small distances (halved each time)??? Perhaps you should try it; the actual result may clear your brain! 🤣

I would suggest you watch this video which debunks Zeno's paradoxes...

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and then this one which may help you to understand infinitesimal increments...

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and this one provides a nice (quite convincing) visual proof of the above result...

​

Finally, I would ask you to consider this (which may also help your understanding):-

\(\displaystyle 0.\bar{9}\) is exactly equal to 1.​

(Just to be clear, the line over the 9 indicates that there are infinitely many 9s, ie: they literally never stop repeating.)

                  So, yes, I am saying that 0.999999999… is the same as 1.

A lot of people have trouble accepting this, and that is, perhaps, understandable. It may seem like \(\displaystyle 0.\bar{9}\) should be just a little bit less than 1 but, if that is the argument, then the next logical question is: "How much less?"

With each additional 9, the difference between \(\displaystyle 0.\bar{9}\) and 1 becomes smaller and smaller, and, since there are a proposed infinite number of nines, then that difference eventually drops to zero.

There are other proofs that \(\displaystyle 0.\bar{9}\) = 1 littered around the Internet (just Google 'em) if you feel you need further convincing but please don't ask us any more questions on this subject unless and until you have watched all of the above videos (maybe more than once?) and given ample thought to what I have said after them.

Hope that helps. 😊
 
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