"The known is finite, the unknown infinite; intellectually we stand on an island in the midst of an illimitable ocean of inexplicability. Our business in every generation is to reclaim a little more land."--Thomas Huxley
Tear a piece of paper in half. Then tear one half into quarters, then that quarter into eighths, etc. When will you have the entire piece of paper elsewhere? When will the piece of paper you are tearing disappear from your hands? The answer of course; lies within the study of the infinite.
Professor Hamman introduced us to the concept of series: basically it is a sum of a bunch of numbers. If we take the series {1/2+1/4+1/8+1/16+...} and never stop adding terms this series will come to equal one. If you never stop, you will never reach the number one. Another series goes like this: {1/2+1/3+1/4+1/5+...} If you continue to add these terms you do not reach one. If you never stop adding; the sum is -----infinity. You may ask,"How can one series add up to a finite number and another series add up to infinity?" I could say "because Prof. Hamman says its so" or "it just does." But I implore you to think of reasons why this is. Intuition leads us there but it is hard to put into words. Here lies the hidden beauty within the math.
We also discussed the concept of a repetend equaling a finite number, such as .9 rep. =1
If we assign "x" to .9rep. then 10x=9.9rep. If we set these two equations up as a subtraction problem such as 10x-x, and 9.99999999-.9999999; we get 9x=9. Therefore x=1. But we assigned x=.9rep. It is both! So it follows that x=1 AND .9rep.. Another way of saying this is an infinite decimal can equal a finite number. But not ALL infinite decimals equal a finite number. An example of this would be if we set x=.01001000100001... no equation would be possible to use and get a finite number.
I hope this helps everyone with their paper!
Catherine, very good summary. However, I do have a question regarding 2 of the sentences that you wrote above: "If we take the series {1/2+1/4+1/8+1/16+...} and never stop adding terms this series will come to equal one. If you never stop, you will never reach the number one." I understand your first sentence, but I feel like the second sentence sort of contradict the first sentence that you wrote.
ReplyDeleteVorleak, your question is what i want to say. Catherine didn't make it contradict. But the "theory" (or whatever it's called) contradicts itself. How come infinite=finite? Yet Professor Hamman said we were trying to prove how infinity connects with finite.
ReplyDeleteThis is the weirdest math ever!!!
weird math, yes very well put Mai Dinh :)
ReplyDeleteYou are very right with this contradiction Vorleak
I guess the possibility of knowing the result before the "experiment" ends is possible...the "willing suspension of disbelief" to suppose you can come to the end while still adding the fractions.
Another way to look at it is from two sides of the equal sign: one side is infinite and it's ok if it never stops (imagine an animation of the numbers scrolling by) and on the right side there is the number one. The BELIEF of infinite fractions being added is there because you see the scrolling numbers on the left. And on the right is the "1"... just chillin'. Hope this helps! Apologies if I confused you more...
"If we take the series {1/2+1/4+1/8+1/16+...} and never stop adding terms this series will come to equal one. If you never stop, you will never reach the number one."
ReplyDeleteI am also boggled by this contradiction. How can it sometimes equal 1 and sometimes not? Is it only possible when we assign an "x" to it? That I understand, but is there another way?
Also, the example we had in class with the paper, in the same way that "If we take the series {1/2+1/4+1/8+1/16+...} and never stop adding terms this series will come to equal one" doesn't that apply to the paper example? If we continue to rip the paper in half and then in half again etc, and never stop adding what we rip in half, won't we eventually have one full piece of paper?
Yumiko,
ReplyDeleteReferring to the last sentence of your post, I am fairly certaiin this is where "what you believe" answers your question. Do you believe space is infinitely divisible or not. Confusting huh?!?!
Hi all,
ReplyDeleteMy internet has been so bad this weekend so I couldn’t comment early. I’m Just adding few words here about the series getting to 1. My understanding is that if you add all of them up or no matter how many series you keep on adding, they are all close to 1. You will realize that each of them will give you 0.9 and the rest follows. if you convert it you will surely get a 1. This is one part of the different kinds of infinity that we have. I keep on laughing when at the beginning of class; I thought infinity no matter how it forms doesn't change. It’s still the same infinity but now I know there are different forms of infinity. Changing the series gives you different kind of infinity. The fist kind was infinity equal to finite as we said in class. Good summary Catherine!
Yaa
I think I may have confused you a bit Yumiko when I said the word "belief". I am pretty sure regardless of how you view time and space regarding Zeno's paradox, the infinite series of constantly dividing in halves always adds to one. It only equals something else when the order is different like Julian states in his summary, or we add up different fractions like {1/3, 1/4, 1/5...}
ReplyDeleteI'm not sure I understand your question about it only being possible when you add an X to it...
I hope this helps.
OK. Thanks for clearing that up. that makes me deciding my argument for my paper much easier.
ReplyDelete