continuum ad infinitum.
infinite series are still a puzzle for even the greatest mathematicians today. the sum of an infinite series is a finite number, a finite number can be divided into infinite finite numbers. but nobody knows what number of infinite numbers it can be broken up into. now, thats contradiciting. number of infinite numbers?? well, why not? how, or why have'nt we been able to go as far as infinity????
i think infinity(or a point at/of infinity) lies only as far as you can go. but we are taught to believe that going further does not necessarily mean we are closer to that supposed infinity. i guess that would defeat the purpose of an 'infinity'. there wouldnt be a point of coining that term if we could get there or reach such a position. but was this term created as a sign of defeat? like as if someone decided, "Oh, we will never be able to go that far, and its so far that it would seem like very far, so there's no point in trying to go there anyway. Lets just tell everyone it extends to emm..say 'infinity' so that no one else can go that far.I dont want anyone else doing something I didnt!"
or have we set up everything else, so that everything is in accordance with this infinite stuff? like numbers? have we created them such, that we made sure they had characteristics that would display the 'infinite', made them in such a way that we were in need of such a term? is it really necessary?do we need something to be infinite? its not like we know of stuff that are infinite (except numbers....we created them), we just dont know if a certain thing is finite. just because we dont know if something is finite (eg, the universe) does it mean that it is infinite?
ok, heres something to ponder over, a very famous 'paradox'. Lets see if anyone can come up with an answer with a logical explanation :
Consider a lamp, with a switch. Hit the switch once, it turns it on. Hit it again, it turns it off. Let us imagine there is a being with supernatural powers who likes to play with this lamp as follows. First, he turns it on. At the end of one minute, he turns it off. At the end of half a minute, he turns it on again. At the end of a quarter of a minute, he turns it off. In one eighth of a minute, he turns it on again. And so on, hitting the switch each time after waiting exactly one-half the time he waited before hitting it the last time.
mathematically, in school, we learned that 1 = 1/2 + 1/4 + 1/16 + 1/32 + 1/64 +1/128 ..... (to 1/infinity)
therefore, applying the above, it is easy to see that all these infinitely many time intervals add up to exactly two minutes.
Now here's the question : At the end of two minutes, is the lamp on, or off?
i think infinity(or a point at/of infinity) lies only as far as you can go. but we are taught to believe that going further does not necessarily mean we are closer to that supposed infinity. i guess that would defeat the purpose of an 'infinity'. there wouldnt be a point of coining that term if we could get there or reach such a position. but was this term created as a sign of defeat? like as if someone decided, "Oh, we will never be able to go that far, and its so far that it would seem like very far, so there's no point in trying to go there anyway. Lets just tell everyone it extends to emm..say 'infinity' so that no one else can go that far.I dont want anyone else doing something I didnt!"
or have we set up everything else, so that everything is in accordance with this infinite stuff? like numbers? have we created them such, that we made sure they had characteristics that would display the 'infinite', made them in such a way that we were in need of such a term? is it really necessary?do we need something to be infinite? its not like we know of stuff that are infinite (except numbers....we created them), we just dont know if a certain thing is finite. just because we dont know if something is finite (eg, the universe) does it mean that it is infinite?
ok, heres something to ponder over, a very famous 'paradox'. Lets see if anyone can come up with an answer with a logical explanation :
Consider a lamp, with a switch. Hit the switch once, it turns it on. Hit it again, it turns it off. Let us imagine there is a being with supernatural powers who likes to play with this lamp as follows. First, he turns it on. At the end of one minute, he turns it off. At the end of half a minute, he turns it on again. At the end of a quarter of a minute, he turns it off. In one eighth of a minute, he turns it on again. And so on, hitting the switch each time after waiting exactly one-half the time he waited before hitting it the last time.
mathematically, in school, we learned that 1 = 1/2 + 1/4 + 1/16 + 1/32 + 1/64 +1/128 ..... (to 1/infinity)
therefore, applying the above, it is easy to see that all these infinitely many time intervals add up to exactly two minutes.
Now here's the question : At the end of two minutes, is the lamp on, or off?
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hmmmm, interesting, this is basically a geometric series related question which I think is rather impossible to solve with the given resourses (ie, im not that smart) but anyway this is how i went about it only to end up scratching my head.
Ok, you start by switching the lamp on (why???? god alone knows IF he exists) after 1 min u switch it off. Then for some reason, which i dont feel like getting into, u decide that after 1/2 a minute u switch it off, then after 1/4 of a minute u switch it on, this keeps going on and on ( atfer 1/8 minute, 1/16 of a minute and then 1/32.....clicking the switch after half the time taken to previously hit the switch is completed) till there is no "half the time" left to click it on or off. Is the switch on or off?
Primarily, the value of 1min, is divided into an infinte number of numbers,
1, 1/2, 1/8, 1/16........1/(n-1), 1/n.
n = infinity
a = 1 (first member of the series)
r = 1/2(differance multipled)
hence SUM of all the numbers, or basically the total time spent on this wastefull activity is
= a/(1-r) = 2 minutes
(caluclate urself)
now we plug these values into an equation and find out if the last member of this series is a odd or even number and LO and BEHOLD u solve an ageless brain wrecking problem.
to find the nth term the formula is a x r^n ( r is raised to power n)
u try sovlving this but ull never get an answer cause u get 0.5^n as the nthe term, not much use as it equals to nothing so u have 2 variables.
SOO i turned the table round, and imagined turning the swich on or off starting backwards, that is i started with 0 time and after every double time passed i hit the switch again till i completed 2 minutes.
here i get a better answer which is 1=n2^n. One equation with one variable.
Try racking ur brains now, im tired off this switch, ill try something more and will be back.
ok, i appreciate your effort, but i guess in your hurry to do something else (less futile??) you mistook/misread some stuff.
first off, the series is : 1/2+1/4+1/8+1/16+32..
therefore making the next term 1/2n and not 1/(n-1). therefore, your nth term is also wrong,therefore your proposed equation is also wrong.
sorry, but you are wrong! but keep trying! but i say, if at first you dont succeed, take a break, however short or long!
and so it begins...
taking Monsieur Albert Newton Faraday's (nishant das) suggestion(idea)consideration, how about working backwards?
lets try and start with two minutes at first and work our way backwards towards '0' minutes?
we could use trial values (i.e. suppose its 'on' after 2mins and/or 'off' after two minutes) and see which one gives a switched off lamp at the '0'th minute.
well, i guess we'll again come across a series, but this time, it would be :
1/infinity - 1/(infinity/2) - 1/(infinity/4).... (to 0). but we know the state of the lamp at 0. so maybe such a trial and error method might work.
on the lines of binary, '1' is considered to be the 'on' state, and '0' to be 'off'. even on, and odd off.so using that methodology, at '0' minutes, the lamp is off.
lets first take 2 mins to be a time when the lamp is on. then working backwards.....
bah humbug!!! im tired already.
more on this later....
(i prefer more 'futile' activities to this now..)
ok man, i really really really appretiate your initiative to spread knowledge and light the "lamp" of wisdom but what this question is trying to do is detrimine if the last number of the numerical system is an even or odd number, i feel its impossible so please finish this post cause its eating my head, and start a juicy debate related to the limits man binds himself to when it comes down to the universe.
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