/sci/ - Science & Math

[Anonymous No.11876121]

somethings been bothering me.

as we divide 1 by larger and larger numbers. it gets smaller.
lets divide 1 by 2^n as n approaches infinity
the presumed result will be 0

But if we look a little closer and start asking some questions things get funny.

in order for 1/2^n to reach zero it has to go through a certain threshold. Imagine you have a sieve that only things that are 1 centimetre can pass through, you could strain materials for an infinite amount of time but nothing wider than 1 centimetre will ever pass through.

in order for 1/2^n to be zero there has to be a number A that is passed through but pass through it leads to problems because we must pass from positive to 0 while dividing by 2. the point between positive and zero is where the sieve lies and it only lets through numbers with certain properties a number must have the property A/2 = 0 and a > 0 to pass through this sieve, it has to be greater than 0 because it is coming from the positive direction.

if you don't pass through the sieve than 1/2^n can't equal 0

but it is impossible to pass through the sieve because the properties a number needs to have to pass through the sieve are impossible.

why does everyone pretend this isn't a problem?

[Anonymous No.11876146]

because its not

[Anonymous No.11876175]

>>11876121
axiom of infinity
https://youtu.be/SrU9YDoXE88?t=14m

[Anonymous No.11876179]

>>11876121
>the point between positive and zero is where the sieve lies
there is no point between positive and zero
given any positive number you can find another positive number closer to 0

[Anonymous No.11876181]

>>11876175
doesn't matter. A sieve willy only let things of a specific size pass through no matter how much time you have. something too large to pass through a sieve won't suddenly be able to pass through it because you have more time.

[Anonymous No.11876185]

>>11876181
>time
wtf does time have to do with this

[Anonymous No.11876189]

>>11876179
1/2^n as n approaches infinity is 0

if this is true then you must pass from positive to zero. either 1/2^n is zero or not. if it is it has to pass through the sieve and to pass through it a number needs to have properties that are impossible

[Anonymous No.11876195]

>>11876121
because we don't exist

[Anonymous No.11876199]

>>11876185
its an analogy. imagine you have a 1 inch long number ling that you cut in half every second. given infinite seconds the line will be cut to zero length presumably. but the same problem arises where the line will need to be a length n such that n/2 = 0 and n > 0. if such a number doesn't exist then it's impossible for 1/2^n to equal zero.

[Anonymous No.11876201]

>>11876189
1/2^n is positive for all finite n
it doesnt matter because that has nothing to do with determining what it approaches

[math]\displaystyle \lim_{x\to \infty}\frac{1}{2^n} = 0 \iff (\exists M\exists \delta >0, m \geq M\implies|\frac{1}{2^m} - 0|<\epsilon)[/math]
This has nothing to do with a sieve

The fact that youre going from the positives, to including 0 by using a limit is just a consequence of something called Closure in topology

[Anonymous No.11876205]

>>11876201
>1/2^n is positive for all finite n

i specifically said as n approaches infinity

[Anonymous No.11876208]

>>11876205
cool, did you read any of the rest of my post?

[Anonymous No.11876221]

>>11876205
>n approaches infinity
you do realize it doesn't really approach inf
inf-r=inf
it gets bigger, that's all

[Anonymous No.11876225]

>>11876199
shitty analogies give shitty conclusions.
no surprise you're confused

[Anonymous No.11876227]

>>11876205
oh. makes a little more sense now. it gets close but it's not equal to 0.

that's the same problem i have with people saying 0.999.... = 1

it gets close but we can't say it's equal to 1

[Anonymous No.11876229]

>>11876227
meant for >>11876221

[Anonymous No.11876241]

>>11876221
>>11876208
but somethings still feels off.

what's the difference between 1/2^n as n approaches infinity and 0? if the have a difference of 0 aren't they equal?

from what little i know about limits it means if we center delta at zero we can make delta any value and zero will always stay within delta so if we make delta zero essentially we're saying it is zero.

[Anonymous No.11876243]

>>11876227
no, you just dont understand what a limit is

[Anonymous No.11876249]

>>11876227
that's different
0.9... is static, it's not growing like in lim-evaluations
the ... means it's already at inf amount of 9s
it's using the axiom of infinity
so 0.9... = 1

[Anonymous No.11876251]

>>11876249
stop saying axiom of infinity, its got nothing to do with this

[Anonymous No.11876258]

>>11876251
sure it does

[Anonymous No.11876271]

>>11876251
sure it does, can't say there are aleph-0 amount of 9s if you don't allow it to exist

[Anonymous No.11876273]

>>11876121
You're just confused. 1/2^n cannot ever equal zero because you can't have a number that equals 1 when you multiply by 0. Numbers are abstractions, they don't necessarily always make physical sense.

[Anonymous No.11876281]

>>11876225
>>11876208
i feel like you're just avoiding the problem by using limits.

if you can't divined by infinity we can still construct situations where you continually divide. so given a 1 inch long number line cut in half every second given infinite seconds
you'd expect that you'd end up with zero but it's impossible to do so despite the pattern 1/n gets smaller as n gets larger which bothers me.

if you could divide 1 by infinity it couldn't be zero but the pattern predicts it would which bothers me.

[Anonymous No.11876287]

>>11876121
You're an idiot, op. Bet this will blow your mind:

https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise

[Anonymous No.11876296]

>>11876281
>as n gets larger
any finite number, sure
infinite isn't a number
1/inf=0

[Anonymous No.11876299]

>>11876281
i guess my problem is more of this.

given a 1 inch long number line with a cut in half per second will the number line be cut to zero length given infinite seconds?

the answer is no. but you'd expect yes.

[Anonymous No.11876302]

>>11876296
1/inf 0.

No you retard. you have to pass through the sieve. you can't reach 0 unless you ass through the sieve and in order to pass through the sieve a number need's certain properties.


1/inf SHOULD = 0 but it can = 0.

[Anonymous No.11876303]

>>11876281
By definition you can't count an infinite amount of time.

[Anonymous No.11876306]

>>11876302
can't*

[Anonymous No.11876312]

>>11876303
its just a thought experiment.

we're imaging you could divide by infinity and discovering the result you'd expect to have is impossible.

[Anonymous No.11876317]

>>11876281
>i feel like you're just avoiding the problem by using limits.
exactly
all the issues you can "find" dont matter because we use limits, which avoid all of the issues

[Anonymous No.11876320]

>>11876312
If theoretically speaking you could divide by a number so large it never ends, you'd end up with zero.

[Anonymous No.11876325]

>>11876320
>number so large it never ends
no such thing

[Anonymous No.11876327]

>>11876325
That's literally what infinity is.

[Anonymous No.11876330]

>>11876302
>pass through the sieve
nope, that's where the axiom comes in

[Anonymous No.11876333]

>>11876327
and infinity isn't a number

[Anonymous No.11876334]

>>11876325
>>11876327
Infinity by definition is such that 1 and a googolplexto the power of [the amount of atoms in the universe]googolplex are equally distant from it.

[Anonymous No.11876336]

>>11876333
That's my point. OP's question makes no sense, he's just unimaginative.

[Anonymous No.11876339]

>>11876320
No. if you need to pass through some threshold it wouldn't matter.

i use the sieve analogy for a reason. you wouldn't be able to pass something from one end of a sieve to the other given an finite amount of time if it was too large.

in the cut the number line in half part to go from a positive to zero there needs to be a number n such that n/2= 0 and n > 0 such a number can't exist so you can never reach 0.

it's literally impassible

there is either a Passover from positive to zero or there isn't. if there is you need certain properties to pass from positive to zero and no number has those properties so even if you could divide 1 by infinity it couldn't be zero

[Anonymous No.11876344]

>>11876334
formal definition:
An unbounded quantity that is greater than every real number.

[Anonymous No.11876346]

>>11876330
how does the axiom solve the problem?

[Anonymous No.11876348]

If you cut in half after one second, another half after 1/2 second, another half after 1/4 second etc. then after 2 seconds you will have zero length. How about that OP

[Anonymous No.11876360]

>>11876346
there simply is no trickery you can do with real numbers to reach infinity
your sieve-thing is just another way of saying that.
Axiom states that the first unattainable exists, and it is aleph-0
that's how the jump happens
it's like the pic: as long as the points A and B aren't the same, the resulting number C ir a normal number. once B meets A, the tangent jumps off the lower line onto the top one, which is infinity

[Anonymous No.11876361]

>>11876346
there simply is no trickery you can do with real numbers to reach infinity
your sieve-thing is just another way of saying that.
Axiom states that the first unattainable exists, and it is aleph-0
that's how the jump happens
it's like the pic: as long as the points A and B aren't the same, the resulting number C is a normal number. Once B meets A, the tangent jumps off the lower line onto the top one, which is infinity

[Anonymous No.11876369]

>>11876348
no . you need to cut a positive to zero.but to cut a positive to zero is impossible so you can't reach zero. if n/2 = 0 and n>0 doesn't exist you can't reach zero because only this number has the dimensions needed to pass through the sieve from positive to 0 otherwise you use have infinite small positive numbers.

[Anonymous No.11876374]

>>11876369
How long is the segment after 2 seconds then?

[Anonymous No.11876393]

>>11876369
1/4


multiply 2 by seconds elapsed then divide by it or do it one at a time. 1/2 = 0.5 0.5/2= 0.25

[Anonymous No.11876396]

>>11876393
>>11876374

[Anonymous No.11876409]

>>11876393
Read >>11876348 once more please

[Anonymous No.11876457]

>>11876409
it doesn't matter how often you cut.every cut is a half cut and to reach zero a cut must cut a length n such that n > 0 and n/2 = 0

such a length/number can't exist therefore it's impossible to reach zero unless we're just pretend such a number doesn't need to exist.

so let me ask you.

is there a number n such that n/2 = 0 and n>0?

if no then you can never reach zero because will always have positive numbers and never pass to zero. you can't just magically skip to zero and skip the sieve pretending it doesn't exist.

[Anonymous No.11876473]

>>11876457
So how long is the segment after 2 seconds?

[Anonymous No.11876523]

>>11876473
1/4

you've asked this twice
what's the significance of this?

[Anonymous No.11876528]

>>11876523
after 1 second, the length is 1/2
after 1.5 second the length is 1/4
after 1.75 second the length is 1/8
clearly after 2 seconds the length is smaller than 1/4. so how much is it, if not zero ?

[Anonymous No.11876553]

Op is retarded, 1/n never reaches 0. There's no "sieve". It's either extremely close to 0, or 0. Anxthing else is bullshit.

[Anonymous No.11876563]

>>11876528
the pattern indicates that you get zero. that's the problem.

is there a number n such that n/2 = 0 and n>0?

yes or no.

if no how do you go from a positive number to zero? what positive number do you divide by 2 to get zero? if this number doesn't exit how do you go from a positive number to zero?

this number must be finite. so all we'd need to do to prove you can get to zero by dividing one is evaluate this threshold number you must pass to go from positive to zero.


STOP looking at the pattern and look at the threshold between zero and positive

imagine you're about to pass from a positive number to zero.by dividing a positive number.

explain how that happens. if it never happens you can never reach zero.
.

[Anonymous No.11876576]

>>11876563
>that's the problem.
there's no problem
>is there a number n such that n/2 = 0 and n>0?
no
>if no how do you go from a positive number to zero?
what does "going from one number to another number" mean ? explain. how do you "go from 13 to 7" ?
>what positive number do you divide by 2 to get zero?
there's no such number
>if this number doesn't exit how do you go from a positive number to zero?
again, what does "going from one number to another number" mean ?
>this number must be finite. ...
this number doesn't exist, so I won't respond to the rest
>STOP looking at the pattern and look at the threshold between zero and positive
what does "threshold between zero and positive" mean ? explain. what is "threshold between 13 and 7" ?
>imagine you're about to pass from a positive number to zero.by dividing a positive number.
dividing a positive number by a positive number never results in zero
>explain how that happens. if it never happens you can never reach zero.
the sequence 1/2^n is never zero, nobody claims otherwise

I have no idea what's your problem OP

[Anonymous No.11876632]

>>11876576
if a/b = 4
we know that a has to be 4 times greater than b. we know the properties a must have to make this equation true.
lets saying you're subtracting 1 from 10 till you get to zero. can you skip one? no you go 9 8 7 6 5 4 2 1 then 0 you have to pass 1 when subtracting by 1 to get to zero. you wouldn't ask me what i mean by passing one.

lets say you're dividing 1 by 2 infinitely many times. to get to zero you have to pass the number n such that n/2 = 0 and n>0

it's really not that complicate. to get from 10 to 0 by subtracting by 1 you pass 1.


do you understand now or is it still too abstract?

[Anonymous No.11876648]

>>11876576
to make it even simpler to prove that we can get to zero by subtracting one from then e show that this is a number

n >0 and n - 1 = 0

this above number has to exist in order for 10 -1 to equal zero if it didn't it wouldn't makes sense to say you can get to zero by continually subtracting 1 from 10 right?

similarly it doesn't make sense to say we can get to zero by continually dividing 1 by 2 if the number n>0 and n/2 doesn't exist

literally the most simple shit ever. i have no idea why this is hard to understand.

[Anonymous No.11876653]

>>11876648
n>0 and n/2 = 0*

[Anonymous No.11876659]

>>11876632
>lets say you're dividing 1 by 2 infinitely many times. to get to zero you have to pass the number n such that n/2 = 0 and n>0
no, there's no "let's say". division is a binary operation and as such can be extended to an arbitrarily large, but finite number of iterations. but strictly speaking there's no such thing as dividing something infinitely many times. the only thing you can do to make sense of it is to take the limit of "1 divided by 2 n-times" for n -> inf. clearly this limit is zero. but it's not a division in the ordinary sense anymore: it's not a result of 2 divided by something, it's a result of taking the limit of 2 divided by something. therefore rules for division don't necessarily apply here.

"it's literally the most simple shit ever. i have no idea why this is hard to understand."

[Anonymous No.11876673]

>>11876659
we weren't talking about limits anymore. i was talking to the guy saying 1/inf = 0

i'm arguing with him. I've already conceded about the fact that limits get around this problem.

i'm trying to see why he think 1/inf = 0 if you could divide infinity many times.
the pattern would lead you to think it would but it can't is my arugment.

[Anonymous No.11876681]

>>11876673
>divide infinity many times

[Anonymous No.11876689]

>>11876659
>>11876528
he's saying you can get to zero by dividing 1 by 2 infinite times you can't.

if you're not this guy i wasn't arguing with you.

[Anonymous No.11876693]

>>11876681
do you know what imagination and thought experiments are?

imagine moving at the speed of light out of this this thread .

[Anonymous No.11876702]

>>11876693
i imagine that you won't stfu
schizos never do

[Anonymous No.11876704]

>>11876121
In your sieve analogy, for every possible size you can provide you can also show that the sieve picks it up

[Anonymous No.11876708]

>>11876189
>if this is true you must find a (real) number n that switches from being a number to infinity
read up on limits please

[Anonymous No.11876730]

>>11876702
how am i a schizo? i think it's a pretty normal average thought that i haven't seen a answer to.

we subtract 1 from 10

9 8 7 6 5 4 3 2 1 0

we have to pass 1 to get to zero

the threshold number is 1 and it has the property n >0 and n - 1 = 0

if the threshold number doesn't exist for 1/inf it doesn't make sense to say 1/inf = 0 but from the pattern 1/2^N as n approaches infinity would make you believe it should be zero.
the number n>0 and n/2 = 0 literally can't exit
so you can't get to zero by diving 1 by 2 even with infinite divides

its just weird to me.

[Anonymous No.11876791]

>>11876730
that’s why we say it approaches 0. This is where the sequence is going with increasing n.
Just like the sum over 1/2^n approaches 2.
There is no magical threshold where the sequence or series becomes its limit, only that you get closer to the limit by increasing n.
I feel like you’re trolling in bad faith here.

[Anonymous No.11876853]

>>11876730
>threshold
this has nothing to do with dividing something infinitely times. you're basically asking for a number which is larger than zero and smaller than every positive number at the same time. no such number exists because real numbers are ordered differently than natural numbers.
imagine going from 0 to 1 through all rational numbers. where's the "threshold" the number needs to pass to stop being less than 1 and starts being 1 ?

[Anonymous No.11876912]

>>11876121
>in order for 1/2^n to be zero there has to be a number A that is passed through but pass through it leads to problems because we must pass from positive to 0 while dividing by 2.
No, that assumes there is a final division. By definition there isn't.

Also, 1/2^n is smaller than any "sieve" for some natural number n, so your point is moot.

1/2^n > x
1/x > 2^n
n > log2(1/x)

Let n = floor(log2(1/x))-1
Then n < log2(1/x)

Thus 1/2^n > x is not true.

[Anonymous No.11876944]

>>11876730
told ya

[Anonymous No.11878253]

>>11876853
>threshold" the number needs to pass to stop being less than 1 and starts being 1

it has the property n<1 and n+(ADDENED) = 1

what are you talking about? . we know a number with the above properties CAN EXIST. SO IT'S POSSIBLE TO GO FROM 0 TO ONE THROUGH RATIONAL NUMBERS.

How many numbers satisfy the propriety N < 1 and N+(addend)=?

infinitely many numbers. that means there are infinitely many ways to go from n to 1 by adding some number to n

for any n < 1 there is is a number a > 0 such that a + n = 1

we're trying to get to a number while performing repetitive operations on it. continuously dividing or continuously subtracting or continuously adding.

just (IMAGINE) for a second the operation 1/2(INF) was possible. would it be 0 or not?

[Anonymous No.11878778]

>>11876189
Any one number n you can think of is much closer to 0 than it is to infinity. I don't see why you'd think there's an n where 1/2^n=0

[Anonymous No.11878956]

>>11876121
>>11876189
>>11876205
>>11876227
>>11876457
>>11876563
>>11876632
>>11876648
>>11876673
>>11876730

Every incorrect statement and false assumption you're making throughout all of your posts just comes from an improper understanding of the definitions of terminology.
you think you're being profound and insightful, but this entire thread comes off as you being hyper-confident about various incorrect premises.
you may be an intelligent person but you don't understand this topic nearly as much as you think you do. There is plenty of opportunity for learning in this thread if you would revise your assumptions and ask questions to clarify your own misunderstanding instead of resorting to argument.

[Anonymous No.11878977]

>>11878956
wrong

[Anonymous No.11879066]

>>11878956
you probably think i'm talking about limits. i'm not. i'v admitted saying a number approaches some limit is not the same as saying it's equal to that number.
i'm imagining you could divide by infinity and what would happen.
if you multiply 0 by infinity it would still be zero.
if you multiply any number greater than zero by infinity it would be infinite
if you divide 1 by infinity it COULDN'T be zero but you'd expect for it to be zero but the property a number needs to have to go from positive to 0 through division can't exist. so no amount of division can get you from positive to zero REGARDLESS if the divisor is infinite.

since the number can't exist it's impossible. we don't even need to imagine dividing by infinity.

but the pattern would make you think it SHOULD equal 0 if you COULD divide 1 by infinity.

[Anonymous No.11879431]

>>11876241
watch 3blue1brown essense of calc videos , or atleast his videos about approaching