limits

Philosophy Forums Science, Technology, and Math
1

I want to travel from point A, to point B which is a distance of 4 feet… Each leap i take is only half the distance of the previous leap…

the first leap is 2 feet.

common sense tells us that we will never equal or exceed a limit of 4 feet traveled. because we will always only travel half way to the finish line from our current position given this leaping factor…

it is a sequence that approaches but never equals 4 (the sum of all integers in the sequence).

what happens when we apply infinity to this concept?

lets say for humors sake that it does indeed eventually tally up to 4… great!.. This is the common face value understanding of limits…

now the reason why this is in adequate is because once a tally of 4 is reached, there is always another leap you can take even if it’s half the distance of the infinitesimally small leap you took to get there. this would put the tally above 4 feet, breaking the supposed limit.

to deny this is to deny the existence of an infinitesimal which by nature is the last addition to the tally witch itself cannot be divided, thus completing the limit… (this is one way out of my assertions but…)

but wait, you can always make another place value which makes infinitesimals capable of “infinite smallness”.

this infinite smallness is what makes a limit un-obtainable by its sequence through even infinite exhaustion…

something to think about which may help you see the problem as i do is this question.

what is one infinitesimal multiplied by infinity :laughing:

(please don’t beg about 50 questions by saying that an infinitesimal = 0)

from what i have gathered, this is how the average mind rationalizes the argument.

you see the 9’s multiplying to the point there are hundreds of them and then your mind envisions fuzziness (something like TV static on a blank channel), and when the fuzziness clears, you are standing on top of a pile of rubbled 9’s saying “I’m finished”.

and here is how my mind works it out.

i see a biker speeding away from me… he gets smaller and smaller… then he goes into trans warp and travels an infinite distance in a straight line away from me…

obviously i cannot see him, but if i had a telescope that could magnify infinitely, i know i could see him…

2

I think there’s some confusion in the idea of infinite you are using. You can’t really multiply anything by infinity. Infinity isn’t a number, it’s more like a direction. So, your question, “what is one infinitesimal multiplied by infinity”, works out to be “the limit as x goes to infinity of x times (the limit as y goes to infinity of 1/y)”. In this case, I think the answer is 1, because it’s as good as writing “the limit as x goes to infinity of x/x”, which goes to one. But then, there are an infinite number of infinitesimals.

Anyway, the point is that if your looking at the limit of your leaps, there is no last leap. The problem only arises when you confound infinity, treating it as infinite as one point, and them employing it as you would a finite number in the next step. The infinite sum in your leaping example sums to 4 over infinite iterations.

3

Let’s look at your 4 foot jump problem:

lim (n–>infinity) 2/2^(n-1) = 4

Try plugging it in your calculator. Add 2, then 1, then 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256, 1/1012…

You’ll see the pattern. If you continue adding up numbers you’ll get 3.99999999… This is because the numbers in the sequence get too small too quickly to add past 9 in every order of magnitude less than 10^0.

There are a number of tests to see if this series converges. One of them is the ratio test, which is a nice way of phrasing what I said above. You just divide the next ‘step’ in the limit of the function by the current ‘step’ (add 1 to n and divide that function by the function with just n):

lim (n–>infinity) (2/2^n))/(2/2^(n-1)) = r

lim (n–>infinity) 2^(n-1)/2^n = r

r = 2^-1

r = .5

If r < 1, it converges. As you can see, r is always .5 which is less than 1.

Yes, you could never actually jump an infinite number of times to get to that fourth foot you’re talking about, but an infinite series shows what you would get if you could jump an infinte amount of times.

The problem with the whole .999… bit is that you’re considering .999… as is. It can’t be written as .999 bar, that’s just shortand for saying this is a number represented by the limit of an infinite series. The rules of infinite series allow you to make it to infinity, even though it’s not possible in reality:

lim(n–>infinity) 9/10^(n)

lim(n–>infinity) 9/10^(n+1)/9/10^n = r

lim(n–>infinity) 10^n/10^(n+1) = r

1/10^1 = r

r = 10^-1

r = .1 < 1

Therefore it converges. You need to use a different expression for the infinite series that represents .999… before you can see what it equals, but that’s already shown on the math site that shows the proof so you can look at that on your own.

4

but you say .999999[R] =1
so we end up with
4
hahaha

5

Exactly. The limit goes to 4. If you physically add them you’ll keep getting 3.999… but if you continue doing it to infinity you’ll get 4.

Try and learn, colin.

6

[quote]
Exactly. The limit goes to 4. If you physically add them you’ll keep getting 3.999… but if you continue doing it to infinity you’ll get 4.quote]

but colin leslie dean has proven
.9999[R] =/1
so where are you now

7

No you haven’t. .999… does equal 1. Many people have proven that. I know you think we’re the ones being stubborn but it’s you.

8

sorry sci logic thinks differently about deans proof

notice the reference to a new mathematical system

talkaboutscience.com/group/s … 99487.html

9

LJ, let’s talk some more about infinity. Let’s analyse the meaning of the word. From The Online Etymology Dictionary: c.1385, from L. infinitus “unbounded, unlimited,” from in- “not” + finitus “defining, definite,” from finis “end.”
So, infinite: in-=not, -finite=ending. An infinite series has no end. When you say on your other board “a 7 at the end of the infinity line”, you are saying something incoherent. Figuring what is “at the end of the infinity line” is a non-question. There is no end to a line with no end.

10

I don’t care what some nutcase on another forum says that agrees with you. He’s wrong.

For God’s sake, you don’t even get how 1/3 * 1/3 = 1/9. you have no business breathing my air. Please stop.

11

thats fine
but
we have at infinity different size infinities ie cantor
and
under your definition
1.00000[BAR] - .333333[BAR] must be indeterminate as there is no end to the 0s and .3s so you can never subtract anything

12

but dean has proven 1/3 =/ .33333[R]
so
1/3 * 1/3 = 1/9
is a different product than .33333[R] * .33333[R]

FROM SCI LOGIC

13

Wonderer - you can’t continually leap only half the distance to point B. Try it. You can’t do it. You are measurable in three dimensions - your foot, for instance, is about a foot long, more or less. Exactly what is the “you” that would be traveling only half the distance? Is it a point? Or is it your body, which contains no actual points. You are real, and points are not.

14

so what’s the pragmatists view on this problem of differing ideas of infinitesimals and infinity?

should an infinitesimal just be the smallest particle we can measure? how about infinity everything we know about in the universe?

i could rearrange my example or bypass it and just use a normal equation, but then perhaps i wouldn’t be so lucky as to get derailing comments…

15

I think I just gave you my “view”. It comes from the defintion of “point”. That might be the “mathematicians” view, as well.

Some people want to make points “real”. I guess that’s their “view”.

Infinity is not a number, as car;eas pojnts out. I guess that just comes from the defintions of “number” and “infinity”.

View this how you wish, I suppose.

16

ok guys this topic has gone askew, lady jane ill ask you to stop spamming my threads with quotes… it is very annoying.

everyone else kindly focus on the notion of the word infinity and infinitesimals, as the differences there are resulting in these conflicting views.

i think that an infinitesimal is something that can always become smaller. in practicality i think it is impossible to see or calculate, but in theory exists.

i think infinity means everything in this universe and beyond. forever and never ending.

particularly in the .333[bar] problem, we all know that due to the base ten system, 1 cannot be split into 3 equal parts conventionally.

the [bar] system is one method that represents the quantity of 1 third through means of approaching a limit.

now when you actually divide 1 into 3 parts a few things may happen…

the infinitesimal remainder of the division into thirds could be split into a 1/3’d infinitesimal which would not be able to quantified above “1/3’d” attached to each portion of the 1.

or

the infinitesimal remainder, because it can always get smaller, in fact equals zero (this is not my belief) resulting in the 3 thirds tallying up to one.

i guess my main issue is that limits have an error margin of - 1 infinitesimal.

when we say .333[bar] equals 1, we are not being entirely truthful (IMO), we are in fact just using the closest number we can plug into a calculator.

in reply to Faust, yes i know it was your view, but you did not have to be trivial in poking some distracting holes in my example… i appreciate the knowledge and opinion, but not in form of cynicism

17

Wonderer - if an infinitessimal cannot be “calculated”, then it cannot be used in mathematics. I did poke holes in your example. In fact, you may have to accept that your thesis is incorrect. That it is bunk, in a word. My use of quotes was not intended to be cynical - it was to point out that you are making all this up as you go along, and that it means nothing. Sorry if that seems harsh. If you had asked that only those that agree with you respond, I would likely have obliged.

I won’t pester you any longer.

18

no i apologize for misconstruing your use of quotes to be cynical :unamused:

indeed i am making this up as i go along but that hardly means it means nothing, thats a suprising thing to hear from you.

here i just made this up

condiser this sequence

2, 1, 0.5, 0.25, 0.125…

i believe the formula for the sum of an infinite series of numbers is n1/(1-r) where r equals the ratio

the ratio equals 1 half so we get 2/(1-0.5) which equals 4.

lo and behold we have an example to which you cannot poke nasty holes in.

so what did the formula do?

it took the first value, n1 (meaning the first number in the sequence) and multiplied it by 1 minus the ratio… why?

well say the ratio was one third, if the first value is 2 then the second one would be 2 thirds and so on.

we see that the first value is 1 third of the entire distance, so it would logically follow that the entire distance is the first leap, plus the missing third. In the case of thirds i could either divide by a 2/3’ds or multiply by a 1.5 . due to the .5 example and the 1 third example i realised that the necessary ratio to mutiply by or divide by can be reached by subtracting the ratio in the sequence from 1 ( 1 - 1 third) and then use that to divide n1 thus i got x= sum of an infinite series

x= n1/(1-(n2/n1))

here is how i mentally construct the problem when trying to determine the limit that led me to the equation.

A-----------------first leap-----B
|----------|----------|----------|

however the problem still remains that once the first leap is defined, the limit will never be attained. the reason why we use limits to represent the sums of infinite series is because thats the only way we can. and a limit is just that, a limit. an unreachable unsurpassable limit. we cannot define the actual number on the count of it has farr too many 9’s

saying that .999[bar] = 1 is like saying you know what 'pie equals…

19

I see no problem with the sequence of formulas. The prblem remains that you are not measuring the arrival of a human at a point in space (on a surface, a plane). Because a human cannot be measured in that way. Points have no dimension - only position. You’re mixing a real object with a mathematical abstraction.

I would not send you to the store for half a bushel of corn, unless I wasn’t really hungry.

20

A limit tells us what would happen if we could reach infinity.

Pi does equal a number. If we could sum an infinite series that represents it we would know exactly what it is.

The problem is the notation. It seems weird that .999… = 1. It shouldn’t.

Did you read anything I wrote?