No. It simply acknowledges that numbers have a left side and a right side, and that precision matters. And the logical proof is back in my monkey and 9s. Norris can’t say, as he did above, that the number of seconds along the line matters, and precision does not in his example. Because they are the same thing. T sub 0 to T sub 1 is a representation of 0.9 to 0.99 with T sub inf being 0.9.. So either I have all the 9s, and the time it takes me to write one, relative to infinite time, is zero, and thus the sum total is zero, which is illogical. Or, I have all the 9s, and the time it takes me to write one relative to infinite time is near zero, and the sum total of all the time it takes me is infinite. Which is logical. It doesn’t go both ways.
This does not make sense. Because something is a smaller and smaller % value of the whole has no bearing on its actual value. 1 is big! 1+1 uh oh, 1 has shrunk because relative to the whole (2) it is now half the size. Please confirm or contradict that this is what you are claiming. Because if that is your premise, we will never, ever see eye to eye on this and, frankly, I think your stance is tenuous at best, because you are trying to contradict what I am saying with some really bizarre logic. EDIT: Upon further reflection, I believe you are saying that I am claiming the above is true, which is patently false. I have said that it takes infinite amount of time, not zero. Each step takes a finite amount of time of x and when repeated infinite number of times it takes an infinite amount of time.
I am using your own definition. You said in a post above that 1/3 = .333... + r where r is some infinitesimal remainder. I did simple algebra to transform it and when I did it showed that r must be zero. Now, your problem with 10 x .333.. = 3.333... is bizarre to me as well. You talk about precision, but are, it seems to me, ignoring the definition of the repeating decimal. by definition of the decimal notation: .33333.... = 3/10 + 3/100 + 3/1000 + ... infinitely precise, infinite threes. It never ends, all threes. You cannot contradict a definition because it is defined that way and then we move forward from there. It is how mathematics works. same as we define the + sign to mean addition. 1 + 1 = 2. You cannot come in behind me and say yeah, but 1 + 1 = 1. It goes against the definition of what the + operator does. You want to make a mathematical construct such that 1 + 1 = 1? Be my guest. In fact, I would encourage it. You might come up with something unique and interesting. But in standard mathematics, 1 + 1 = 2 and .3333... is infinitely 'precise' by definition.
You are using circular logic, which is illogical. You cannot claim math is logical and use math to prove it. Math in this is illogical, and here is the proof: A monkey doing long division for infinite time will produce an infinite repeating decimal of when doing 1/3. And will produce infinite remainder. Thus: there is infinite string of 3s and infinite remainder. Thus there is infinite remainder. Full. [uck fay]ing. Stop. Nobody is arguing that Mathematics doesn't incorrectly believe that 1/3 = 0.3.. The argument is that math is wrong. Here is why: Let me repeat, infinitely: A monkey doing long division for infinite time will produce an infinite repeating decimal of when doing 1/3. And will produce infinite remainder. Thus: there is infinite string of 3s and infinite remainder. Thus there is infinite remainder.
I am not ignoring the repeating decimal. I am saying that because there is a defined value on the left side that differs from the defined value on the left side of a similar number, they are not equivalent. One is more precise than the other, and that significant digits matter. This holds up, in all mathematical formulas, meaning, I can do this math with fixed numbers, and every calculation will equal every calculation. Except infinitely repeating digits. Why is that? Why do you think that there is this difference? It's because the premise is flawed in thinking that finitely repeating numbers can be treated as finite numbers.
And compared to infinite, that finite number is very, very, very small, and getting smaller. In fact, as time moves on, proportionally it'll look like this 1s 1/10s 1/1000s ... .00000000000001 .000000000000000000000000000001 .00000000000000000000000000000000000000000000000001 .00000000000000000000000000000000000000000000000000000000000000000000001 So you are telling me, you don't believe that the sum of (1/10 + 1/100 + 1/1000 + .. ) is equal to 0?
Mathematics starts with axioms. And from those axioms it follows logical, rigorous methods to prove and disprove things. Which is why I have said and always will say, if you have a problem with mathematical system A, create mathematical system B. Introduce a different set of axioms. But you cannot compare A and B and claim A is wrong because it produces different results than B. It makes no sense. On a different tact, and to steer this more away from the actual calculations, but to philosophy: Do you have a philosophical problem with infinite calculations?
1/10 + /100 + 1/1000 + ... = .1 + .01 + .001 + .0001 + ... = .111111111... That is definitely NOT zero. Now, if you have a function f(x) = 1/10^x the limit of that is zero. lim 1/10^x as x -> infinity = 0 But that is not remotely the same as sum (1/10^x) where x = 1 -> infinity.
The math is the math. There is nothing wrong with the system, only the way it is understood. Math lives on the principle of "proof" not understanding. I don't have to understand integrals, just prove that they work. I don't have to understand infinite, just regurgitate a proof. Here is my new math system: take the current one, recognize that 0.9.. and other infinitely repeating decimals cannot be complete, and that 0.9.. < 0.99.. < 0.999.., thus require that one must take the limit of infinitely repeating decimals, and convert to whole numbers. There in that line, I've created a new mathematics system with a billion proofs, axioms, and rigorous methods.
Now, over infinite time, the zeros are now infinitely large, such that [0..]1 That number is not zero? Then why do you think this one is: cotton's Mexico problem: x = 100 50 25 12.5 .. 0.0000000000001 .. 0.0000000000000000000000000000001 .. 0.0000000000000000000000000000000000000000000001 .. 0 ?? <--
To make a point. Eventually a number will becoming [0..]1. Infinitely repeating zeros, and then a 1. And you would say that value is 0. If that one value is zero, over infinite time, all those other values must also be zero, because over infinite time, they too will get to [0..]1 And the sum of zeros cannot be 0.1..
0.99.. - .9.. = (limit (0.9..)* = 1) -( limit (0.9..)* = 1) = 0 * all infinitely repeating decimal places require taking the limit when doing operations. it was in the handout.
That is not what we are doing in Cotton's mexico problem (the same thing as Zeno's paradox). we are adding distances together in the Cotton Paradox: distance to Mexico = 1 unit. Day 1: 1/2 unit Day 2: 1/4 unit Day 3: 1/8 unit . . . So the total distance Cotton travels over infinite days is: 1/2 + 1/4 + 1/8 + 1/16 + ... And this summation is equal to 1 over infinite days. The amount of time between each iteration is 1 day. No matter where you look it is 1 day. I don't get where you are saying I am claiming that time goes to zero. It doesn't. The only thing changing each day is how far Cotton travels.
Then in your system the following property cannot be true: a - a = 0 because by your definition of subtraction: .99... - .9... = 0 add .9.. to both sides .99.. = .9... which contradicts our axiom that .9... < .99... Which is OK, but it is going to be hard to do any kind of manipulations on anything going forward.
Any time you deal with infinitely repeating decimals you have to take the limit. Any time. Every time. Follow the rules: a - a = 0 because by your definition of subtraction: limit(.99..). - limit(.9...) = 0 add limit(.9..) to both sides limit(.99..) = limit(.9...) which does not contradict any axiom, 0.9.. < 0.99.. < 0.999.. still holds true, but if you manipulate them in any way, you must take the limit.
No, there are multiple things changing each day. There is distance traveled, and the time it took to travel, amongst other things. And the distance cotton travels will never get to 1, because it constantly taking the half distance, it is incomplete. The limit is 1. The time it takes for each travel, compared to infiite is : [0..]1. Do you not believe that value to be zero? Time Day 1: 1 hour Day 2: 1 hour Day 3: 1 hour Compared to infinite Day 1: 1 hour / infinite = ? Day 2: 1 hour / infinite = ? Day 3: 1 hour / infinite = ? .. Infinite Proportion to infinite sum of time traveling: sum day 1 + sum day 2 + sum day 3 .. + => ?
Ok. So .99... - .9... = 0 by your definition of operations. and .9... < .99... lets manipulate that: .9.. < .99.. lim (.9..) - lim(.9..) < lim(.99...) - lim(.9...) 0 < 0 so by saying .9.. < .99.. your system also states that 0 < 0. I am not sure we can use a system where this is true. But keep it coming so we can hash out your math system.
I know you guys are having great fun in here, but @fl0at_ is there anyway to ignore a thread so it doesn't show up on "New Posts" Also, since I'm asking about board functionality in a thread that shouldn't have it, is there anyway to change the # of posts per page?
