The large council
How many times a day (= 24h00m00s) do the second hand, the minute hand, and the hour hand of a clock meet each other?
The answer is 2.
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1 solution
Right , but if you include both starting and final moments of the day (12 and 24) leading to answer 2 in the count you should change the solution for the first problem to 23 anyway.
That's because considering this count you have 23 moments when the hands meet since there will always by such an argument be taken into account the last (24:00:00) anyway.
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12h00m00s isn't the starting point of the day. So I didn't count both the starting and the end points, I only counted 24h00m00s and didn't count 00h00m00s. Also it's up to you if you count 00h00m00s or 24h00m00s but if you're consistent you can only count one of the two as the other one will be part of the next or previous day. A day is 24h00m00s and not 24h00m01s or 24h00m00.00000....1s.
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Oh haha right if the starting point would have been coutned the answer should have been 3 indeed anyway.
Yea , ok then it's consistent with your other answer or otherwise said the reasoning given here and the one for the other problem start from the exactly so to say same interpretation or data but it's not correct that it's inconsistent to count just one of the starting or ending points as the ending point of a day and the staring point of the other day coincide hence not being that if you include the ending point you should exclude it from the beginning of the other I think anyway.
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@A A – As far as I know a day is 24h00m00s and not 24h00m01s or 24h00m00.00000....1s. If it's 24hr00m00s you can only count the starting point or the end point but not both.
Edit: I actually also don't agree that 00h00m00s is part of this day, as far as I'm concerned the day starts a fraction of a second later ;-). But you can argue that it starts exactly at 00h00m00s but in that case it has to end a fraction of a second before 24h00m00s. Either way the answer stays the same.
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@Steven De Potter – Yes , a day is 24 hours and last moment of the day is in the first moment of the next day and yes it makes much more sense if the day starts at that point anyway.
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@A A – If a day is exactly 24 hours then the last moment of the day can not be the same as the first moment of the next day, because than your day would be more than 24 hours. Therefore you can't count both 00h00m00s and 24h00m00s.
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@Steven De Potter – Well , it's not the 00 period that's counted but the say 1st second that happens between the moment 0 and the moment 1 second.
Counting in this way you can say that a second passes when I am at 1 and also 2 seconds when you're at 2 and so on. So if I start at hour 0 1 hour passes therefore when I am at moment 1 hour , and by analogy 24 hours at 24. You need to look at those things as cumulative units in a period of time. Like grains of sand in a hourglass or like meters run in a race. I will not count 0 because it's the accumulation between the 0 and something until that something which matters anyway.
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@A A – If for one second from 1 to 2 you count both 1 and 2 you get an infinity small amount more than a second. The second is either from 1 up to 2 or from between 1 up to and including 2. The same goes for a day, therefore if you count both 00h00m00s and 24h00m00s your day is an infinitly small amount more than 24h00m00s. They teach this concept in mathematics when they teach limits (or at least in my country they do).
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@Steven De Potter – Well then if they do so when they teach limits let's see how right that is anyway. That's merely and artifice but let's look at the matters in this way first though.
I am in a race and will run 9 km starting from point 0 and I want to know what the point in which I shall stop to travel exactly 9 km. Therefore I will have to measure the quantity between the interval 0 9 including or not the 9. If I am to take 9 as a limit in the way it is done in calculus I should say I will never reach the 9 but get infinitely close to it because in order to travel the 9 km I need to get infinitely close to a 9th km. I can't say that I shall travel 9-0.1 because then I don't really cover 9 km nor 9-0.00001 will suffice or any limited quantity because for any limited quantity traveled I shall have also a limited quantity at a distance from the 9 km that are to travel and I don't want that. Therefore the quantity which I travel say M in the calculus interpretation will be so to say M=9-0.00000000..1 right ? But may I ask you , if anyway you you will think naturally at our problem what does that 0.00000000..1 means fact and if there is any difference between such an infinitely small quantity and 0 so to say though ?
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@A A – The thing is if you run 9km, you'll get exactly to the 9km point (as you don't count the 0 point). And there is a difference in math between zero and an infinity small number. Dividing by an infinity small number for example yields a result while divining by zero is undefined.
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@Steven De Potter – Yes , therefore applying this for hours you get at 24:00 to have 24 hours. That point therefore coincides with the first moment of the next day too anyway.
Think at them as it's seen in the thinking. Consider the infinity you speak of. Regarding the equality between 0.00000..1 and 0 from my point of view if you say that they are equal then they really are the same thing. Nonetheless in order to actually say they are or not equal you need to say clearly in what way you think at them and the quantity 0.00..1 because if you indeed accept the way infinity currently is thought of in math even for such a matter then of course we are going to make a lot of superficial and irrational claims like the one 0=0.00..1 in some cases but there is a difference because one is somethign and the other nothing.
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@A A – You reply doesn't make much sense to me, since I said they aren't equal. Also, the end point of the day still doesn't coincides with the starting point of the next day. If you apply what I said to days you get 24:00 to have 24 hours but 00:00 is still part of the previous day, therefore they don't coincide. Anyway, I don't see much sense in keeping this up as we just seem to go around in circles.
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@Steven De Potter – Well let me show you the sense.
I showed that if we are talking of 9 km in a run we need to arrive at point 9 km. By what I see in your second last message you agreed that this is right despite initially telling that when we are thinking of limits point 9 shall never be reached and there should be an infinitesimal quantity to reach it that proved to be 0 though. If therefore this is applied analogously to hours and the analogy should work because it speaks of an accumulation too , for 24 hours we should arrive at 24 the same way in which for 9 km to be done we should reach the point 9 km though. Imagining the hours on an axis you need to reach therefore the point 24 for the said 24 hours to pass in the same way you need to reach point 9 to travel 9 km. My 9 km run was just an example provided to conceive better eventually the accumulation of quantity in a precise way and to justify by this that in order to reach the nubmer of km you need to travel you need to get to the point 9 km which doesn't have an infinitesimal quantity (other than 0 which is irrelavant) from anyway.
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@A A – Like I said, you don't count the starting point when you run 9km (you didn't run that infinity small distance at the start, you just happened to start there). So if you translate this to our problem with the clock the starting point of a day still doesn't coincides with the end point of the previous day.
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@Steven De Potter – Ah I see where it wasn't clear.
Yes you don't take the starting point into the count for calculating any amount of distance or time covered but that doesn't affect the conclusion. What's beign said is that if you want to travel 9km and start at 0 you should reach the 9th km. If the analogy where to hold the same is with 24 hours start there and end at 24. But you seem to say that for the clock problem this is not analogous. If it is analogous the conclusion is clear , and the only way by which you seem to find that the problem is not analogous is by saying that we should take the satrting point in the count but this dosen't make sense to me so can you pelase explain your point regarding the analogy and why such an analogy doesn't completely apply though or if I miss something else ? As I said if the analogy holds then to travel so to say 24 horus you need to reach point 24 and include it in the count anyway.
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@A A – The clock problem it is analogous, but your conclusion is incorrect in my opinion. Whether or not you count the starting point DOES affect the conclusion. If you don't count it, it isn't part of this cycle, while of you do count it you get more than 9km (or 24h00m00s).
The analogy holds, you can also see this if you look at the 22 times I wrote in my solution; 24h00m00s is included there. But in that case the end point of this day and the first point of the next day still don't coincides.
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@Steven De Potter – It's irrelevant if you count or not the starting point in a counting of amount increase for the simple purpose of counting alone though.
If you want to count it you should count it as 0 because it doesn't affect with anything the increase since in the starting point you covered no distance. In case you don't count it that doesn't affect the counting either because you eliminate the starting point when there is no distance and all the other points which give increases in the quantity are counted. Therefore I think that with this point or thing you agree with me that for the purposes of counting alone an increase in amount of quantity doesn't matter if you take or not the starting point into account but the problem put here is not that. The problem here is if the starting point of calculating an increase in amount of hours for this problem belongs to the day or not anyway. Considering this by the very definition of a starting point , a priori , we anyway deduce that the starting point belogns to the day for which we measure the hours. Therefore point 0 belongs to the day , it's irrelevant for counting the amount of the passed time so to say. We shall not confuse the problem of calculating the amount of increased quantity with the problem of inclusion in the day anyway. Therefore so to say the starting point belongs to the day and if it is counted or not doesn't matter because the ending time is still 24. I don't find at this point unless you would show otherwise that counting changes the amount of time increased and therefore that we should either exclude it or conclude that the last moment of the day is at 23 hours 59 seconds. If you say that the difference between the hour 24 and and the last moment of the previous day would be infinitely tiny that';s right. But infinitely tiny which means that the day will end at 23:59:5959595959.. though means that the difference becomes so small it ultimately is of course 0 though.
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@A A – The starting point does have an infinitely small distance, therefore it does matter if you count it or not. And I call the starting point the starting point but as I told you earlier, I don't think the day starts there. You started call it the starting point, I followed because otherwise our discussion would be ridiculously hard to follow. Also, if I didn't call it the starting point but called it what it actually is I'd have to say "a point that is an infinity small amount before the real starting point" (note that this isn't even an exact point as there are an infinite amount of points that fulfill this definition) which is too much of a bother to me.
Also, the problem of counting and including it are not independent problems.
And the ending point is still 24h but in one case the starting point of the next day does collide with this end point and in the other case it doesn't.
Finally, the difference becomes so small but it will never become 0 of course.
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@Steven De Potter – Firstly let's clarify about the starting and ending points because it truth between the last and first moment of two days there is no difference or are identical though.
You mean that the starting point is the instant before the actual starting point. But we're itnerested in the actual starting point so let's make precise this term though. We'll name a point a starting point if it is the first moment of the day or belongs to the set of hours as the first point of quantity in the set. When you say that the starting point(1) is the instant before the actual starting point (2) you consider the actual starting point (2) to be the first point which matters in the count of accumulation of quantity so you define that point in accordance with the rule of counting that takes only the points that are relevant for accumulation anyway. That's why you exclude the instant before , the 0 and consider the 0.11.. moment as the one of the next day I think.
This problem is a problem about the limits case (not in the calculus sense) about the last moment when a day begins and the first when another ends so to say. Therefore predication about it is at this limit case which rises paradoxes and problems because at such limits case it's complicated to perceive things right.
When we try to identify such a moment ,for example if my problem will be where does the final moment of a day is we find that we have to talk about infinity because for each moment I describe I can find one even less than the quantity which I offered as being the tinniest quantity. Now let's take our problem as identifying the points of transition between a day and another. know that the moment of transition between days happens in the last moment of the first day which by definition belongs to that day as the last quantity of the set of quantities of it but when that last moment will happen immediately after it will come the first moment which belongs as a relevant quantity of the next day , but that moment is at an infinite distance fromthe last of the next day. Therefore at a 0.111111.. distance which when thought intutively not jsut shown mathematically sot o say equivalates 0therefore the first moment of the next day and the last moment of the last day equivalate and it's a convention where you choose them to belong. Sorry for the rough explanation I have to go now anyway.
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@A A – Ok, the day starts right after 00h00m00s, the starting point is an infinite small time after 00h00m00s. The end point is 24h00m00s. This way a day is exactly 24h00m00s.
(You could also say it starts at 00h00m00s but than it has to end right before 24h00m00s to have a day that is exactly 24h00m00s long.)
If 00h00m00s was the starting point and 24h00m00s the end point, your day would be more than 24h00m00s. There would be a moment when it's both Monday and Tuesday. If this was the case most of the logic problems on this website which involve the days of the week would be incorrect.
If there was an overlap in days and I told you today is Tuesday. And asked you what day it was yesterday, there could be two, possibly even three answers (Sunday, Monday, and possibly even Tuesday).
Anyway, I've added that a day is 24h00m00s to the problem, since technically it could have been a day on which a change is made from or to daylight saving time.
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@Steven De Potter – I don't know how many problems would be incorrect but you have to consider what I said there attentively. I'm not saying your model or proposal is wrong necessary , as it still works correctly but the initial suppositions which underlie the counting you propose doesn't seem completely right in my opinion and in fact I'll try to give a more formal proof of my claims though.
Proposition 1 : Let D1 be the set of all moments which are included in a day and D2 be the set of all moments which are included in the day after the previous day. Then considering this two days the last moment of the first day (say L) and the first moment of the day after that day (F) are the same moments anyway.
Proof : suppose for the sake of contradiction that the moments differ , that L different from the moment of the other day F. We know that the two days are immediately one after the other or continuous which means that between all the elements which are between the first moment of D1 and the last moment of D2 the elements belongs either to D1 or D2 ( inclunding (D1 and D2). If L and F differ then there are 2 cases to consider : either is a moment between L and For there is none.
For the first case then we would have a moment between L and F which is neither the first moment of D2 nor the last of D1 and therefore doesn't belong to any of our 2 sets but this contradicts our initial claim that all elements between the first moment of D1 and the least of D2 all elements belongs to either D1 or D2. So because this case leads to an inconsistency we can legitemately conclude that there is no such moment between L and F and therefore that case 1 considered here is inconsistent.
For the second case there is no moment between L and F but then the element between L and F is none or 0 therefore contradicting our initial claim anyway. Because both cases give contradictions we conclude that the initial moment of a day and that the first moment of a day are equivalent or the same.
Therefore counting the moments by starting with 00 and ending with 24 is a so to say consistent system of counting the moments and the day transition so to say. That there is an infinitesimal distance between the moments is just an other way of saying the distance is 0 anyway.
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@A A – I don't see any contradiction with the 2nd case where there is no time between L and F. Please explain what the contradiction is.
However, by counting both 00 and 24 there is an inconsistency with defining that a day is exactly 24h. How infinity small this extra time is, it isn't zero. Calling it infinitely small is absolutely not a way of saying it is zero, it is the exact opposite. By calling it infinitely small we are saying it absolutely never can be zero.
In about every field of science, in a cycle the points can only belong to one cycle, this holds true for an absurd amount of cases.
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@Steven De Potter – Ok let me point it as I didn't indeed anyway.
If there is no time between L and F but they are not the same element then we can always find some period between L and F i.e some elements between the sets. For there to be no element therefore we need to have the L and F not different say. If L and F are distinct there will always be because we count a continuum some elements between them as we can always find an element between them anyway.
It is right that the counting of the elements of any m number of ordered cycles should exclude the components which belong to both , their intersection so to say. This gives the idea that the number of elements do not repeat in any cycle and their intersection is void of elements anyway.
But , in truth the same result (of counting the same number of elements) so to say the same counting can happen if you exclude intersections or common anyway elements between the elements of a cycle and elements of the next cycle because those intersections will be counted just once as they really repeat once. This counting will yield the same result until it count or doesn't the last component of the sequence of cycles being counted , the last component of the last cycle which can be counted or not depending on when the cycle actually ends therefore on what rule we choose and how we think of this increase so to say.
But let's turn to the infintiely small. You mean it is so small than any imaginable quantity implies it is even a smaller unit , right ?
I suppose such a small quantity would be a 0.000000000..1 for 1 is the smallest quantity imaginable though it can also be some other like 0.000000000000..96. But from saying like this don't we say actually that the quantity becoming therefore so small in such a way that ultimately equals 0 and not that it never reaches 0 ?
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@A A –
- They can totally be different times without having any time in between them; one right after the other, there is no contradiction in that.
We can approximate it to zero, but it never is or becomes zero.
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@Steven De Potter – The idea was that they should have time between them if they are different moments because if they are different isn't there a tiny moment between L and F ? If I choose any 2 quantities such that a < b there will always be some other quantity between the a and b because for any quantity a and any quantity b there will always be a quantity bigger than a and lesser than b since the moment a and b satisfy the equality b =a + something if not b = a+0 , either one or the other. On the other hand though , you may have a point because if it were for this to work then for any starting quantity all other quantities should coincide to it. Therefore the after in what you said can be interpreted as an instant after it so to say being equivalent with the element itself which is different. I think i got things now though if I mam attentive a little. Indeed the distance between our quantities a and b considered will be so to say 0 but there are more interpretations for distance between the quantities is 0 either that they are the same or they are different where the next is immediately the different quantity. This is the interpretation the right way done for 0 if we admit there are different quantities. In other words the validity or wrongness of the system depends on whether we have different quantities or not and in direct thinking ,in intuition we see there are different quantities therefore concluding that the second interpretation is also valid. Nonetheless though this doesn't seem to say that such an interpretation necessarily sound and therefore that the first with 0 wrong. Therefore such a stuff doesn't conclude if the L and F are the same or not but shows my proof wrong. This makes things consistent under the interpretation provided by you but we have to look also at the starting point problem. For your interpretation to work you need to so to say start the counting with the first relevant quantity to be counted because if you start with the quantity which is not being counted , that being L you should include the poitn as an element of the set of the day or not. Actually this seems to be arbitrary you can either include the starting point or not in the day because this is not dependent on the quantity measured on the "amount" and the problem of including it or not is independent of the problem of counting rightly though. Therefore indeed it can be concluded that from the point of view of counting the system proposed by you with L and F different is consistent. Nonetheless the qualitative problem of wether the starting point belongs or can belong to the set of components of the measuring not of the measured units from which it is excluded should say that it belongs because it defines the measure by giving the start. Therefore we should say that the starting point belongs to the measuring if by measuring we understand measuring procedure but for the amoutn claculated it doesn't belong in such a system but considering that your problem asks nonetheless of a day you should say if it is quantitive or qualitative day anyway. Therefore day in what you say in the problem is to be understood as a measured qauntities that makes a day (in your system) this shouldn't be included. But if day means qualtiative day which is measuring , the measure procedure you include the 00 so you did very good to say there 24 hours as you explain youa re refring at the quantity of day Observe though that this happens in your system which seem consistent. Nonetheless there should be that only one system is right.
For the second part , suppose by your way therefore you should also say 0.99999999999.. is infinitely close to 1 but never 1 so to say like with 0.11.. right ? But then if I ask you to conceive those infinite set of 9s or 0s shall we conclude that the number 0.000..1 or 0.9999.. means what ?
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@A A – "The idea was that they should have time between them if they are different moments because if they are different isn't there a tiny moment between L and F ?"
=> Nops, there doesn't have to be. One can follow the other, we wouldn't be able to type both values as there would be an infinite amount of digits, but one can be right after the other without there being any time between them.
"If I choose any 2 quantities such that a < b there will always be some other quantity between the a and b because for any quantity a and any quantity b there will always be a quantity bigger than a and lesser than b since the moment a and b satisfy the equality b =a + something if not b = a+0 , either one or the other."
=> That is only if you choose an a and b which do not exactly follow each other up. If one is right after the other than b = a + an infinity small amount. And time goes by, by adding this infinity small amount for an infinite amount of times, as an infinite amount of moments pass.
The second part of my previous reply ("We can approximate it to zero, but it never is or becomes zero.") was an reply to:
"I suppose such a small quantity would be a 0.000000000..1 for 1 is the smallest quantity imaginable though it can also be some other like 0.000000000000..96. But from saying like this don't we say actually that the quantity becoming therefore so small in such a way that ultimately equals 0 and not that it never reaches 0 ?"
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@Steven De Potter – Well , those first parts which you cited me were just the first phrases of what i said there and not the conclusions. If you understood correctly the continuation of those first phrases you should have seen the actual conclusion. Nonetheless for the second things of the first part , the one with a = b+0 or a = b+something. The claim applies for a being b+infinitesimal small quantity as that is also something. But , please note that as I forgot to point in the previous message the claim a=b+some infinitesimal part which you sustain is inconsistent with that tiny party not being 0 unless you have some explanation of that tiny part being soemthing different from a quantity as 0 and still be nothing between a and b. If you admit the fact that there is something between a and the immediate b after and that something is different from 0 then it should be between a and b as something therefore being no escape on saying either the infinte small stuff is something or 0. You have to choose either something or 0 for it. This is not a calculus stuff necessary in whcih the intepretation of limit to be applied I think.
I understood your reply for the second part. That's why I asked in continuation you if you consider that , in analogy and coherence with 0.0000001.. not being 0 0.9999.. considered so isn't 1 and also after dealing with the answer to this problem though what's your interpretation or what you think 0.000001.. and 0.99999.. means.
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@A A –
- As each moment in time has an infinitely small length, b can equal a+infinitely small (and therefore something) without there being anything between a and b.
Well.. you're asking me what a number means, it is a very specific value. Nothing more, nothing less. If you put s behind it, it means that many seconds, if you put m behind it, it means that many meters, etc. :P You can represent all the real numbers by a line and then any specific value would be a point on this line.
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@Steven De Potter – Unfortunately the first point doesn't seem correct o me but it's maybve just anyway me. An infinitely tiny amount is either something or 0 so a+infinitely small means either one of the two a+something or a+0.Therefore if you say b=a+infinitely small and you say there is nothing between them then that is b=a+0. There's no problem in considering that the infinitely small is not a different quantity from 0. Considering though an infinitely small quantity/magnitude as different from 0 it means that the infinitely small is something but then you should explain how it can be something between b and a while still be 0 between them. If you sayinfintiely small is something which is between a and b but 0 is between them that looks for the msot cases like a simple contradiction because there can't be both something which would be between a and b but at the same time admit there is nothing between the two b and a and if in this case it is so , you should show how can this contadiction be avoided in the way of thinking and claim you say I think.
Emmm , but you didn't say if 0.9999999...=1 or not considering the same way of thinking which makes you say that 0.0000..1 doesn't equal 0. That was so to say the first of the questions. For the second part I just asked you to say how you think at the value 0.00000..1. I meant that there is something in your mind which happens when you say and think it and try to makethat articualte and explicit. Defining a number and what a number is , may be pretty complicated letting alone the difference between the number as the unit of measure of something empriical and the simple abstraction of number which can be considered either quantitatively either qualitatively in it's a priori content. Saying that a nubmer is a value for example is too broad because values are also truth values or parameters or of course a lot other stuff and if I would have to risk a preliminary definition of the term I'd say a number is a quantity or the abstract concept of an accumulation. People used set theory to define numbers as sets which would be cardinal numbers. There are of course apart from the cardinal nubmer the ordinals which are a more qualtiative expression because they are number used in definiing a order or in a counting of things but even this type of nubmer has the notion of cardinal number and qauntity implicit in the thinking of it anyway.
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@A A –
- "Therefore if you say b=a+infinitely small and you say there is nothing between them then that is b=a+0."
.2. I previously said: "We can approximate it to zero, but it never is or becomes zero." The same can be said of 0.999... : we can approximate it to 1, but it never really is or becomes 1.
.3. "Saying that a nubmer is a value for example is too broad because values are also truth values or parameters or of course a lot other stuff and if I would have to risk a preliminary definition of the term I'd say a number is a quantity or the abstract concept of an accumulation."
Well there's nothing wrong with saying a number is a value; it doesn't mean that a value is therefore a number. Also calling it a quantity is too strict; what about all the cases where it isn't a quantity. It becomes a quantity by placing units behind it, 5s, for example means 5 times the defined quantity of 1 second; but is 5 on itself a quantity? 5 what? What does it quantify, without units?
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@Steven De Potter – You'd really have to be more articulate to make your point clear. If there is an infintiely small duration then there is a duration , if not there is none.
So you state that both 0.9999.. and 0.00000..1 are not 1 or 0 But , nonetheless you maybe should know that it's considered that 0.9999.. = 1.
I didn't said it's wrong to say it is a value. I said it's too broad , that not all values are numbers and therefore the definition isn't good. Nor do i said the definition which I provided is too good , but I think it is more close to the right understanding. Though speaking of this is not necessary for this discussion and can be very complicated I'll just reply to your objection.
Considering your example of 5 seconds my definition wasn't about an empirical quantity when i said "quantity". The concept of quantity is a category of thought independent of empirical content and when I say that a number is a quantity though this can't be thought without the empirical things since noentheless the "quantity" is different from them. The fact thatr I can't perceive a quantity or a number without empirical data doesn't mean that such a concept is necessary derived from empirical content since the category itself is found in the faculty of thought and has nothing to do with the empirical things but at least in the classic view proposed by transcendental idealism is activated by those things to work. Nonetheless for this point so to say it isn't necessary to speak of this though its maybe important to mention it. If letting transcendentalism so to say alone I have to reply you I'd say that "5" as an quantity or magnitude implies it being a magnitude of anything I want abstarcted from their sensorial content as a unit so 5 means formally the quantity of 5 units. Considering you'd say that this means that I take the formal units as necessary to perceive 5 when in fact they are not and 5 in itself is something else which i can perceive independent of any sensorial content different from a quanity I'd reply to such an objection by saying that from all we know 5 is a result of counting so it is a counted amount of things which implies the notion of aggregate/set and therefore that if you want to point where the reasoning is wrong you need to be more articulate regarding this last point because saying 5 without formal units may just be like saying a square circle in Euclidian geometry and the reason for why the nubmer is wrongly defined along asking what's 5 without is a logic contradiction.
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@A A – Ok, moment a and b are each 1s long. A = (0,1] and B = (1,2]. Since they are 1s long b can equal a + 1 without there being any time between a, and b. Now instead of 1s the moments a and b each have an infinitely small duration. B can equal a + infinity small amount, without there being any time between a and b.
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@Steven De Potter – Yep , you got it somehow I think. I proposed an explanation too some messages upper and I'll re-explain that in my view too anyway.
There is possible that between two different points or moments or whatever which are to be though as elements of a set in general that the distance between is 0. But this has to be interpreted by virtue of fact that the moment are different from which you c n derive that they are also continuous , one after the other. The example you provide has a problem with the notion of infinitely small amount because it's not clear or mathematically explicit what you mean by infintiely small since as I said if you propose it to be 0.000..1 that is 0 as shown upper. Of course the quantity considered here that is 0 is no quantity anyway.
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@A A – Well there's a mathematical way of expressing infinitely small (just as there's one for infinity); it's the limit of x for x => 0.
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@Steven De Potter – Right. And we're interested of what is that smallest value , is it 0.0000..1 ?
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@A A – Sure, if ".." represents an infinite amount of zeros.
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@Steven De Potter – But didn't we see that quantity is 0 ? Therefore that can't be the infinitely small if you prefer it to be greater than 0 anyway.
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@A A – When did we see that? I said it can be approximated to zero but never truly is zero.
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@Steven De Potter – Well , you agreed with me than 0.99999.. is by analogy also not equaling 1 but then you have to prove that is right because officially 0.9999.. = 1 anyway.
In one of my upper comments I remarked that and thought you agree with me therefore of the fact that 0.9999.. = 1 which would imply also that 0.00000..1=0. Indeed you never said yu agree that 0.999999 is 1 but sicne you didn't reply to that comment in which I told you officially 0.9999..=1 is valid I thought you do anyway.
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@A A – I did say it isn't 1. Here's a copy paste from a previous comment I made:
.2. I previously said: "We can approximate it to zero, but it never is or becomes zero." The same can be said of 0.999... : we can approximate it to 1, but it never really is or becomes 1.
Also, there's no such thing as "officially" 0.999... = 1. That's an approximation, since in the vast majority of cases the little difference doesn't matter. In our problem however, it does since it's the difference between the answer being 2 or 3, which is a 50% error, not a small approximation.
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@Steven De Potter – Ok. Indeed saying it's official accepted is very unsuitable when speaking of the validity of a predication or concept so please excuse me I ever said that. What meant nonetheless was that most mathematicians , the very alrge part of the community thinks 0.9999999.. is actually 1 and while I do not agree with all that is at large accepted in the community nor I being a mathematician I agree with it. You can inquire related to that stuff on Brilliant or on internet about that and you'll find some pretty convincing proofs of the truth of such a claim well i do actually have a proof quite hard to dismiss which I think is the easiest from all the proofs seen anyway.
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@A A – Well, I'm pretty sure they don't think it actually equals 1 but approximate it to 1.
Anyway there are several ways to proof that 0.000...1 = 0 is incorrect.
1: If this equation was correct than you can multiply both sides by the same number and it should still be correct. If we multiply both sides by infinity the equation is incorrect, therefore 0.000...1 cannot equal 0 (we basically multiply that very small error by infinity to make it a bigger error). The same can be done for 0.999... by subtracting 1 from each side first.
2: If 0.000...1 = 0 than also -0.000...1 = 0, and also 0.000...2 = 0.000...1, and -0.000...2 = -0.000...1, and we could keep doing this for an infinite amount of values, and in the end all the values would equal each other, which is clearly incorrect.
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@Steven De Potter – Well , but don't be so sure. Rather find out those proofs before dismissing the proofs to see what they say anyway.
Your proof of the invalidity of the equation relies on very doubtful claims regarding the computation of the troublesome parts , the 0.0000..1 and 0.9999.. anyway. Therefore you should show that you have sufficient reasons for atesting such a calculation to be valid. Explciitly you don't know for sure if 0.00000..1 multiplied gives you anything else that 0 or 0.999999.. gives you a negative value. Such a question is set up rather only after findng what value they do have and therefore to say so is dependent on the value not the other way around the quantity or value dependent on the computation. When saying for example that 0.99999..-1 so to say gives you a negative result you do this on the basis that 0.9999.. is a negative value which is the first thing you consider to calculate the result therefore the proof being incoherent as it starts from something which is determined A to prove that that thing which determines it B does determines it (A) which of course seen by such is obvious wrong anyway.
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@A A – I edited my reply before you submitted yours, can you re-read it?
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@Steven De Potter – Yes. Indeed as I said in an upper comment even if you have 0.000000..96 or so to say whatever other value that should be also 0 anyway.
Also , there is no reason for why 0.00000000..1 multiplied by infinity doesn't give 0. If 0.00000000..1 means that the 0s never end then that excludes the 1 anyway.
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@A A – Well, we don't need to know what infinitely small multiplied by infinite is, the equation is still incorrect.
Here's another way of showing it btw:
Claim: 1/∞=0
Multiply both sides by ∞ and you either get 1 = 0. This is incorrect therefore the claim is incorrect. Or you get an equation which is undetermined, therefore the equation is still incorrect.
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@Steven De Potter – When you divide something at infinity doesn't it mean that you divide that stuff in an infinite number of parts though ?
Therefore , again for this proof we can't start from the claim that 1/infinity is 0 in the first place but letting that alone 0.0000001is rather 1/10^infinity not 1/infinty though.
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@A A – Well, "1/∞" literally divides 1 in an infinite number of parts, so I'm not sure why 1/10^(∞) is more correct. But you can do the exact same thing with this equation:
Claim: 1/10^(∞)=0
Multiply both sides by 10^(∞) and it becomes either 1 = 0, or undetermined. In both cases the equation is incorrect. (Oh, you can say that this doesn't proves 0.000...1 = 0 wrong, but it does show that infinitely small =/= 0. For 0.000...1 you can do the exact same thing.)
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@Steven De Potter – Multiplying 0 by infinite is incorrect. This doesn't make therefore that the proposed equation is incorrect as such anyway.
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@A A – Hmm.. Yeah, you're correct there I think, both ∞/∞ and 0*∞ are undefined, I guess.
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@Steven De Potter – Well , yes it's something like that I suppose therefore concluding that because they are unedfined so to say we can't say too much about the equation anyway. I'm not sure if the term "undefined" is the most suitable here and if instead of naming it so actually we shouldn't use the term "inconsistent" but nonetheless the point that using an inconsistent expression doesn't help you getting a valid proof shall apply in both situations and this is whatw e are interested here though an analysis of the reasons of inconsistency of infinite/infinite dividing numbers use powers are infinities by themselves is quite a cute after thought.
Moreover supposing that theya re not inconsistent the result should still be consistent. Most likely 0*infinite if it wouldn't be undefined as well as infinity/infinity equal infinite. Therefore applying that stuff in the equation should lead you to the so to say equation that after all infinite = infinite which is correct. Of course considering things this way we are forcing thinking which is a least curious though.
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@A A – Not sure what you're saying but 0*infinite doesn't necessarily equal infinite.
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@Steven De Potter – Before speaking of what operations with infinity do and don't I think it's better to know what we mean by saying that infinite does operation "a" to be afterwards pretty sure we know what's the right way of interpreting the meaning of such a predication , to say clearly what we have in mind or whatis the meaning we so to say conceive when we say that an operation with infinity does that or that and perceive it clear also when we say it. Of course this implies we have a clear conceiving of the notion or concept of "infinity" to use that and say about the way it happens to be manifest in thought , in the same way in which for you to perceive the idea of "movement of a point" to say so you need to have clear the point which is the object that moves since otherwise your perception of "movement of a point" would be very unclear while here even this is object (infinity) is pretty unclear anyway.
Please try to solve my other similar problem first , since the solution to this problem will spoiler my other problem.
The minute hand and the hour hand will meet each other twenty-two times a day. To see how many times out of these twenty-two the second hand will meet them as well, we need to know the exact times at which the minute hand and hour hand meet each other. Because the minute hand and the hour hand meet each other exactly 1 2 h o u r s 1 1 t i m e s it takes 1 1 1 2 or 1,090909... hours to meet. This equals 1h5min 27,272727... seconds. The times at which they meet are than:
01h05m27s
02h10m55s
03h16m22s
04h21m49s
05h27m16s
06h32m44s
07h38m11s
08h43m38s
09h49m05s
10h54m33s
12h00m00s
13h05m27s
14h10m55s
15h16m22s
16h21m49s
17h27m16s
18h32m44s
19h38m11s
20h43m38s
21h49m05s
22h54m33s
24h00m00s
Only at 12h00m00s, and at 24h00m00s will the three hands meet, so the answer is two.