I understood that to be the definition but how do you actually apply that definition in practice to construct say a meter-stick that is exactly one meter??
yes, but through which medium?? Light can be slowed down/delayed/absorbed & re-emitted or however you want to say it, which can have a significant effect on the linear distance it travels.
Um, I thought an ampere was defined as a specific amount of electrons passing a particular point in one second. How does that have anything to do with kilograms??
Thinking that everyone is so easily capable of killing or harming is making a serious mistake
I didn't even imply that it would be so easy for people to make this choice but when a person's life is on the line, I think you'd be surprised how quickly they'll make up their minds and act, especially if they know they are not alone (in being armed). This is the survival instinct, those us prepared and willing to survive instances like this, probably will.
Fact is, though, that if you try to stop a shooter, the chances of you ending up on the victim list is a lot higher than you ending up the hero who stopped him.
I guess that depends on whether you are one who exercises their second amendment rights and is also carrying. If everyone was like that, the one (with a gun) vs. many (w/o guns) paradigm that is currently the norm would cease to be. One (w/ a gun) vs. Many (w/ guns) is a losing proposition to start with and soon criminals and the criminally-insane would realize it and not bother. This is why I believe everyone should be trained proper firearm handling techniques in public school (every year, like health class/gym) until high school where they receive their final exam and which upon passing their government issued firearm. It would do a lot to bring all sorts of crime rates down as well as accidentally shootings since everyone will be properly educated. Not to mention, I'd like to see the country that tries to invade a country where just about every citizen is trained on how to use and required to own a firearm.
Any number that has an infinite, never ending string of decimal places is an irrational number. Hence, 0.333... is an irrational number, just like pi, e and 0.999...
Please tell me this number that is closest to 1 without being
I did, 0.999... (repeating nines to an infinite number of decimal places) which is also why it is an irrational number.
No, it does not. Decimal notation cannot represent 1/3 exactly, it can only attempt to do so with the irrational number you listed. This is a cheat, a hack, an exception to the rule. And because of this people accept that when they see 0.333... that it most likely is 1/3 because they understand the limitations of decimal notation. I cannot see anyone trying to use 0.333... in any other manner, so I have no real problem with this hack. However, someone could use 0.999... to represent the number closest to 1 without being 1 very logically (how else would you represent it). Yes, it is an irrational number, but so what? Irrational numbers are used all the time in math.
But my point is allowing people to be accustomed into thinking 0.999... = 1 when that notation can also be used to define the irrational number I mentioned is just bad form. It also only serves to confuse people, unnecessarily.
I will argue once again that in some cases 0.999... represents 1 (when rounding 1/3 or 1/9 to decimal and multiplying it by 3 or 9 respectively). however 0.999... also represents the number closest to 1 without being 1. Two valid theories of math conflicting. Since the first is an obvious correction on the limitations of precision of decimal numbers and the second is not, I believe the second to be more true.
In my understanding of infinity, you can never accomplish a infinite number of things, as you would continue having to do them, ad nausium, forever, hence infinity, the never ending series of events.
I do appreciate your respectful debating of this subject, it is a pleasure to see it outside of Plato's The Republic, especially on Slastdot:).
In the end I think we are splitting hairs. I know to solve your given equation, you would solve for the limit as it approaches infinity, and you could argue exactly what you've argued and be correct, in theory. However, in reality, the ball would not have bounced an infinite number of times before stopping as a ball bouncing an infinite number of times can never stop. That situation is impossible (a ball bouncing an infinite number of times and then stopping) regardless of the amount of time it takes. It can bounce a whole mega-ton-ass-load of times, but not an infinite amount of times.
I do, as they are not equal. one is the approximation of the other. The decimal system can't handle thirds and other fractions very well, it has to cheat. Either by using imaginary numbers with infinite amounts of decimal places, or rounding off to a nearby real number. This is fact. What is being done here is confusing people with a hack that manipulates/corrects a fundamental flaw in using decimal numbers.
It is only because decimal notation cannot truly represent 1/9, that we must allow ourselves to believe that 0.999... = 1 even though it is not absolutely true. This is because we believe 1/9 = 0.111... or 1/3 = 0.333... when they are not. Those are only approximations and thus using them in place of their fractions will result in a close but not absolutely correct answer. I will agree that in some instances 0.999... can be equal to 1, but not in all instances. In some instances it represents the number closest to 1 without being 1.
but drawn to its logical conclusion, it becomes a=x=b, since there will be no number y between a and x or number z between x and b. It's ludicrous to think that for one number to be greater than another number there must exist a third number in the middle.
It's a limitation of the decimal system, not my understanding of infinity. Saying 0.999... is the same as 1 and believing it is true, highlights this limitation. If someone meant 1 they would have said 1 not bother implying the number closest to one without being one which is what 0.999... is.
Also, I mentioned this somewhere else, 1/3 does not equal 0.333..., it approximately equals it. Again highlighting the limitations of the decimal system. This is why you always do you calculations in fractions and compute the decimal result once at the end so the error in this approximation doesn't multiply itself.
Let me turn the situation around and see if it you see my point. The question asked was how long will it take it to come to a complete stop? If that ball bounced an infinite number of times, that means it bounced without end, it never reached a complete stop.
My point isn't that you can't calculate that the ball will stop bouncing in 30 seconds (or whatever), my point is that it will not have bounced an infinite amount of times before it stopped. Those two statements are diametrically opposed, if it bounces without end (infinite number of bounces) it never stops. If it stopped, it could not have bounced without end.
no, they are so close that we can't comprehend it. But there is a space. In dealing with our physical world on this planet, this rounding off (so to speak) of infinite smallness works fine, gets us close enough to be considered true. But applied to gargantuan numbers like the size of the universe, it may become pretty significant.
You must be using some other form of logic that I am not aware of. You are talking about a bouncing ball, that is a physical object. If it is going to bounce an infinite number of times, it will take an infinite amount of time because it will never stop, ever. Hence, infinite amount of bounces. Once you cede that the ball will stop bouncing, you have also ceded that it did not bounce an infinite amount of times.
The math may have involved infinity in the terms of as the number of bounces approaches infinity, but not an infinite amount of bounces.
The proof is correct as are the others (1/9 = 0.111111..., multiply both sides by 9, simplify fraction on the left, 1 = 0.9999999
Sorry 1/9 is approximately =.111.... it is not equal to it. Just like pi is approximately equal to 3.1417 (to however many decimal places you need at the moment). These numbers, however, are not actually equally to the decimal representation of them which is why people use the symbols pi, e, i, etc. in their place.
You know I like the shirt that say 2*2=5 (For incredibly large values of 2) as well. It does open minds for possibilities outside of the normal but it is not solid math nor should it be considered to be. A number with an infinite amount of decimal places cannot be defined and thus does not exist. Trying to prove that an integer (which does exist) is equal to an imaginary number which does not exist is both pointless and asinine.
Infinity is a potential number it is not a number. I have a very good concept of infinity. I just choose to disagree on some of the points used when dealing with it.
I'm not missing the point. This proof just exposes a flaw in how we currently envision and deal with infinity. Just like infinity * 0 being undefined when it should logically be zero. Understanding multiplication to be a series of a additions this becomes painfully obvious. No matter how many times you add nothing to itself you will still have accumulated nothing. No matter how big a number you think you might start with, if you never begin writing the equation, you still have nothing.
I swear some theoretical mathematicians really just over-complicate things for no good reason.
If the amount of actions are infinite, then they never stop. It would be logical fallacy to say an infinite amount of bounces will *ever* complete. If it stops bouncing, than it can logically be concluded that an infinite number of bounces had not occurred.
I get that, however, once you multiply it by 10, the resulting number has (infinity-1) decimal places, not infinity. I know the theories of math say that infinity - 1 = infinity but I don't buy it as infinity will always be greater than infinity - 1.
Why is it faulty, because the proof is obviously wrong. 9a can't be equal to 9 and 8.99999..991 at the same time.
First off, by multiplying by ten, they lost one 9 at the end of the series. thus if using 6 decimal points (a = 0.999999, 10a = 9.999990. 10a - a = 9.999990 - 0.999999, 9a = 8.999991).
if you want to argue infinite repeating decimals, than yes, 0.9999... is approaching 1. It's limit as we approach an infinite number of decimal points would essentially make it equal to 1. But you cannot reach infinity so this is a moot point.
They implemented this code/functionality so that when requested they have an automated way to provide the entire details an interested parties account to whatever law enforcement agency requested it. In a grand PR scheme, they figured that it would eventually be leaked this functionality exists, so they present it as a feature to users who then get used to the idea of it being possible. So finally, later on, when it is discovered that they send those pretty profile-encompassing zip files to law enforcement its not completely new.
Slashdot Mirror
I understood that to be the definition but how do you actually apply that definition in practice to construct say a meter-stick that is exactly one meter??
yes, but through which medium?? Light can be slowed down/delayed/absorbed & re-emitted or however you want to say it, which can have a significant effect on the linear distance it travels.
Um, I thought an ampere was defined as a specific amount of electrons passing a particular point in one second. How does that have anything to do with kilograms??
It's still unfair.
What's unfair? The situation?
Thinking that everyone is so easily capable of killing or harming is making a serious mistake
I didn't even imply that it would be so easy for people to make this choice but when a person's life is on the line, I think you'd be surprised how quickly they'll make up their minds and act, especially if they know they are not alone (in being armed). This is the survival instinct, those us prepared and willing to survive instances like this, probably will.
Fact is, though, that if you try to stop a shooter, the chances of you ending up on the victim list is a lot higher than you ending up the hero who stopped him.
I guess that depends on whether you are one who exercises their second amendment rights and is also carrying. If everyone was like that, the one (with a gun) vs. many (w/o guns) paradigm that is currently the norm would cease to be. One (w/ a gun) vs. Many (w/ guns) is a losing proposition to start with and soon criminals and the criminally-insane would realize it and not bother. This is why I believe everyone should be trained proper firearm handling techniques in public school (every year, like health class/gym) until high school where they receive their final exam and which upon passing their government issued firearm. It would do a lot to bring all sorts of crime rates down as well as accidentally shootings since everyone will be properly educated. Not to mention, I'd like to see the country that tries to invade a country where just about every citizen is trained on how to use and required to own a firearm.
Please tell me this number that is closest to 1 without being
I did, 0.999... (repeating nines to an infinite number of decimal places) which is also why it is an irrational number.
Using this notation, 1/3 = 0.333... exactly.
No, it does not. Decimal notation cannot represent 1/3 exactly, it can only attempt to do so with the irrational number you listed. This is a cheat, a hack, an exception to the rule. And because of this people accept that when they see 0.333... that it most likely is 1/3 because they understand the limitations of decimal notation. I cannot see anyone trying to use 0.333... in any other manner, so I have no real problem with this hack. However, someone could use 0.999... to represent the number closest to 1 without being 1 very logically (how else would you represent it). Yes, it is an irrational number, but so what? Irrational numbers are used all the time in math.
But my point is allowing people to be accustomed into thinking 0.999... = 1 when that notation can also be used to define the irrational number I mentioned is just bad form. It also only serves to confuse people, unnecessarily.
I will argue once again that in some cases 0.999... represents 1 (when rounding 1/3 or 1/9 to decimal and multiplying it by 3 or 9 respectively). however 0.999... also represents the number closest to 1 without being 1. Two valid theories of math conflicting. Since the first is an obvious correction on the limitations of precision of decimal numbers and the second is not, I believe the second to be more true.
You can accomplish an infinite number of things
In my understanding of infinity, you can never accomplish a infinite number of things, as you would continue having to do them, ad nausium, forever, hence infinity, the never ending series of events.
:).
I do appreciate your respectful debating of this subject, it is a pleasure to see it outside of Plato's The Republic, especially on Slastdot
In the end I think we are splitting hairs. I know to solve your given equation, you would solve for the limit as it approaches infinity, and you could argue exactly what you've argued and be correct, in theory. However, in reality, the ball would not have bounced an infinite number of times before stopping as a ball bouncing an infinite number of times can never stop. That situation is impossible (a ball bouncing an infinite number of times and then stopping) regardless of the amount of time it takes. It can bounce a whole mega-ton-ass-load of times, but not an infinite amount of times.
I have no problem whatsoever with 1/3 = 0.3333...
I do, as they are not equal. one is the approximation of the other. The decimal system can't handle thirds and other fractions very well, it has to cheat. Either by using imaginary numbers with infinite amounts of decimal places, or rounding off to a nearby real number. This is fact. What is being done here is confusing people with a hack that manipulates/corrects a fundamental flaw in using decimal numbers.
It is only because decimal notation cannot truly represent 1/9, that we must allow ourselves to believe that 0.999... = 1 even though it is not absolutely true. This is because we believe 1/9 = 0.111... or 1/3 = 0.333... when they are not. Those are only approximations and thus using them in place of their fractions will result in a close but not absolutely correct answer. I will agree that in some instances 0.999... can be equal to 1, but not in all instances. In some instances it represents the number closest to 1 without being 1.
but drawn to its logical conclusion, it becomes a=x=b, since there will be no number y between a and x or number z between x and b. It's ludicrous to think that for one number to be greater than another number there must exist a third number in the middle.
It's a limitation of the decimal system, not my understanding of infinity. Saying 0.999... is the same as 1 and believing it is true, highlights this limitation. If someone meant 1 they would have said 1 not bother implying the number closest to one without being one which is what 0.999... is.
Also, I mentioned this somewhere else, 1/3 does not equal 0.333..., it approximately equals it. Again highlighting the limitations of the decimal system. This is why you always do you calculations in fractions and compute the decimal result once at the end so the error in this approximation doesn't multiply itself.
Let me turn the situation around and see if it you see my point. The question asked was how long will it take it to come to a complete stop? If that ball bounced an infinite number of times, that means it bounced without end, it never reached a complete stop.
My point isn't that you can't calculate that the ball will stop bouncing in 30 seconds (or whatever), my point is that it will not have bounced an infinite amount of times before it stopped. Those two statements are diametrically opposed, if it bounces without end (infinite number of bounces) it never stops. If it stopped, it could not have bounced without end.
no, they are so close that we can't comprehend it. But there is a space. In dealing with our physical world on this planet, this rounding off (so to speak) of infinite smallness works fine, gets us close enough to be considered true. But applied to gargantuan numbers like the size of the universe, it may become pretty significant.
obviously 0.000...01. Or as undefined in math. It certainly wouldn't be zero.
You must be using some other form of logic that I am not aware of. You are talking about a bouncing ball, that is a physical object. If it is going to bounce an infinite number of times, it will take an infinite amount of time because it will never stop, ever. Hence, infinite amount of bounces. Once you cede that the ball will stop bouncing, you have also ceded that it did not bounce an infinite amount of times.
The math may have involved infinity in the terms of as the number of bounces approaches infinity, but not an infinite amount of bounces.
so according to you, in the set of integers, 1=2 because there exists no x where 1 LT x LT 2. That is just stupid.
The proof is correct as are the others (1/9 = 0.111111..., multiply both sides by 9, simplify fraction on the left, 1 = 0.9999999
Sorry 1/9 is approximately = .111.... it is not equal to it. Just like pi is approximately equal to 3.1417 (to however many decimal places you need at the moment). These numbers, however, are not actually equally to the decimal representation of them which is why people use the symbols pi, e, i, etc. in their place.
You know I like the shirt that say 2*2=5 (For incredibly large values of 2) as well. It does open minds for possibilities outside of the normal but it is not solid math nor should it be considered to be. A number with an infinite amount of decimal places cannot be defined and thus does not exist. Trying to prove that an integer (which does exist) is equal to an imaginary number which does not exist is both pointless and asinine.
Infinity is a potential number it is not a number. I have a very good concept of infinity. I just choose to disagree on some of the points used when dealing with it.
I'm not missing the point. This proof just exposes a flaw in how we currently envision and deal with infinity. Just like infinity * 0 being undefined when it should logically be zero. Understanding multiplication to be a series of a additions this becomes painfully obvious. No matter how many times you add nothing to itself you will still have accumulated nothing. No matter how big a number you think you might start with, if you never begin writing the equation, you still have nothing.
I swear some theoretical mathematicians really just over-complicate things for no good reason.
If the amount of actions are infinite, then they never stop. It would be logical fallacy to say an infinite amount of bounces will *ever* complete. If it stops bouncing, than it can logically be concluded that an infinite number of bounces had not occurred.
I get that, however, once you multiply it by 10, the resulting number has (infinity-1) decimal places, not infinity. I know the theories of math say that infinity - 1 = infinity but I don't buy it as infinity will always be greater than infinity - 1.
Why is it faulty, because the proof is obviously wrong. 9a can't be equal to 9 and 8.99999..991 at the same time.
First off, by multiplying by ten, they lost one 9 at the end of the series. thus if using 6 decimal points (a = 0.999999, 10a = 9.999990. 10a - a = 9.999990 - 0.999999, 9a = 8.999991).
if you want to argue infinite repeating decimals, than yes, 0.9999... is approaching 1. It's limit as we approach an infinite number of decimal points would essentially make it equal to 1. But you cannot reach infinity so this is a moot point.
My thoughts on this are: ::don tin-foil hait::
They implemented this code/functionality so that when requested they have an automated way to provide the entire details an interested parties account to whatever law enforcement agency requested it. In a grand PR scheme, they figured that it would eventually be leaked this functionality exists, so they present it as a feature to users who then get used to the idea of it being possible. So finally, later on, when it is discovered that they send those pretty profile-encompassing zip files to law enforcement its not completely new.