Only if you round it up.Quote:
Originally posted by GlenW
I think there is some confusion here between equality and equivalence.
0.999........ is not equal to 1.
0.999........ can be considered equivalent to 1.
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Only if you round it up.Quote:
Originally posted by GlenW
I think there is some confusion here between equality and equivalence.
0.999........ is not equal to 1.
0.999........ can be considered equivalent to 1.
I'm sorry but I don't get the problem with this.
If you can accept 3/9= 0.3333...
Why can't you accept that 9/9 = 0.9999...
George Cantor (the founder of modern set theory) defined the real number system as having an infinite number of digits after the decimal place.
Thus,
The sum of an infinite series equals it's limit. Those who say that it only approaches it's limit (never actually reaching it) are not talking about an infinite series.Code:_
1.0 = 1.000...
_
0.9 = 0.999...
And those who are thinking it's some sort of illusion with the base 10 numbering system, think again.
I was one of the most adament people opposing the notion that 1 = 0.999... but I have come to realise I was wrong...big time.
One way people attempt to disprove it is to try to demonstrate that there is a number in between 1 and 0.999... using a base x number system where x > 10.
This fails because no matter which number system you use, the series still converges to the same value.
No. Then it becomes equal.Quote:
Originally posted by Nightwalker83
Only if you round it up.
If you keep the number as it is, you can say that for all intents and purposes it can be considered equivalent to 1.
But if you do no rounding up it can never be equal.
Because it = 1 not 0.9999...Quote:
Why can't you accept that 9/9 = 0.9999...
But do you consider
Code:_
0.3 = 1/3
Nine goes into nine once.
9 divided by 9 = 1.
anything divided by itself = 1.
0.3333...... is equivalent to, not equal to 1/3
I meant to say it aprox equals 1. Of course .999 will never equal one because then you'd have to change number and it will lose its meaning.Quote:
Originally posted by GlenW
No. Then it becomes equal.
If you keep the number as it is, you can say that for all intents and purposes it can be considered equivalent to 1.
But if you do no rounding up it can never be equal.
Why is 0.3333... not equal to 1/3?
It is only inequal if you consider a finite number of digits after the decimal place. Since we are denoting [...] to mean an infinite number, you have no basis for saying that they are not equal.
*bangs cell wall*
You and I are making a total tits-up of being on the same side.:)Quote:
Originally posted by Nightwalker83
I meant to say it aprox equals 1. Of course .999 will never equal one because then you'd have to change number and it will lose its meaning.
That would equal 0.9 not 1.Quote:
Originally posted by simonm
But do you consider
Code:_
0.3 = 1/3
The basis for saying that is;Quote:
Originally posted by simonm
Why is 0.3333... not equal to 1/3?
It is only inequal if you consider a finite number of digits after the decimal place. Since we are denoting [...] to mean an infinite number, you have no basis for saying that they are not equal.
0.333... is infinitely close to 1/3, but not equal.
Sorry about that ;)Quote:
Originally posted by GlenW
You and I are making a total tits-up of being on the same side.:)
This is absolutely correct. You can banter it back and forth all you want, but simon has the right answer here.Quote:
Originally posted by simonm
x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 9/9
x = 1
There you go.
Ok, i am prepared to believe that .999... is equal to zero, if someone answers this question.
What is the number closest to 1, that is not equal to 1 and is less than 1?
WHY Won't anyone answer that?!?!?!?!
I will.Quote:
Originally posted by sail3005
Ok, i am prepared to believe that .999... is equal to zero, if someone answers this question.
What is the number closest to 1, that is not equal to 1 and is less than 1?
WHY Won't anyone answer that?!?!?!?!
It's 0.9999......
Apparantly so. What does it matter what 0.99999 equals?Quote:
Originally posted by rjlohan
Do programmers all get this way at some time, where simple issues get challenged logically, and your brain explodes because you wasted your life trying to prove an apple is a banana??
:confused: :p
it doesn't :p
Quote:
Originally posted by simonm
I'm sorry but I don't get the problem with this.
If you can accept 3/9= 0.3333...
Why can't you accept that 9/9 = 0.9999...
George Cantor (the founder of modern set theory) defined the real number system as having an infinite number of digits after the decimal place.
Thus,
The sum of an infinite series equals it's limit. Those who say that it only approaches it's limit (never actually reaching it) are not talking about an infinite series.Code:_
1.0 = 1.000...
_
0.9 = 0.999...
And those who are thinking it's some sort of illusion with the base 10 numbering system, think again.
I was one of the most adament people opposing the notion that 1 = 0.999... but I have come to realise I was wrong...big time.
One way people attempt to disprove it is to try to demonstrate that there is a number in between 1 and 0.999... using a base x number system where x > 10.
This fails because no matter which number system you use, the series still converges to the same value.
The problem you're having is assuming that 0.111... is actually a definable number. You can not simply multiply an infintite number like this by 10 and move the decimal point.
Look, 1/9 = 0.111... on a calculator. This is an inaccurate representation of this equation, since we can not define infinity as a set number. Hence, you can not perform regular arithmetic on such a number.
Considering the fact that this is an infinite recurring number, we can conclude that 1 can not be divided into 9 equal parts and represented by a decimal system. It must be represented by the fraction 1/9 only to make an accurate representation. This fraction can not be defined as a decimal number, because it isn't.
Hence, 1/9 <> 0.111.., and 2/9 <> 0.222...., but 9/9 = 1.
You following me? You can philosophise all you want, but the simple fact is, that infinity is an abstract concept, and abstraction can not be applied to maths in this way. We can use abstration to estimate (very accurately at times) some mathematical function/number/etc. But it will never be a true representation of the number, and never =.
No it's not. Infinity can not be defined, so you can't use it in a logical context.Quote:
Originally posted by cafeenman
This is absolutely correct. You can banter it back and forth all you want, but simon has the right answer here.
3/A = q³
it must be ... surely ... beyond the shadow of a doubt ... probably ... errrm ...
:rolleyes:
There is no absolute value that is closest. If you are dealing with finite accuracy, it would be 0.999... up to however many digits you have. When you have infinite digits, though, you have infinite accuracy, and as such there is no longer a 'closest to' just as there is not a 'farthest from'.Quote:
Originally posted by sail3005
Ok, i am prepared to believe that .999... is equal to zero, if someone answers this question.
What is the number closest to 1, that is not equal to 1 and is less than 1?
WHY Won't anyone answer that?!?!?!?!
RJ, the method using recurring numbers is part of a proof that any recurring number is rational. It's not an undefined number, it's rational.
sail3005
You're not going to get a satisfactory answer I'm afraid.Quote:
Ok, i am prepared to believe that .999... is equal to zero, if someone answers this question.
What is the number closest to 1, that is not equal to 1 and is less than 1?
WHY Won't anyone answer that?!?!?!?!
George Cantor prooved that there are uncountably many real numbers and so you cannot count sequantially through the real numbers.
Cantor's "Uncountability" proof can be found on this page:
Godel
'Zactly. Like the Av Keddie dear :) S'very cool.Quote:
Originally posted by kedaman
it doesn't :p
Look. 0.999 . . . . = 0.999 . . . .
End of story.
:p
Thanks Vicky :)
I'm not up on all these terms, I'm merely going on what I see as simple 'logic'. The number 0.11111.... can not be defined simply because of the fact that the traling 1s are infinte. Infinity has no definition, or bounds.
The number 1/9 on the other hand is a concept we can use. I'm simply saying that 1/9 <> 0.1111.... because 0.11111 is not a definite number. It is infinite.
rjlohan
Go tell that to Achimedes, Newton, Cantor and many other of the great mathematicians who used a definition of infinity in a logical context.Quote:
No it's not. Infinity can not be defined, so you can't use it in a logical context.
Similarly, you can't use infinite numbers as part of a mathematical equation to arrive at an absolute equality. You can get an approximation, and that's all you ever need.Quote:
Originally posted by simonm
sail3005
You're not going to get a satisfactory answer I'm afraid.
George Cantor prooved that there are uncountably many real numbers and so you cannot count sequantially through the real numbers.
Cantor's "Uncountability" proof can be found on this page:
Godel
Well this is a technical subject RJ. You can't get away from the fact that, technically, recurring numbers are rational. Or, 'definite', if you like.
Infinities cause a lot of problems when you want to look for proofs, but you can often eliminate them from the issue with a little effort and carry on unburdened by them.
I think many of those people had some pretty stupid ideas at times. Just because something they said was genius, doesn't mean it all was. It's all very fascinating philosophically, but I don't think in this case that something like that is true. BTW: I have no idea what exactly you are referring to.Quote:
Originally posted by simonm
rjlohan
Go tell that to Achimedes, Newton, Cantor and many other of the great mathematicians who used a definition of infinity in a logical context.
The fact that infinity by definition has no limit should be enough to convince anybody to see that you can't use that number. You can't just multiply it by 10, because that assumes that there was some digit somewhere down the track, 1billion places after the d.p if you like at which to begin such an equation. And since there isn't, you can't.
1/9 is a rational number. 0.1111.... is an approximation of that number. If you wish to assume that number is infinite, which it is, then you can no longer use it in a mathematical context, except to approximate.Quote:
Originally posted by HarryW
Well this is a technical subject RJ. You can't get away from the fact that, technically, recurring numbers are rational. Or, 'definite', if you like.
Infinities cause a lot of problems when you want to look for proofs, but you can often eliminate them from the issue with a little effort and carry on unburdened by them.
Here's another puzzler :
When and how do you know a number has an infinite number of 9's after the comma ? There is a certain uncertainty about the concept 'infinity'.
How many times does a pattern have to recurr before you can conclude that it will recurr an infinite amount ?
What if the pattern breaks after the 1.000.000.000.000.000.000.000th digit (instead of a 9 an 8) ?
You dismiss it on the basis of logic but what if your logic theorem is wrong ?
Are you ever a 100% sure about your statements regarding infinity ? If not, how much of an approximation do you think you hold ? IS it enough to accurately predict or not ?
In this case the logic of long division is enough to prove that. And shut up will ya - it's confusing enough in here... :pQuote:
Originally posted by Wally Pipp
Here's another puzzler :
When and how do you know a number has an infinite number of 9's after the comma ? There is a certain uncertainty about the concept 'infinity'.
How many times does a pattern have to recurr before you can conclude that it will recurr an infinite amount ?
What if the pattern breaks after the 1.000.000.000.000.000.000.000th digit (instead of a 9 an 8) ?
You dismiss it on the basis of logic but what if your logic theorem is wrong ?
Are you ever a 100% sure about your statements regarding infinity ? If not, how much of an approximation do you think you hold ? IS it enough to accurately predict or not ?
Sorry RJ, but I just don't think you know what you're talking about ;)Quote:
Originally posted by rjlohan
1/9 is a rational number. 0.1111.... is an approximation of that number. If you wish to assume that number is infinite, which it is, then you can no longer use it in a mathematical context, except to approximate.
Is it ?
How can you be sure that the 1.000.000.000.000.000.000.000.000.000.000th digit is the same as the 1000.000.000.000.000.000th ?
The point is that you work with approximations. It's no use guessing the 1.000.000.000.000.000th digit of a number if you use the rounded version in your common calculations.
While it makes up for an interesting philosophical debate it's simply impractical to be used on a daily basis.
The approximations work just fine.
Hear hear. Most of what they said doesn't make sense at all, but they say one intelligent thing, and suddenly, their geniuses, and everything they say must be right :rolleyes: I wish my life was like that.Quote:
Originally posted by rjlohan
I think many of those people had some pretty stupid ideas at times. Just because something they said was genius, doesn't mean it all was.
No doubt :p, but I'm yet to be convinced that 0.99999... = 1Quote:
Originally posted by HarryW
Sorry RJ, but I just don't think you know what you're talking about ;)
Well you'd have to say something intelligent first :pQuote:
Originally posted by rinoaheartilly
Hear hear. Most of what they said doesn't make sense at all, but they say one intelligent thing, and suddenly, their geniuses, and everything they say must be right :rolleyes: I wish my life was like that.
Quote:
Originally posted by rinoaheartilly
Hear hear. Most of what they said doesn't make sense at all, but they say one intelligent thing, and suddenly, their geniuses, and everything they say must be right :rolleyes: I wish my life was like that.
Mine is. :)
.....I think.... :confused:
:p