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The meaning of infinity
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untermensche
Contributor
Rational number:
Wikipedia said:
1/9 is just one way of writing a particular rational number. Nothing "just happened". No one has to sit down with pen and paper and evaluate it out by long-hand. It simply is the number. Recognizing facts like this is what has allowed science and technology to advance. The religion is demanding that facts be rejected because they don't match your intuition, no different than rejecting evolutionary theory because it seems too complicated or the heliocentric solar system because you can see the sun circle overhead.
No.
1/9 is shorthand to say "1 divided by 9". It is an operation, not a number.
The result of the operation is the rational number.
Since people already know the result they can use the two interchangeably.
But they are not the same thing.
Jokodo
Veteran Member
1/9 is just one way of writing a particular rational number. Nothing "just happened". No one has to sit down with pen and paper and evaluate it out by long-hand. It simply is the number. Recognizing facts like this is what has allowed science and technology to advance. The religion is demanding that facts be rejected because they don't match your intuition, no different than rejecting evolutionary theory because it seems too complicated or the heliocentric solar system because you can see the sun circle overhead.
No.
1/9 is shorthand to say "1 divided by 9". It is an operation, not a number.
The result of the operation is the rational number.
Since people already know the result they can use the two interchangeably.
But they are not the same thing.
In that case, just assume everyone has been using "1/9" as shorthand for "the result of the operation of dividing one by nine". It doesn't change anything. It remains true that that number's decimal representation is 0.111..., that basic arithmetics let us derive that a decimal representation of that number times 9 is "0.999...", and that therefore, the number described by the decimal notation of "0.999..." and the number described by the decimal notation "1.000..." are the same entity.
steve_bank
Diabetic retinopathy and poor eyesight. Typos ...
I said it several times. pay attention that the ... means an infinite series not a series of finite algebraic addition. In the third step 10x = [9 + .9 + .09 + .999...] is an infinite series. you can not then jump to 10x = 9 + [ x] amd leave the 9 dangling to be algebraically manipulated any more than you can pick the first 9 out of 0.999.... 9 is the first term in a series not a lone digit. 10x/x = 9 is not a valid operation, you can not algebraically operate on subsets of the series.
Also x = .888... = .8 + .08 + .008... is not an algebraic value, the series does not represent a finite number.x evaluated at a finite number of digits is an algebraic variable which can be evaluated.
And after thinking last night I see 0.999.. as a repeating decimal not subject to a convergence test to begin with,.
1/3 = 0.3333..it does not converge on anything. Keep adding 3 to the end.
To apply an algorithm there should be no difference between 0.333... and 0.999... So my question, if 0.999... = 1 what does 0.333... equal? Please answer.
x = 0.999..
10x = 10[.9 + .09 + .009 + .0009...]
10x = [9 + .9 + .09 + .009...]
10x = 9 + [x]
Infinity can not be reached. To say as 0.999.. goes to infinity it equals 1 is a contradiction. Apply the same reasoning to 0.888... where does it get you.? Sooner or later you have to expand your thinking.
You keep say the same thing over and over again.
Explain why you cannot subtract a finite number from an infinite series?
Explain WHY you cannot ”algebraically operate on subsets of the serie”
Admit that you have no clue whatsoever.
The truth is that it is no problem to add or subtract finite number.
It is in fact no problem to rearrange the term as long as it is absolutely convergent (that means that the sum of the absolute values of the terms) are convergent.
A geometric serie are absolutely convergent.
I can certainly pick the first 9 out of 0.9999...
I will then get 0.0999...
0.333... = 1/3
0.888... = 8/9
You promised to check up on the geometric series... do your homework!
untermensche
Contributor
1/9 is just one way of writing a particular rational number. Nothing "just happened". No one has to sit down with pen and paper and evaluate it out by long-hand. It simply is the number. Recognizing facts like this is what has allowed science and technology to advance. The religion is demanding that facts be rejected because they don't match your intuition, no different than rejecting evolutionary theory because it seems too complicated or the heliocentric solar system because you can see the sun circle overhead.
No.
1/9 is shorthand to say "1 divided by 9". It is an operation, not a number.
The result of the operation is the rational number.
Since people already know the result they can use the two interchangeably.
But they are not the same thing.
In that case, just assume everyone has been using "1/9" as shorthand for "the result of the operation of dividing one by nine". It doesn't change anything. It remains true that that number's decimal representation is 0.111..., that basic arithmetics let us derive that a decimal representation of that number times 9 is "0.999...", and that therefore, the number described by the decimal notation of "0.999..." and the number described by the decimal notation "1.000..." are the same entity.
You can pretend you are multiplying but you are not.
You know where you want to go and pretend you have gotten there.
9 * 0.1111.... is an operation that never finishes. It is a series of infinite multiplications.
You never actually ever reach 0.999.... That is where the rounding takes place. In the pretending the operation has a final amount.
DBT
Contributor
I am not ignoring what you write, I am pointing out that you are repeating things that are not contested, that events are finite - but ignore the larger context in which objects exist and events happen, the possibility of infinite space, multiverse, etc.
untermensche
Contributor
DBT
Contributor
First off....who has made that claim? Quotes and links please.
untermensche
Contributor
DBT
Contributor
No, the problem is that you appear to be arguing with yourself, making up stuff that nobody has claimed and calling it absurd.....which in your own words has ''nothing to do'' with me or anyone else. Which is an exceedingly strange thing for you to do, I'm sorry to say.
untermensche
Contributor
AT ANY PRESENT MOMENT ALL the events in the past have completed.
NO more events in the past will take place AT THAT PRESENT MOMENT.
No more events can take place in the past at a present moment.
They are ALL completed.
They could not have been infinite.
Time in the past could not have been infinite.
NO more events in the past will take place AT THAT PRESENT MOMENT.
No more events can take place in the past at a present moment.
They are ALL completed.
They could not have been infinite.
Time in the past could not have been infinite.
Jokodo
Veteran Member
You got it almost exactly backwards. Precisely because the series is not finite can you take out the first term, multiply the remainder by an appropriately chosen factor such that what used to be the second term has the same magnitude as the initial first term etc, and be guaranteed that the new number is identical to the old one. If it were a finite sequence of nines, the new x what be one term shorter than the old one and thus not identical.
Of course it represents a finite number: the number 8/9 (or in untermensche-speak: the number we get as the result by performing the division 8/9, doesn't make a difference). In base 6, that number is 0.52 exactly, in base 12 it's 0.A8, and in base 9, it's simply 0.8. The fact that it cannot be exactly represented without ellipsis in base 10 is a bug of base ten notation, not an intrinsic property of the number.
it converges to, or rather is 3/9, which can be simplified as 1/3. Just like 0.999... is 9/9, which can be simplified to 1/1.
It gets yo to 0.888... = 8/9. Just like 0.999... = 9/9
One of those two has a simpler alternative but equivalent representation in base 10, the other one doesn't. That's all the magic there is.
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Jokodo
Veteran Member
Like I said to steve above, you got it exactly backwards. I know that the operation should give 1.0 as a result since I'm doing nothing but multiplying 1/9 by 9. If I were simply pretending to have gotten were I want to get, I'd simply write 0.111... = 1.0. It's only by sticking slavishly to the rules that I get the (counterintuitive and, to some apparently, sacriligeous, but nonetheless necessarily correct) result that 9 * 0.111... = 0.999... = 1.0
It doesn't matter how many "1"s there are. As long as the (tautological) rule that "1 * 9 = 9" doesn't change and the input is defined as containing no non-zero digits other than "1", I can see at a glance that I'll get the exact same number of "9"s in the output as I had "1" in the input, in the exact same positions. Whether that number is finite or infinite doesn't need to concern me.
If I were rounding, whether at the 10th or at the 5000th "9", I'd get 1.0 as a result. 0.999... is what I get from not allowing myself to round.
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steve_bank
Diabetic retinopathy and poor eyesight. Typos ...
jokodo
0.999... represents a finite number? By definition an infinite number of decimal places is mot finite. That is what ... means, continiues without end.=
10x - [9 + .9 + ..09 +.009...] is an infinite sequence. When you go to [9 + x ...] 9 is the first term in an infinite sequence. You cannot separate the 9 and x.
You are confusing x0.999... and 8/9 with rounding and truncation. In an infinite sequence there is no term to round at. When you round you are picking a finite number of terms. Calculate 8/9 or any fraction on a calculator and it will round according to the floating point standard to a finite number of digits.
You are conflating finite math with an infinite number. Again, at what point in 0.888... or 0.999.. does the sequence round up?
it converges to, or rather is 3/9, which can be simplified as 1/3. Just like 0.999... is 9/9, which can be simplified to 1/1.
You demonstrate why tour approach fails. 0.999.. converges to 1 while 0.333.. converges to 1/3 which is 0.333... ?
Bordering on gibberish.
0.999... represents a finite number? By definition an infinite number of decimal places is mot finite. That is what ... means, continiues without end.=
10x - [9 + .9 + ..09 +.009...] is an infinite sequence. When you go to [9 + x ...] 9 is the first term in an infinite sequence. You cannot separate the 9 and x.
You are confusing x0.999... and 8/9 with rounding and truncation. In an infinite sequence there is no term to round at. When you round you are picking a finite number of terms. Calculate 8/9 or any fraction on a calculator and it will round according to the floating point standard to a finite number of digits.
You are conflating finite math with an infinite number. Again, at what point in 0.888... or 0.999.. does the sequence round up?
it converges to, or rather is 3/9, which can be simplified as 1/3. Just like 0.999... is 9/9, which can be simplified to 1/1.
You demonstrate why tour approach fails. 0.999.. converges to 1 while 0.333.. converges to 1/3 which is 0.333... ?
Bordering on gibberish.
Jokodo
Veteran Member
I'm afraid you're clearly the one who is confused.
"[A]n infinite number of decimal places" is not a property of the number, it's property of (one of) its string representation(s) in the decimal system of notation. You're confusing the number and its string representation in an arbitrary system of notation. Any halfway decent programming language will throw you a type error when you do that. What are you, a javascript programmer? You might as well be saying that twelve can't possibly be a multiple of four since the two strings "twelve" and "four" don't even share a single letter.
Yes, and that's an unavoidable and well-known bug that occurs whenever the set of prime factors of the number you're dividing by is not a subset of your base's prime factors. It is not a property of the number. 8/9 is the same fucking number whether you choose to write it down in duodecimal or decimal notation. Just like it's the same fucking number when you write the operation down in reverse Polish.
It doesn't.
It doesn't converge to 1. It is 1 or 1/3 respectively. To repeat: The fact that 0.333... cannot be represented in a finite string without ellipsis in base 10 is a bug of base ten, not a property of the number. in base 12, 1/3 is 0.4 exactly, or 0.2 in base 6. Are you claiming that the result of dividing 1 by 3 depends on how many fingers the nearest sapient has?
What you're in effect saying is that the result of of 1/3 can be either a finite or non-finite number depending on the language in which you code the result. You are confusing the referent and the sign. You might as well say that in German, Antarctica ("Antarktis", nine letters) is a smaller continent than Australia ("Australien", ten letters).
"[A]n infinite number of decimal places" is not a property of the number, it's property of (one of) its string representation(s) in the decimal system of notation. You're confusing the number and its string representation in an arbitrary system of notation. Any halfway decent programming language will throw you a type error when you do that. What are you, a javascript programmer? You might as well be saying that twelve can't possibly be a multiple of four since the two strings "twelve" and "four" don't even share a single letter.
Yes, and that's an unavoidable and well-known bug that occurs whenever the set of prime factors of the number you're dividing by is not a subset of your base's prime factors. It is not a property of the number. 8/9 is the same fucking number whether you choose to write it down in duodecimal or decimal notation. Just like it's the same fucking number when you write the operation down in reverse Polish.
It doesn't.
It doesn't converge to 1. It is 1 or 1/3 respectively. To repeat: The fact that 0.333... cannot be represented in a finite string without ellipsis in base 10 is a bug of base ten, not a property of the number. in base 12, 1/3 is 0.4 exactly, or 0.2 in base 6. Are you claiming that the result of dividing 1 by 3 depends on how many fingers the nearest sapient has?
What you're in effect saying is that the result of of 1/3 can be either a finite or non-finite number depending on the language in which you code the result. You are confusing the referent and the sign. You might as well say that in German, Antarctica ("Antarktis", nine letters) is a smaller continent than Australia ("Australien", ten letters).
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steve_bank
Diabetic retinopathy and poor eyesight. Typos ...
0.999... = 9/10 / 1 - 9/10 = 9/10/9/10 = 1
0.888... = 8/9
0.777.. = 7/9
0.333.. = 3/10 / 1- 1/10 = 1/3 = 0.333...
Apllied to repeating decimals the purpose is to reduce the series to a fraction, not resolve a finte number. Consideering the difference between 0.999... and 0.333.. I have to conclude geomteric series is not a proof that 0.0999.. = 1 as a finite number. The technique may be useful, but it is not consistent. If in an actual series of calculations an intermediate result was 0.999.. , carrying it forward as 1 or a truncated number of digits would depend on the situation. Bedore applying a technique and drawing a conclusion, fully evaluate it over differnt conditions. Way back I learned that the hard way. Any technique may have the potential to draw a wrong conclusion going only by a limited application.
I can think of one udefull apllucation real world. Converting decimals to fraction for drill bit sizes and fractional inches. Before calculators there werdecimal to fraction tables for machinists and engineers.
https://www.helpingwithmath.com/printables/tables_charts/cha0501dec_equivalent01.htm
https://en.wikipedia.org/wiki/Geometric_series#Repeating_decimals
0.888... = 8/9
0.777.. = 7/9
0.333.. = 3/10 / 1- 1/10 = 1/3 = 0.333...
Apllied to repeating decimals the purpose is to reduce the series to a fraction, not resolve a finte number. Consideering the difference between 0.999... and 0.333.. I have to conclude geomteric series is not a proof that 0.0999.. = 1 as a finite number. The technique may be useful, but it is not consistent. If in an actual series of calculations an intermediate result was 0.999.. , carrying it forward as 1 or a truncated number of digits would depend on the situation. Bedore applying a technique and drawing a conclusion, fully evaluate it over differnt conditions. Way back I learned that the hard way. Any technique may have the potential to draw a wrong conclusion going only by a limited application.
I can think of one udefull apllucation real world. Converting decimals to fraction for drill bit sizes and fractional inches. Before calculators there werdecimal to fraction tables for machinists and engineers.
https://www.helpingwithmath.com/printables/tables_charts/cha0501dec_equivalent01.htm
https://en.wikipedia.org/wiki/Geometric_series#Repeating_decimals
Jokodo
Veteran Member
WTF are you talking about???
3/10 / 1- 1/10 = 0.2
What the fuck are you talking about? 0.111..., 0.777... and all the other guys we are talking about are numbers, not series, not iterables, not lists of digits. Trying to determine whether or not they're finite by looking at the symbol used to refer to them in the arbitrary language of decimal notation makes exactly as much sense trying to determine whether Asia or Australia is bigger by doing a character count of the English words "Asia" and "Australia".
What "difference"? What's "inconsistent" about it? 0.333... comes out as 3/9, exactly like 0.999... comes out as 9/9. The fact that 3/9 cannot be represented as a finite string in decimal notation without ellipsis while 9/9 can is a property of decimal notation, not of the numbers themselves. (Also, nobody claimed that 0.0999... is 1.0)
And you'd be wrong to do so. (At least when you have independent confirmation that 0.999... actually stands for an unending series of nine in the idealised string representing the number as a decimal, rather than an unknown real number that has been rounded to 0.<as many nines as your float precision allows>)
When you don't fully understand why your engine or program does what it does, that is very good advise. In my day job as a software developer, I try to stick to it. But here we are talking about pure math. Whether 0.999... is or isn't 1.0 doesn't depend on hardware implementation and memory load. It directly follows from how we defined the decimal system to work.
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Jokodo
Veteran Member
Sure it will. Sometimes it will even appear to round wrongly (correctly in the underlying binary, but not fitting the decimal output).
Your calculator will also round large integers when you cast them to floating point numbers. Here's the output of python from casting an integer to large to fit the normal integer byte buffer to floating point and back to integer (python does automatic casting from normal integer to long integer, and indicates that it did so with an "L"-suffix):
>>> a = 12548785845454454568456
>>> int(float(a))
12548785845454453604352L
These two numbers are different by almost a million:
>>> int(float(a)) -a
-964104L
Saying "a calculator will round 8/9, therefore, it doesn't have a final result" is like saying Python rounds 12548785845454454568456 represented-as-real to 12548785845454453604352.0 (which makes a lot more sense in binary then in decimal) means that 12548785845454454568456.0 is a almost a million less than 12548785845454454568456
untermensche
Contributor
That's not what you're doing.
You're dividing 9 by 9.
You are not dealing with 1/9 at all.
A stupid worthless trick.
You are pretending there is a finish to an infinite process.
And in that pretending is where you are rounding off.
You wanted to know where you are doing it so now you know.
Another stupid worthless charade.
Pretend an infinite process ends to prove an infinite string of 9's ends.
Worthless.