- #36
okidream
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No. There aren't. See my reply to choot. 1 is the boundary.Integral said:Fine, there is a finite number of 9s before the one. There must then be an infinite number of zeros AFTER the 1.
No. There aren't. See my reply to choot. 1 is the boundary.Integral said:Fine, there is a finite number of 9s before the one. There must then be an infinite number of zeros AFTER the 1.
Absolutely! You can easily show it directly with long division. 9 goes into 9.0 0.9 times with a remainder of 0.09 - repeat forever and you get [itex]9/9 = 0.\bar 9[/itex].okidream said:Now, if you examine this case: Is 9/9 = 0.999.../0.999... ?
okidream said:Who says it's impossible? Tell me what do you get when you multiply 9 with 9. You get 81, don't you?. The last, of the last of the last place is the number 1, isn't it? Isn't that different from 9? So isn't the boundary the number 1?
the issue here is, how are you going to write the infinite number 8.999...1, (encapsulated with the last of the last value that ends with '1') in the definition given in all of maths? I can't think of any way, except like I wrote it--- 8.999...1.
Which is the reason I think the decimal point system, even if it has been worked on for centuries old, as Integral put it, which case I don't really give a toss, is pretty darn useless and only creates fallacies of its own.master_coda said:It's impossible by definition. If there is a last value, then by definition there are only a finite number of values. If you have an infinite number of digits, there is no last one. Saying "1 is the boundary" doesn't magically change this fact.
okidream said:Which is the reason I think the decimal point system, even if it has been worked on for centuries old, as Integral put it, which case I don't really give a toss, is pretty darn useless and only creates fallacies of its own.
Just as how you point it's impossible for me to do so by definition, why can't I point out that it's impossible too by definition that:
0.333... is the sum of the G.P of the decimal point additions which reaches the sum 1/3, is all but plain fallacy.
The same argument I can say 1/3 is the sum of the G.P with a = 1/4 and r = 1/4, (the series looks like 1/4, 1/4^2, 1/4^3, ... ), and then I conclude that my number base was 4 and the number was 0.111...
okidream said:Which is the reason I think the decimal point system, even if it has been worked on for centuries old, as Integral put it, which case I don't really give a toss, is pretty darn useless and only creates fallacies of its own.
okidream said:Just as how you point it's impossible for me to do so by definition, why can't I point out that it's impossible too by definition that:
0.333... is the sum of the G.P of the decimal point additions which reaches the sum 1/3, is all but plain fallacy.
okidream said:The same argument I can say 1/3 is the sum of the G.P with a = 1/4 and r = 1/4, (the series looks like 1/4, 1/4^2, 1/4^3, ... ), and then I conclude that my number base was 4 and the number was 0.111...
Think about it, and stop shooting me with what I did wrong, because from the start (which is possibly the definition itself) it seems already wrong.
ohwilleke said:This is definitional, but once you introduce infinite strings, it is necessary to define operations on those infinite strings, and the definition conventionally used has the virtue of being analogous in all respects to those same arithmetic operations when defined on finite strings. Any other definition would not preserve a host of standard algebric properties of real numbers.
For example, this definition preserves the relation of B*(A/B)=A, and it is difficult to imagine any other definition which would preserve this property.
If you choose any other definition division would not have a well defined inverse function for numbers on the real number line, which would be a very undesirable feature for most ordinary mathematics.
Likewise, this definition is necessary to preserve the relation that A/B=C/D
Where A and B are in one base number system and C and D are in another base number system and A=C and B=D in parts of each number system where the mapping from the AB number system to the CD number system are the well defined (for example, where A and B are whole numbers in one base number system and C and D are whole numbers in another base number system and there is a definitional rule that establishes a mapping from whole numbers in the AB system to whole numbers in the CD system).
If the standard definition of the infinite string is not adopted, you have done the equivalent of adopting a preferred reference frame in GR.
Alternately one could define 0.33333. . . as the limit as the number of digits approaches infinity of the series 0.3, 0.33, 0.333, . . . which would uniquely produce the same natural definition of 0.3333. . . . which can be restated:
The limit a the number of digits approaches infinity of 1/3-0.3, 1/3-0.33, 1/3-0.333, . . . which is zero.
I agree.ohwilleke said:The limit a the number of digits approaches infinity of 1/3-0.3, 1/3-0.33, 1/3-0.333, . . . which is zero.
Hurkyl said:...Which is a familiar geometric series whose sum is (3/10) / (1 - 1/10) = 3 / (10 - 1) = 3/9 = 1/3.
You're tired of explaining this because what you are saying makes no sense (seriously that post means nothing to me).okidream said:well, I can divide 3 by any integers, (although by the defintion of base,
not < 3, since we all love definitions, don't we?) and that number sums to 1/3. The fraction 1/3. For any base - the fraction 1/3 of that base. Period.
I'm already getting tired explaining on this.
1) Do you know what a number is?okidream said:well, I can divide 3 by any integers, and that number sums to 1/3.
okidream said:I agree.
To answer this question from the beginning:
The reason why 1/3 *3 = 1 where 0.333 (repeating) *3 is not:
is because 1/3 is perfectly rational (or simply as an elementary school kid would call it a fraction) while 0.33(repeating) is not.
0.33(repeating) = 1/3 is iff 0.33(repeating) is taken as sum to infinity, regardless of ANY BASE. Get it... ANY BASE.
Further, but at the same time, 0.111(repeating) is also = 1/3 is iff is taken as sum to infinity, becuase I simply had chosen 4 as my base.
Those who will continue to say this doesn't make sense, please don't just shoot those words. Prove it.
okidream said:...
To answer this question from the beginning:
The reason why 1/3 *3 = 1 where 0.333 (repeating) *3 is not:
is because 1/3 is perfectly rational (or simply as an elementary school kid would call it a fraction) while 0.33(repeating) is not.
...
matt grime said:Why the hell have you introduced this spurious analogy?
I know how to make the decimals a model of the real numbers, thank you.