0.999..., does or does not equal 1 thread

[nonconformist]

0.(3) is not an irrational number, it has a pattern.

Sorry, not irrational, fractal or whatever it's called. Haven't studied algebra in 3 years.

 

[ParadigmShifter]

Well it's definitely not fractal (Which means fractional dimensional).

And 1/3 representing something that can't be written on paper? How about 1/3

Or, in base 3

0.1

 

[azzaman333]

Sorry, not irrational, fractal or whatever it's called. Haven't studied algebra in 3 years.

The word you are looking for is "annoying"*.


*may not actually be the word you are looking for

 

[nc-1701]

I think it does for the main proof I've seen is the 10*0.99...=9.999 etc. one already posted.

Another idea would be to write 0.999... as an infinite sum (perhaps an easier definition).
infinity
Σ9*(1/10)^i=0.9+0.09+0.009+... =0.999...
i=1

Since this is a geometric series we know the sum of an infinite geometric series is equal to a/(1-r) where a is the first term and r is the ratio, thus.

infinity
Σ9*(1/10)^i=0.9+0.09+0.009+... =0.999...=0.9/(1-(1/10))=0.9/0.9=1
i=1

 

[nonconformist]

Well it's definitely not fractal (Which means fractional dimensional).

And 1/3 representing something that can't be written on paper? How about 1/3

Or, in base 3

0.1

Eh, 1/3 is just a way of more easily representing a number that is unbeleivably hard, if not impossible to represent, in the saem way as, say, a log, pi, Graham's number etc.

 

[ParadigmShifter]

Nope that's wrong.

The numbers you mention (apart from Graham's number, which is just enormous but is an integer, comes from Ramsay theory I believe), are all irrational. pi is trasncendental also (logarithms normally are too).

0.1 is EXACTLY EQUAL to 1/3 IN BASE 3.

It's just recurring in base 10. It is still a rational number.

 

[mdwh]

It's interesting how this question seems to come up so often - something that some people love to argue that it's not true, despite not having a clue about the maths. I see it as the mathematical equivalent of Evolution

I think a lot of people get tripped up by the notation of "...", thinking it just means "a lot of". The problem is that ellipses arent really well defined. It's better to use notation such as the bar notation, or brackets (especially as ellipses can be ambiguous if there's more than one recurring digit - e.g., 0.45454... - do I mean to repeat the (45), or just the (4)?) I previous arguments, there have been times when I start by giving the rigorous definition of a decimal expansion (i.e., as an infinite sum), to which they respond "Oh, well now that's cheating if you choose define it like that"!

Eh, 1/3 is just a way of more easily representing a number that is unbeleivably hard, if not impossible to represent, in the saem way as, say, a log, pi, Graham's number etc.

I'm not sure that using the recurring notation is "unbelievably hard", and it clearly isn't impossible. The term you are looking for is repeating or recurring decimal ( http://en.wikipedia.org/wiki/Repeating_decimal ).

Also this is just a property of the decimal expansion notation, and is specific to the base you are using. As ParadigmShifter pointed out, in base 3 it's 0.1.

Do you think that 0.5 is also impossible to represent? Because in base 3, that's a recurring decimal ( 0.(1) ).

 

[mdwh]

but i still don't get that hehehhehe one and PS's explanation.

my anser to PS is .111111111...... makes sense, eh!

So 1 - 0.(9) = 0.(1).

Therefore 0.(9) = 1 - 0.(1)

But surely 0.(1) > 0.1? In which case, 0.(9) < 0.9, which doesn't make sense.

(0.(1) is actually equally to 1/9, which according to you, would make 0.(9) = 8/9. I'm not sure how that makes any sense...)

In line 5 of hehehe's 9 x 0.9999.... = 9 thats what i want to know why thats considered correct! cuz then it would mean o.999999..=1! but doesn't prove anything cuz i already know that i want to know y!

Well, you're right that that proof isn't really rigorous. See wer's post http://forums.civfanatics.com/showpost.php?p=8995564&postcount=8 for a better proof.

0.(9) is defined as an infinite sum; and the definition of infinite sum itself has a strict mathematical definition, called the limit of the sequences of the finite sums.

Basically, it's the equivalent of the classic 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... . If you look at each finite sum in turn, you have 1, 1.5, 1.75, 1.875, 1.9375, ... . It can be proven that the limit of these is 2.

 

[GoodGame]

So the convention is that 0.XXX is a repeating number to infinity,
while 0.XXX to the yth decimal position is just goofing off?

 

[Leifmk]

The equivalence follows trivially from a proper definition of what the real numbers are, and what the notation "0.999..." is supposed to mean. It is a lack of understanding of these points that leads some people to incorrectly reject the equivalence, and it is said lack of understanding that needs to be adressed to correct the issue.

 

[madviking]

Sigma(i=1, infinity) of 9/(10^i) = 1

e.g.
9/10 + 9/100 + 9/1000 + ...

 

[Black Bart]

Yes.

.99999999999... = x
10x = 9.999999999...
9x = 9
x = 1