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0.9~ = 1 ?


geko

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Hi everyone. just want to say first that this site's great. Been looking for a forum like this and reckon i found a good one

 

Anyways look if you would - much stuck my head is on this:

x = 0.9999...

10x = 9.9999...

10x - x = 9.9999... - 0.9999...

9x = 9

x = 1.

 

i apologise if this explains it simply but i just cant get my head around the idea that .9~ is 1. It's a whole .1~ off for a start ('~' = recurring - in case that's not an accepted symbol).

 

Surely you have to 'round-up' .9~ for it to be 1? Is this just playing with numbers? I see lines 3 and 4 as this for some reason but havent got the vocabulary to exaplain i feel lol

 

Of course this is a lot like 1 minus .3~ (1/3) = .6~ (2/3), but then .3~ + .6~ = .9~.......... has the .1~ just been lost because you've messed with the number?

 

My real question is does 0.9~ = 1? Also, if it does, would someone explain why it does in another way but the above if it can? Much thanks!

 

p.s. i found this about it as well but it doesnt exaplin much to me (im just nosy about this stuff ). by the way i dont think formulas and equations etc are copyrighted so im gonna reprint as i found it:

 

lim(m --> ?) sum(n = 1)^m (9)/(10^n) = 1 0.9999... = 1

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Hi geko, welcome to our forums! And thanks for your comments.

 

I'm not a math wiz but I think this is just a simple trick which shows some problems when working with infinite numbers.

 

The trick lies in the result of the multiplication not being shown correctly.

 

It's better to simplify in order to understand it:

 

What you get when you multiply, say, 0.9999999999 (10 decimals) with 10, is 9.999999999 (nine decimals).

 

Then subtract:

 

9.9999999999 - 0.999999999 = 8.9999999991

 

Multiplying by ten simply moves the comma one place to the right, so subtracting the product from the original number will automatically give you a close approximation of 9 (but never 9).

 

Line 3 and 4 are only valid when you round 8.9999999999999~1 up to 9 (so your assumption is correct).

 

Someone correct me if I'm wrong...

 

Tormod

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Originally posted by: geko

i apologise if this explains it simply but i just cant get my head around the idea that .9~ is 1. It's a whole .1~ off for a start ('~' = recurring - in case that's not an accepted symbol).

 

One error is here.

It's a whole .1~ off for a start

 

It is NOT .1~ off. It is .~1 off. If you add .9~ +.1~= 1.1~, NOT 1.0

 

I think Tormod took care of some of the rest.

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Originally posted by: Freethinker

Originally posted by: geko

 

It's a whole .1~ off for a start

 

 

 

It is NOT .1~ off. It is .~1 off. If you add .9~ +.1~= 1.1~, NOT 1.0

 

 

Yeah, i did think of this last night whilst lying awake but what an oversight! I was always like that with math. too quick to guess and not thorough enough in the sorting. Thanks for the replies but ........... is 0.~1 valid? If so, .9~ does not equal 1 - no matter how close it gets? Freethinker? plz......

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ok, here, dont know if its a proof.

 

.999~ = lim n--> infinity (n/(n+1))

 

when n is infinity, n+1 is the same as n, thus its equal to one.

 

 

this is same idea to 1/0=infinity........(i guess...)

 

remember, .999~ has infinity number of 9. Multiplying by ten or even 1 billion still makes it inifinity. no matter how great you increase the number, once it is 10^n, there will be still infinity number of 9 in it. (except when you multipying it by infinity.....lol.....it will actually becomes infinity...which cannot be express into regular numbers....really confusing)

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hmmm, some ppl might think that .999~ is .00~1 off from 1

 

but actually, .000~1 is equal to zero.

remember, .00~1 doesnt mean .0000000000000000000000001 or anything like that. it has INFINITY zeros in it. it is always a bit less than .0000000000000000000000000000000000000000001.....

 

.00~1=1/infinity=0

 

in the graph y=1/x

it can never reach the x-axis....

but when x is infinity, it reaches. but we cannot see this in the graph. assume that you reach the point, but there will be a point beyond that... which means that this point is not infinity....and it will get closer and closer to 0....when it is infinity, it reaches, but can be never seen...

(hope you understand this) ; )

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  • 3 weeks later...

I remember posing this to my Yr 12 Maths teacher some year ago, and he couldn't explain it.

 

lol

 

So I asked my friends father, who had a Degree in Applied Mathematics, Majoring in Statistics about it.

 

He proceded to go through a really ugly derivation that, at the time, I had absolutely no idea about.

 

From a vague recolection, he used limits and some sort of integral to prove that 0.9 recurring DOESNT equal one.

 

So I would assume that Tim_Lou's explanation, involving the graph is close enough.

 

 

EDIT

 

Gee my typing leaves a lot to be desired ....

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