View Full Version : "0.9999..." = 1
Uthman
06132008, 10:02 PM
:sl:
I hope this hasn't been posted before. I just wondered how many of you know that the decimal number 0.9999...(recurring) actually equals 1.
Allow me to prove it for you:
Let
x = 0.9999...
Multiply both sides by ten:
10x = 9.9999...
Subtract x from both sides:
10x  x = 9.9999...  0.9999...
9x = 9.0000...
Divide by nine:
x = 1.0000...
Anybody want to challenge me on this? Or perhaps somebody would like to post a different proof or counterproof? :D
:w:
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truemuslim
06132008, 10:06 PM
I think my math teacher told me bout this...
i remember it from LI tho....straaange...
Oooh yehH! lool i was on LI while he was talking! :X hehe
Reply
aysenil
06132008, 10:08 PM
:thumbs_upwoow gr8 mashallah.... who can challenge u :D
and i dont no any other proof :?
Reply
Mikayeel
06132008, 10:14 PM
:sl:
Hmm, seems interesting!, Barakalahu feek bro:D
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Tornado
06132008, 10:19 PM
Originally Posted by
Osman
:sl:
I hope this hasn't been posted before. I just wondered how many of you know of the fact that the decimal number 0.9999...(recurring) actually equals 1.
Allow me to prove it for you:
Let
x = 0.9999...
Multiply both sides by ten:
10x = 9.9999...
Subtract x from both sides:
10x  x = 9.9999...  0.9999...
9x = 9.0000...
Divide by nine:
x = 1.0000...
Anybody want to challenge me on this? Or perhaps somebody would like to post a different proof or counterproof? :D
:w:
Let
x = 0.9999...
Multiply both sides by ten:
10x = 9.9999...
Subtract x from both sides:
10x  x = 9.9999...  0.9999... < You subtracted x on one side, and .999 on the other so it doesn't make sense.
Since 0.3333 + 0.3333 ) 0.3333.. = 0.999..
and since 0.333.. is 1/3, then 1/3 + 1/3 + 1/3 = 1
Reply
Uthman
06132008, 10:34 PM
Originally Posted by
Tornado
Let
x = 0.9999...
Multiply both sides by ten:
10x = 9.9999...
Subtract x from both sides:
10x  x = 9.9999...  0.9999... < You subtracted x on one side, and .999 on the other so it doesn't make sense.
I don't see why it doesn't make sense, since x = .999...
The same thing is still being subtracted from both sides.
Originally Posted by
Tornado
Since 0.3333 + 0.3333 ) 0.3333.. = 0.999..
and since 0.333.. is 1/3, then 1/3 + 1/3 + 1/3 = 1
Good one! Of course, that's provided that 1/3 does actually equal 0.333....
Regards
Reply
Tornado
06132008, 10:35 PM
Originally Posted by
Osman
I don't see why it doesn't make sense, since x = .999...
The same thing is still being subtracted from both sides.
Good one! Of course, that's provided that 1/3 does actually equal 0.333....
Regards
:[ Yes I was totally wrong. 1/3 does equal 0.33333333333333 : ).
Reply
Uthman
06132008, 10:38 PM
Originally Posted by
Tornado
1/3 does equal 0.33333333333333 : ).
Why? Because your Maths teacher told you? :D
Reply
Tornado
06132008, 10:52 PM
Originally Posted by
Osman
Why? Because your Maths teacher told you? :D
No, my trusty calculator :)
Reply
Uthman
06142008, 09:19 AM
Originally Posted by
Tornado
No, my trusty calculator :)
Oh. :[
Sure, but aren't calculators programmed by us notso trusty humans anyway? :?
Reply
Idris
06142008, 09:53 AM
New to me.
Reply
'Abd alBaari
06142008, 09:56 AM
Assalamu Alaykum
Me too^
How about 1  0.999999 = 0.000001
So 1 is not equal to 0.99999 :D
WaAlaykumus Salaam Warahmatullah!
Reply
Ayoub
06142008, 10:04 AM
Originally Posted by
Osman
:sl:
I hope this hasn't been posted before. I just wondered how many of you know of the fact that the decimal number 0.9999...(recurring) actually equals 1.
Allow me to prove it for you:
Let
x = 0.9999...
Multiply both sides by ten:
10x = 9.9999...
Subtract x from both sides:
10x  x = 9.9999...  0.9999...
9x = 9.0000...
Divide by nine:
x = 1.0000...
Anybody want to challenge me on this? Or perhaps somebody would like to post a different proof or counterproof? :D
:w:
If x = 9, 9x does not = 9. If x = 9, 9x = 81. :)
Reply
Whatsthepoint
06142008, 10:42 AM
Originally Posted by
Ayoub
If x = 9, 9x does not = 9. If x = 9, 9x = 81. :)
Wha..?
:[
Reply
Abdul Fattah
06142008, 10:47 AM
The problem is the surrealism of infinity. In the real world where infinity doesn't exist, then in step 3 where you subtract x, there you'll see that the 9.999... always has one decimal less behind the point than 0.9999...
That way the subtractions can never become 9. So the calculations themself are fine, it's the mathematical axiom of infinity that is dodgy.
The same happens for 1/3 = 0.333... that only works with an infinite number of factors. In reality it's impossible to calculate with that, so we calculate in approximations (so does your calculator).
Reply
Uthman
06142008, 10:57 AM
:sl:
Originally Posted by
Abdul Baari
How about 1  0.999999 = 0.000001
So 1 is not equal to 0.99999 :D
:D
Yes, but remember that the 0.999... is a recurring decimal with an infinite number of nines. For this reason, 1  0.9999.... would result in 0.0000000......
There would be an infinite number of zero's in the answer so there would be no 1 at the end because there is no end!
:w:
Reply
Uthman
06142008, 10:59 AM
:sl:
Originally Posted by
Ayoub
If x = 9, 9x does not = 9. If x = 9, 9x = 81. :)
That certainly makes sense, but the argument here is that 0.9999... = 1, where x = 0.99999...
Nobody claimed that x = 9! :)
:w:
Reply
Abdul Fattah
06142008, 11:54 AM
To clarify the previous post, consider the same calculation, only this time with finite numbers instead of infinite numbers:
x = 0.9999
Multiply both sides by ten:
10x = 9.999
Subtract x from both sides:
10x  x = 9.999  0.9999
9x = 8.9991
Divide by nine:
x = 0.9999
Reply
Muhammad
06142008, 12:20 PM
:sl:
Subtract x from both sides:
10x  x = 9.9999...  0.9999...
9x = 9.0000...
How come
10x  x becomes
9x, when
x=0.9999...? :exhausted
Reply
tetsujin
06142008, 01:19 PM
Originally Posted by
Osman
:sl:
I hope this hasn't been posted before. I just wondered how many of you know of the fact that the decimal number 0.9999...(recurring) actually equals 1.
Allow me to prove it for you:
Let
x = 0.9999...
Multiply both sides by ten:
10x = 9.9999...
Subtract x from both sides:
10x  x = 9.9999...  0.9999...
9x = 9.0000...
Divide by nine:
x = 1.0000...
Anybody want to challenge me on this? Or perhaps somebody would like to post a different proof or counterproof? :D
:w:
Who ever showed you that needs to get a good smack in the head.
I suggest you go to a good math teacher and allow them to properly educate you on the rules of mathematics.
Your "equation" boils down to ((x * 10)  x)/9 = something
You've allowed x to be an irrational number, which is fine, but that's where you are having difficulties. Logically speaking, ((x * 10)  x)/9 should equal x again.
Try .888888888 and see what you get with your logic.
(0.888888888..... * 10  0.888888888..... )/8 = ?
(8.888888888.....  0.888888888..... )/8 = What do you think it should be?
Please don't make a mockery of mathematics.
All the best wishes,
Faysal
Reply
crayon
06142008, 01:40 PM
Originally Posted by
tetsujin
Please don't make a mockery of mathematics.
Please don't be such a jerk.
All the best wishes,
Faysal
...
Reply
Uthman
06142008, 01:49 PM
x=0.8888....
10x = 8.888888
9x = 8
x = 8/9 = 0.8888...
:)
Logically speaking, ((x * 10)  x)/9 should equal x again.
And it does. In the proof that I showed, x = 0.9999... and x = 1. The argument is that both are therefore the same.
Please don't make a mockery of mathematics.
I assure you that I am doing nothing of the sort. Mathematics is based upon proof, not intuition. What I posted was one such proof, there are many others. If you disagree, then please argue with a counterproof. This particular case is believed by many professional mathematicians themselves, who I would assume know more than you and I about the 'rules of Mathematics'.
Finally, I would request that we please stay respectful. This is posted in the 'puzzles and humour' section and it isn't to be taken too seriously. :)
Regards
Reply
tetsujin
06142008, 01:52 PM
Originally Posted by
Osman
Finally, I would request that we please stay respectful. This is posted in the 'puzzles and humour' section and it isn't to be taken too seriously. :)
Regards
I apologize, I love math too much for my own good.
All the best wishes,
Faysal
Reply
Uthman
06142008, 01:54 PM
Haha, no worries. I love maths too. :)
:[
Reply
Abdul Fattah
06142008, 04:22 PM
Hi Tetsujin
Originally Posted by
tetsujin
Please don't make a mockery of mathematics.
All the best wishes,
Faysal
I don't think this is making a mockery out of it. Some advanced math classes even study these false proofs. If anything, they help people have a better understanding of math, or in this case a better understanding of the problems with working with infinity.
Edit: I see I was to late with my reply, just ignore then ^_^
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Uthman
06152008, 04:46 PM
I had a long debate about this with my brother but he has inadvertently provided me with yet another proof which strengthens my case.
He asked me to calculate 0.99999.... + 0.11111....
I worked it out and the answer comes to 1.11111....
That is the same as 1 + 0.1111....
With me? :D Anybody got a problem with that?
Reply
'Abd alBaari
06152008, 04:50 PM
:sl:
He asked me to calculate 0.99999.... + 0.11111....
I worked it out and the answer comes to 1.11111....
That is the same as 1 + 0.1111....
Wow i just tried that, and it worked..You learn something new everyday..:p
JazakAllah Khayr akhee. :)
:w:
Reply
Abdul Fattah
06152008, 04:57 PM
Again the same problem as before.
With a finite number of factors we would get:
0.99999 + 0.11111 = 1.11110
1.11110 = 1 + 0.11110
The problem lies with this axiom:
∞+1=∞
So that means that 1.1111...  0.0000...1 = 1.1111...
The numbers are only equal because the difference is ignored next to infinity. Remember: infinity is not a number, its' a concept.
Reply
Uthman
06152008, 05:06 PM
I agree with you that inifinity is what poses the problem here. But Akhee, you are speaking of infinity as though it is a number when it isn't. It is a concept, not a quantity!
EDIT: Ok, never mind.
Reply
Abdul Fattah
06152008, 05:30 PM
Originally Posted by
Osman
I agree with you that inifinity is what poses the problem here. But Akhee, you are speaking of infinity as though it is a number when it isn't. It is a concept, not a quantity!
EDIT: Ok, never mind.
It seems I edited my post just a few minutes before you were able to make yours :p
Anyway, on another note, irrational numbers really are
irrational in the sense that they aren't necessarily real quantities either! In math we can have infinite terms in an irrational number. Like we can divide the number 1 by the the number two an infinite number of times.
1/2=0.5
0.5/2=0.25
0.25/2=0.125
...
But is that realistic? Say we have a lump of clay, can we divide that into two smaller lumps an infinite number of times? eventually you'll be dividing molecules, atoms, quarks, strings? Is there any guarantee you can repeat this division infinitely? So you could say numbers which are infinitely small or have infinitely small parts (like 1.11111... ) are also concepts rather then quantities just as we both pointed out is the case with infinity itself. This again shows the problem with these proofs, the calculations used treat the irrational numbers like rational numbers.
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Uthman
06152008, 05:37 PM
Originally Posted by
Abdul Fattah
the calculations used treat the irrational numbers like rational numbers.
Very good point! I agree with your post. :) Also on a more general level, more often than not, theory differs from reality anyway.
I love to tease people with this. It's so funny seeing my brother know full well that 0.999... and 1 are different numbers but not being able to prove it. :D
I get a sick sense of pleasure out of it. Maybe I need to get my head checked. +o(
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Uthman
06182008, 12:04 PM
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