Thread: 0.9999..=1 proof

Hire some DIFIRENT numbers:
1
0.9...9=x
0.9...8
0.9...7
.......
0.9...1
0.9...0
0.9..-1
.........
0.0...1=y
0
x=0.9.. / lets *10
0.9..+0.9..+0.9.. ten times =(1-y)+(1-y).. ten times SO
10x= 10-10y / -0.9...
10x-x=(10-10y)-(1-y)
9x=9-9y
9x=9-9y
x=(9-9y)/9 and this is NOT 9 !
x=9(1-y)/9
x=0.9....

This last one clearly shows that0.9...9 is not some "magic" number its number like anyother. This 0.00...1 is more "special" but is stillordinary number.

Originally Posted by rt20

Hire some DIFIRENT numbers:
1
0.9...9=x
0.9...8
0.9...7
.......
0.9...1
0.9...0
0.9..-1
.........
0.0...1=y

I have no idea what any of these mean. I know that "0.999..." means the decimal number whose only digits are "9". But I don't know what you mean by "0...9". How many digits does that "..." skip over? It can't be an infinite number since there is no "last" digit in that case.

0
x=0.9.. / lets *10
0.9..+0.9..+0.9.. ten times =(1-y)+(1-y).. ten times SO

"ten times"? Do you mean 0.9+ 0.9+ 0.9+ 0.9+ 0.9+ 0.9+ 0.9+ 0.9+ 0.9= 10(0.9)= 9? I don't know what you mean by 1- y because I don't know what you mean by "0.0...1" which is apparently how you have defined y
And so nothing beyond here makes sense.

10x= 10-10y / -0.9...
10x-x=(10-10y)-(1-y)
9x=9-9y
9x=9-9y
x=(9-9y)/9 and this is NOT 9 !
x=9(1-y)/9
x=0.9....

This last one clearly shows that0.9...9 is not some "magic" number its number like anyother. This 0.00...1 is more "special" but is stillordinary number.

Should I conclude that you do not understand how "decimal notation" works?

If you really want a proof that 0.999...= 1, do this: 0.999...= 0.9+ 0.09+ 0.009+ ...= 0.9+ 0.9(.1)+ 0.9(.01)+ 0.9(.001)+ ... In other words, it is a geometric series, of the form with a= 0.9 and r= 0.1. It is easy to show that the sum of such a geometric series is which, here, is .

"It can't be an infinite number since there is no "last" digit in that case."
Well there is. I just write it. In mathematics you can have any number you can think of! Why not ?

Saying that that there is not "last digit" is like saying 0=1 witch is not true.
0= 0 + 0 + 0 + ...
= (1 - 1) + (1 - 1) + (1 - 1) + ...
= 1 - 1 + 1 - 1 + 1 - 1 + ...
= 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ...
= 1 + 0 + 0 + 0 + ...
= 1


I am saying that 0.0..1 is the closest to zero number. By definition. I don't have to prove it - i just to define it : )
0.9.. is the closest to 1, with make 1-0.0...1 =0.9.... Its very simple.