⅓=0.333... ⅔=0.666... 3/3=1=0.999...
0.999...ad infinitum does NOT Equal 1!
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Aug 28, 2023
0
#28
What is the difference between 1/3 and 3 times 0.3333?
Experts reveal what to do about it. Clearly, 1/3 lies between 0.3333 and 0.3334, since 3 times 1/3 is 1, while 3 times 0.3333 is 0.9999 and 3 times 0.3334 is 1.0002 Clearly, 1/3 lies between 0.33333 and 0.33334, since … (well, you can write that down yourself, isn’t it?) Clearly, 1/3 lies between 0.333333 and 0.333334.
Experts reveal what to do about it. Clearly, 1/3 lies between 0.3333 and 0.3334, since 3 times 1/3 is 1, while 3 times 0.3333 is 0.9999 and 3 times 0.3334 is 1.0002 Clearly, 1/3 lies between 0.33333 and 0.33334, since … (well, you can write that down yourself, isn’t it?) Clearly, 1/3 lies between 0.333333 and 0.333334.
If it is not equal to one, there is an infinite difference.
Which isn't.
Aug 28, 2023
0
#32
3/3=1 that's a fact, another fact is that 1/3=0.3 recurring. Therefore 3/3=0.9recurring=1
Aug 28, 2023
0
#35
.3333...4 is impossible as it would require a number after an infinite number of 3s
One_Zeroth wrote:
Here is the proof.
Let n be any whole number, then
((10)^n)-1))/(10^n)
Never converts to 1, or any other power of 10 !
Example:
9/10, 99/100...((10^n)-1)/(10^n)
There is a flaw in that proof. According to that expression , to get an infinitely repeating 9's , n cannot be a whole number . That assumption is wrong. So the only way of getting .999 repeating is if value of 'n' is infinity.
Applying that in the above expression, the expression ((10^n)-1)/10^n basically becomes infinity/infinity which is 1.
There is a method to converting reccuring numbers into fractions so just apply that here and that's the proof :
let .999. . . = x (equation 1)
(equation 1)*10 implies ; 9.999. . . = 10x (equation 2)
equation 2 - equation 1 implies ;
9.000. . . = 9x
x = 9.000. . . /9 = 1.000. . .
so .999. . . = 1
there is another proof :
In a similar way , .333. . . = 1/3
let .333. . . = x
multiply with 10 ;
3.333 = 10x
subtract eq 2 and 1 ;
3 = 9x , x = 3/9= 1/3
so we know that .999. . . = .333. . . *3
we just now proved that .333. . . = 1/3
therefore .999. . .= 1/3 * 3 = 1
so there are plenty of ways to prove this , even your expression is proof of it . you came to the wrong conclusion due to the wrong assumption that 'n' is a whole number.
this is the same with the logic of saying 2=4.
Reverse translation error(!)