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is 0.999… equal to or less then 1
Equal in terms of the limit definition. (look up the phrase “limit in calculus”)
Not equal, in terms of there will always be a small difference between 1 and 0.9999…
mathematically it is less than 1
but it’s so close that you’d probably never know the difference anyway.
Paradox: 1/3 = 0.3333333333….
0.3333333333…. x 3 = 0.9999999999….
yet 1/3 x 3 = 1
0.999999… limits to 1 but has a place-holding value of less than 1.
.999 is .001 less than one. .999… until infinity is equal to 1.
It is equalt to one. It has been proven in many ways it is not debated.
Here is a simple easy to follow mathematical proof:
1/9 = 0.111…
( 9 x 1/9) = (9 x 0.111…)
1 = 0.999…
None of the above statments are up for debate.
The problem people have with grasping the above is that infinity is hard for the mind to grasp. Many suppose the difference between 0.999… and 1.0 is a difference of
0.000…(infinite 0’s)…1. In other words it is viewed as a really long chain of zeros with a tiny 1 way out there. However in reality this is not the case. there is no end to the 0’s, as soon as you imagine that 1 at the “end” of your long string of 0’s, get rid of it and replace it with yet another 0. Do this forever until you are convinced that in fact there is no 1 out there and the difference between 0.999… and 1 is exactly 0. in aother words they are equal.
RT- Wow. I never thought about that.
They are equal. Consider this:
hence,
1=0.999…
The reason this is difficult to grasp is because people have a hard time conceptualizing infinity. On some level, everyone imagines that at the end of the .999… on the outskirts of infinity, there is one final 9; when in reality, that final 9 does not exist.
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