Why 0.999… = 1| क्यों calculus की ये trick छिपाते है mathematicians

There are many different proofs of the fact that "0.9999…" does indeed equal 1. So why does this question keep coming up?

It is a bit odd, really, that we have trouble with this equality. I mean, we don't generally argue with 0.3333… being equal to 1/3.
​But then, one-third is a fraction, and we're used to fractions being equal to non-terminating (that is, infinite) decimal expansions. So maybe it's easier for us to accept the equality when we're dealing with thirds? (I'm guessing, but psychology may play some part in this.)
Maybe it's just that it feels wrong that something as nice and neat and well-behaved as the number 1, "the unit", the basis of so much of our arithmetical and mathematical understanding, could also be written in such a messy form as 0.9999…. Whatever the reason, many students (me included) have, at one time or another, felt uncomfortable with this equality.
The infinite repeating decimal 0.999… is equal to the whole number 1. This is very different from the value of, say, 0.999, which is equal to 999/1000
​(which is close to 1, but not equal to 1). The ellipsis (ell-IPP-siss) — that is, the "dot, dot, dot" — after the last 9 in 0.999… makes all the difference.
In other words, the dot, dot, dot says that 0.9999… never ends. There will always be another 9 to tack onto the end of 0.9999…, no matter how many 9s you already have. So don't object to 0.9999… = 1 on the basis of "however far you go out, you still won't be equal to 1", because there is no "however far" to go out to; you can always go further. There is no end to an infinite decimal expansion; there will always be more 9s.

"But", some say, "there will always be a difference between 0.9999… and 1." Well, sort of. Yes, at any given stop, at any given stage of the expansion, for any given finite number of 9s, there will be a difference between 0.999…9 and 1. That is, if you do the subtraction, 1 − 0.999…9 will not equal zero.

But the point of the "dot, dot, dot" is that there is no end to the 9s; 0.9999… is inifinte. There is no "last" digit. So the "there's always a difference" argument betrays a lack of understanding of the infinite. (That's not a criticism, per se; infinity is a messy topic, and you haven't yet been given the tools with which to try to understand it.)

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Why 0.999… = 1| क्यों calculus की ये trick छिपाते है mathematicians

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