0.999 repeating forever is exactly equal to 1

This is not an approximation. Multiple rigorous proofs show the infinite decimal 0.999... and the number 1 are the same value, just written differently. An infinite decimal is defined as the value its ever-growing string of digits closes in on. The gap between 0.999... and 1 shrinks below any number you could name, and since no gap survives, the two expressions describe the same point. The simple demonstrations hold up: a third is 0.333..., and three thirds is both 1 and 0.999.... Resistance usually comes from picturing the decimal as a process that never quite arrives, when it is really a fixed value defined by where that process leads. Nothing is being rounded. The deeper lesson is that decimal notation is not unique — every terminating decimal has a second form ending in repeating nines, so 2.5 can equally be written 2.4999.... The number is one thing; the way we write it is another.

0.999 repeating forever is exactly equal to 1

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