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why does 0.999 repeating equal 1
People say 0.999 repeating is a hair under one, forever creeping up. That's a myth: it is exactly one, the same number written two ways.
Zoom in on the stretch from 0.9 to 1. The mark at 0.99 is one hundredth short of one, and every new nine closes the gap tenfold.
Zoom again: 0.999 is a thousandth short. With nines forever, any gap would have to be smaller than one over ten to the n, for every n.
Now the algebra. Call the number x. Ten x is 9.999 repeating. Subtract x, and the endless nines cancel exactly: nine x equals nine, so x is one.
Or use thirds. Long division gives one third as 0.333 repeating. Triple both sides: three thirds is one whole, and triple 0.333 repeating is 0.999 repeating.
One more way. Suppose they differed by some gap g above zero. But the gap is smaller than a tenth, a hundredth, a thousandth, for every n. Only zero fits.
And add digit by digit: nine tenths, plus nine hundredths, and so on. Each partial sum is one minus ten to the minus n, so the total is one.
The dots do not mean almost: they mean nines forever, and forever leaves no gap, so 0.999 repeating and one are the same number.