Archive for the ‘repeating decimal’ Category

Is there some number that’s larger than .999… but smaller than 1.0?

October 4, 2007

We can show that the repeating decimal .999… is equal to 1.0 with only a couple of legal moves:

If x = .999…, then 10x = 9.999…

Now subtract x from 10x:

9.999999….

– 0.999999…

You get 9x = 9.0 and thus x = 1

So the repeating decimalĀ  .999… equals 1.0!

Is there some number that’s larger thanĀ  .999… but smaller than 1.0?

Such a number would involve an infinitesimal, meaning a literally infinitely small mathematical entity.

— From Everything and More by David Foster Wallace