We’re back with another math fact of the week! Last week, we talked about the surprising fact that .9999… = 1 — thanks again to the excellent Polymathematics blog for that particular fact — and this week’s fact examines a similar property of the Real Numbers.
There is no smallest real number greater than zero.
Unfortunately, while I think this is a really interesting property of the real numbers, it’s not something that requires an extended discussion to demonstrate. In fact, this somewhat surprising property of the real numbers stems directly from their definition. The real numbers are dense, meaning that for any two real numbers x, y with x < y, we can always find at least one real number z such that x < z < y. Hence, there is no smallest real number greater than zero; we can always find another number that is both smaller than our “smallest number” yet greater than zero.
See you next week for another math fact! Until the, don’t forget to e-mail your math questions to jason AT bostongeek DOT com. 
Yet another fine example of infinity being a terrible, terrible concept to deal with.
I wonder, then if something is ‘infinitely small’, does it exist at all?
A black hole is, in theory, a mathematical singularity after all.
Also, this brings me back to last week’s point about 0.999…=1. If so, and we take this new trivia point to mean that 1-0.999…!=0, than I am thoroughly confused.
How in the hell do you combine “0.9…=1″ and “there is no smallest real number greater than zero” and come up with “1-0.9…!=0″?
it really isn’t that hard. Saying there is no smallest real number greater than zero implies that there is no smallest rational or irrational number in the scale down to zero. No matter how small you go, you can go smaller. One assumes you can do this infinitely.
Maybe I am expressing it wrong, but doesn’t that just about equate to 1-0.999… != 0?
Yes, but, 0.999… has an infinite amount of 9s trailing it, not some finite amount. The trick about the density of the real numbers implies that if you stopped after any finite amount of 9s, then yes, you would have a number smaller than one, and there would be another number greater than your number but smaller than one.
I think the easiest way to understand both of the problems is to consider the following question. If 0.999… (where the 9’s are an infinite decimal expansion) is less than one, then what number is greater than 0.999… but less than one? The density of the real numbers implies that we must be able to find such a number.
So, what you are saying is that 1-0.999…999 != 0, but 1-0.999…=0?
So, as long as you allow the decimals to run infinitely… heh. I hate infinity.
No, what I’m saying is that 0.999… - 1 = 1 - 0.999… = 0. How could 0.999… - 1 not equal 1 - 0.999…? Are we doing some kind of strange, non-commutative addition here?