# Q & A: 0.9999999....

Q:
why does .9- (repeating) equal 1? my math teacher told us this and didn’t explain it
- James
U.S
A:
You agree that if 0.99999999...=x, then x can't be bigger than 1.

Can it be smaller than 1? Imagine it is. Then 1-x=y, where y is some positive number. Each time you add a 9 to 0.999... you get 90% closer to 1. So after a while you are sure to be closer than y. So we can't have any y>0. So if 0.9999...=x, we know that x isn't bigger than 1 and we know that x isn't less than 1. So x is 1.

Of course, we haven't shown that 0.9999... has to equal ANY number, just that if it does, that number must be 1. For the proof that 0.999... has a well-defined limit (to which we say that it's equal) you can look at any introduction to infinite limits or beginning calculus book.

Mike W.

Here's another way of looking at this:

0.9999999... is just 9 times 0.11111111.... But 0.111111... is just 1/9 (work out the long division until you satisfy yourself that this keeps going on the same forever). 9/9 is equal to 1.

Tom

(published on 10/22/2007)

## Follow-Up #1: grainy universe?

Q:
Refering to your recent answer for "0.9999999....", I suggest that this is in essence just a limitation of Mathematical decimal notation. Maths is just a language after all. That leads me on to another recent question/answer "grainy universe?" If the physics of the Universe is continuous how can we explain this in terms of movement and space/time? - Let us say I have a piece of paper and I draw a straight line on it with a pencil. If space/time are continuous, then theoretically we could divide that line I drew into an infinite amount of smaller pieces. How then could I draw it in a finite amount of time with a finite amount of graphite on my pencil? If the only weapons we have to attack this are "converging series" or "calculus" then I am exceedingly confident the Universe is "grainy" or our language of Mathematics is once again lacking.
- Drake
A:
 For sure the graphite from the pencil is grainy. You deposit a finite number of grains. The question is whether the space in which you deposit them is really grainy. I don't see any way an informal verbal argument could settle that question. Mike W. You may want to look up "Zeno's paradox" in a calculus textbook. Zeno asked the question that if a runner runs a finite distance in a finite time, you can think of him running a series of small distances, all added together. If you slice up his racetrack into infinitely many small pieces (and the example was one of cutting each piece in half -- no matter how close the runner got to the end, he would have to travel halfway there and then the other half, which could itself be broken into halves and so forth). Zeno reasoned, erroneously, that to travel across an infinite set of these intervals "must" take an infinite amount of time. For the last few hundred years, we realize that the sum of Zeno's geometric series converges to a finite number which is easily calculated to be the length of the original racetrack. Relativitity and quantum mechanics when interpreted together (and we are not good at doing this) seme to indicate that space and time may not be continuous and flat all the way down to very very very tiny distance scales. On very short distance and time scales, quantum fluctuations can bend space and time around. Ordinary calculations of quantum corrections to observable quantities involve adding up all kinds of processes involving virtual particle exchange. If you add up all of these without upper bound on their momentum (that is, without a lower bound on their wavelength), you sometimes get absurd, infinite answers. So people stop their integrals at some small distance scale and say that the theory breaks down, hoping that we will discover eventually what kinds of interactions take place between particles on these small distance scales. Tom

(published on 10/22/2007)