Yesterday at work we had a pretty heated debate over whether .999~ (repeating) is equal to 1. I can’t believe that in all of my years of math, I had never heard of this argument before. Basically, there are a couple “proofs” for this, which seem to make sense to me and verify the legitimacy of this statement. Somehow though, there is just something about it that doesn’t seem right. Here’s one proof.
.999~ = 1
So let’s set x = .999~:
x = .999~
Multiply both sides by 10:
10x = 9.999~
Subtract x from both sides:
10x – x = 9.999~ – x
9x = 9.999~ – x
Substitute .999~ in for x on the right side, which was our original declaration:
9x = 9.999~ – .999~
9x = 9
x = 1
But we said initially that x was equal to .999~!
I know that I have a lot of good math people that read my blog, so I’d be interested in hearing your opinion on this. This whole discussion started after someone came across this blog entry on the topic. There are tons of comments there arguing both sides. It just seems to me that .999~ will never quite get to 1, but these arguments for it really do make sense to me. I just still can’t believe that yesterday was the first time I ever heard this.