Polymathematics has posted another excellent essay on the subject of whether .9999…repeating equals one. This time he is responding, very effectively, to various counter arguments raised by commenters.
One small comment of my own, though: The name of the blog is “EvolutionBlog.” One word.
Reading the various comments left to the post reveals two kinds of skeptics. Some are people who are perfectly willing to accept that .9999…=1, but find the logic justifying this conclusion to be hard to follow. No shame in that. The idea of the limit of an infinite series is not a simple one, which is why most students find calculus to be rather difficult the first time they see it. As with most things, however, a bit of practice makes the strangeness go away.
Other skeptics, however, are outright cranks. Those are the one who say things like:
You must be a public school teacher, and I fear for your students. You don’t know enough math to teach it. Stop filling their heads with nonsense.
.999… is clearly less than 1, but mathematics isn’t advanced enough to handle infinity, so you can’t prove it. My intuition isn’t flawed, math is.
Cripes. I frequently get e-mail from creationists written at a comparable level. Do not engage. Do not engage.
Many of the commenters offered counterarguments based on misunderstandings of the role played by notation in mathematics. Polymathematics writes:
This is a common mistake among my students. You’re mistaking notation for mathematics. Notation is not mathematics. Mathematics is the study of ideas about patterns and numbers, and we have to invent notations to communicate those ideas. Just because you can write something that looks mathematical, that doesn’t imply that what you wrote has meaning. (Emphasis in Original)
Mathematicians have a little saying that captures the point being made here: Notions, not notations.
The first time I heard that .9999…=1 I didn’t believe it either. As I recall, I was a sophomore in high school at the time. I was a member of the math team, and one of the senior members of the team mentioned it during the bus ride home after one of our competitions. The following day I mentioned it to my math teacher, Mr. Manzer. I expected him to tell me that I had misunderstood what the person had said. Instead, he whipped up a quick proof to show me that .999…=1. I don’t recall which argument he chose, but I do recall that it was completely convincing. I was left shaking my head over not having realized it for myself.