Psychology of 0.99999...
February 17, 2011
Are the seventh graders that are discussing this actually trying to get at the foundations of the real numbers, etc? Or is it really just the case that some people don't like the idea of .99999... equaling 1 because of non-intellectual discomfort with the way it's written?
Here's my question:
Why doesn't this come up with .33333... being equal to 1/3?
I'm willing to grant some room on the fact that they can divide one into three using long division and see that the algorithm is going to keep giving them a three, so they've got something a little more hands-on that makes them comfortable with that idea.
But I think that if you accept .333... = 1/3 and you don't accept that
Underneath the .9999...= 1 question, I think, for most people, is the "but it never GETS there" feeling that they have when they see .99999999... . It's thinking finitely about an infinite thing. They aren't sophisticated enough to get the same feeling about
.999... gives people the heebie jeebies because they are used to 99% not being a perfect score and 5.99 being a cheap trick a retailer does to make something look less expensive, which they can technically do because it's not 6. "Ninety nine point nine percent of the time"...how many times does that phrase get used in arguments when you know there is some contrived or extremely unlikely situation where the condition doesn't hold, so you know you _can't_ say "always", so you use 99.9 percent to insulate yourself from the people who are otherwise going to jump on you with the accurate-but-in-practical-terms-irrelevant exception.
In all these cases the difference between the .99 and 1 is critical--the retailer would not be able to write the five there, to try to convince you "well it's less than six dollars" if he didn't have that critical one penny gap to make it true. The 99% is a great score but that 1% that you could have got still sticks in there as something you could have gotten to be perfect. The 99.9% phrasing protects you immediately from having to waste time with detractors who really are not bringing up anything of substance. You are _very_ used to .99 or .999 being _very fundamentally and importantly_ not 1 in all these high-emotional-value situations like your grades, purchasing, and arguing.
So I guess what I'm saying is this:
99.9% of the time when someone objects to .9999... = 1 it's because they don't understand the math.
Ahh, it would be so much funnier to leave it there. But there is a point to be made, and it's this--telling someone "well, there is another formulation of numbers in which .9999... is different from 1", when their fundamental issue with it is unwillingness to accept a truth that doesn't match their emotional prejudice, isn't necessarily doing them any favors.
I don't have a problem with using the "controversy" as a jumping off point to discuss some extremely interesting mathematics that would otherwise NEVER occur to 99.9% of the population as being interesting. But I think it does the students a disservice to present it as "see? You were just as right as the people on the other side of the argument!", because those formulations are not really what they were getting at with their objection. Their objection was rooted in discomfort with a counterintuitive result, which is something that prevents mathematical progression. And I mean that historically as well as individually, like resistance to the idea of zero or imaginary numbers caused whole societies to stumble in their development of mathematic; and the same basic principle applied to an individual.
I think people should be encouraged to recognize their discomfort but accept (or accept the idea that at some point in the future perhaps they should accept--I'm not saying there is a reason to rush it) that sometimes the correct answer just isn't what we wanted it to be. And, yes, they're going to be very interested at that point in alternative formulations where things work differently--full speed ahead on that front, any excuse to show someone cool math is a good one. I just thing we should do it without compromising on/glossing over the other principle.
Short version: if they're accepting .333... = 1/3 and not
With 99.9% certainty.
Oh, and here's another one that I think maybe one of the kids came up with
.1111111... = 1/9
.2222222... = 2/9
.3333333... = 3/9
.8888888... = 8/9
.9999999... = ?
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