I still remember a math trick my brother used to play on me when I was younger. He would ask me what one divided by nine equaled.
1/9 = 0.1
Then he would ask me what two divided by nine equaled, and so on and so forth.
2/9 = 0.2
…
8/9 = 0.8
Finally he would ask me what nine divided by nine equaled. Here, of course, is the trap. According to our logic:
9/9 = 0.9
But everyone knows that a number divided by itself equals one. Therefore:
0.9 = 1
Not being a mathematical genius, this little trick was enough to drive me crazy. (Why did you insist on doing it, Steve?) I used to argue over and over that:
1 = 0.9 + 0.01
Note that the overline is only above the zero and not the one. I do like to pretend that my ability to extend the mathematical notation in order to solve my problem has to indicate some small glimmer of brilliance in me.
My brother had some extremely rational way of explaining the difficulty that I have since forgotten. He also had some other similar puzzle which I can’t remember. But I still like to write this one on white boards and watch people puzzle over it. Try it sometime.
P.S. If you’re wondering, I think the problem is best explained by a failure of the brain to grasp exactly what the mathematical notation means. Unable to comprehend an infinite expansion of the number under the overline, our brain assumes that it must be ever so slightly short of the next step up. In reality, the overline is simply a means of writing in decimal something that is better described in a fractional form.
P.P.S. I get the feeling limits fit in here somehow too, but I can’t put my finger on how. Steve, do you care to enlighten us on the many questions raised by this post?
Actually, you’re right about this last equation as well:
1 = 0.(9) + 0.(0)1
The thing is:
0.(0)1=0
And so everything goes into the right places again..
I believe I learned about this mathematical conundrum in Pre-Calc class in high school. We were talking about limits and series in the class, and I think the example was
Lim (terms -> inifinity) of 0.9 + 0.09 + 0.009 … = 1
Or something like that. (I don’t know how to do the fancy overbars and junk, so bear with me.)
Anyway, that’s the only place limits came into the story at all. Our prof. went on to show us that using simple algebra, you can prove that 0.(9) = 1
x = 0.(9) (parentheses representing the overbar)
10x = 9.(9) (multiplying by 10 just moves the decimal point)
Now, subtract x from both sides.
-x … -0.(9)
9x = 9
Now, divide by 9
x = 1
And yes, effectively 0.(9) is just an imperfect decimal representation of 9/9, which is of course 1.
On second thought, I believe our Prof’s story was that while there is a complicated Calculus-level proof for 0.(9) = 1, there is also the simple algebraic proof. Thus, the point was: don’t make something harder than you have to.