Originally posted by: MadRat
9*0.9r = (10-1)*0.9r = 9.9r - 0.9r = 9 = 9*1
9*0.9r = 9*1
0.9r = 1
For one, the idea that .9r can be divisible by 10 is laughable. What happens is the final decimal place of the last nine is no longer at infinity, but actually it would become "(infinity-1)" by your argument. The "mathematicians" of the forum seem to conveniently ignore the idea of a limit to the 9's in .9r at the infinite position.
You also cannot factor out the 9 from .9r and presume to remain accurate to what you have defined in your original terms. The argument becomes flawed at this point.
*sigh*
Please explain to me how a number like 0.9r is ant different then, say, 0.3r. I can multiply that by 10. That operation is well defined within the real number system.
This also holds for division. PI/10 = .3141592653589........
The only argument that you can use to say that division/multiplication is not valid for a number like 0.9r is to show that it is not part of the real number system. There has already been numerous arguments showing that it is well defined in this system, but please go ahead and present your argument as to why it is not.
As to my original "proof" I can't see why there is any confusion in the first steps. It does require a bit of algebra, but I hope that I'm safe in assuming that everyone arguing here has taken that class or learned it at some point in time. If not then you are in the wrong thread.
1) x = 0.9999...
2) 10x = 9.9999...
3) 10x - x = 9.9999... - 0.9999...
4) 9x = 9
5) x = 1.
step 1) is called an assumption and/or a definition. You are assuming this will hold true throughout the proof. In this case we are telling you, the reader, that x is 0.9r. At any point in this proof, if you see x, it is symbolically defined as 0.9r. This is one way that algebra works, by defining symbols to have numeric meaning.
step 2) is done using a simple algebraic rule. We are multiplying both sides of the equation by 10. If you have the equation x^2+y^2=1 then A*x^2+A*y^2 = A also holds true. This operation is well defined for any number on the real number line. Since 0.9r is on the real number line and x is a symbolic notation of a number on the real number line, there is nothing wrong with this step.
step 3) is also well defined for the real number system. you are combining equations with mathematical operations to form new equations. Take this system: If a=b and c=d then a-c=b-d. This is a theorem that you probably should have learned in grade school, and is what we have done in this step.
step 4) is where the equation is reduced. What is 10x-x equal to? 9x. If you have 10 oranges and take away 1 of them how many do you have left? 9.
The sticking point in this whole "proof" seems to be in this part of step 4). 9.9999... - 0.9999... What does this reduce to? A number of people suggest that 0.9r has one more 9 after the decimal then 9.9r. The fact that it is repeating means that the string of 9s go on forever. They never stop. Ever. For every 9 in 0.9r there is a matching 9 in 9.9r. The opposite is also true. For every 9 in 9.9r there is a matching 9 in 0.9r. These numbers both have the same "length". It is infinite. 9.9r doesn't have a slightly shorter infinite length because that would imply that it was not infinite to begin with. When you subtract them you are left with 9. Please don't tell me that you can't do this. You can. What is 1.(21)r - 0.(12)r? It is 1.(09)r. What is 3.3r - 0.3r? It is 3. What is 9.9r - 0.9r? It is 9.
Ok, now that we have that out of the way, on to step 5) In this step we are dividing both sides by 9. We can do this for the exact same reason that we can do step 2) We are applying the exact same operation to both sides of an equation so the outcome will still be equivalent.
So what does this show? We started by defining/assuming that x was equal to 0.9r. We have shown mathematically that x is equal to 1. If a=b and a=c then b=c, so 1 = 0.9999......
QED!!
On a side note: My post per day average is ~equal to PI.