sleepy said:

"Surely if 9x = 9, then x = 1. But since x also equals .9999999... we get that .9999999... = 1. The algebra is impeccable."

thats his entire proof (from the original site). please do whatever you have to to understand that proof, because i don't.

he proved that .999999...=1 because .999999....=1?

thats not proof thats saying it is because it is

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Quote:

picture a circle graph with .999 filled in

now

fill in .9999
now fill in .999999

each time you are adding 1 more step

.9
.99
.999

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Look at 1/x ratios:

1/1 = 1
1/2 = 0.5
1/50 = 0.02
1/1234 = 0.0008 (aprox)
1/123456789 = .000000008 (aprox)

Do you see the trend?  As the value of x gets large, the value of the ratio gets small.  As we continue this trend, making x larger and larger, the value of the ratio approaches 0.  If we were to make x as large as possible, the number next to infinity, then the value of the ratio would be as small as possible, the number next to 0.  If we were actually able to make x equal infinity, then the ratio would actually equal 0.  This is known as a limit and is written as:



I won't go into the proof of limits, but if you care you can look it up.  It is called the epsilon-delta proof and is one of the foundations of calculus.

Now lets go back to your circle concept.  Assume the area of your circle is 1 (exact).  Now subtract off a little bit from the area,  say 1/x.  The formula for the area of the circle is now:



Now, what happens as you take the limit of the area as x approaches infinity?



We already showed that the limit of 1/x as x approaches infinity is 0, and 1 is not dependent upon x, therefore 1 doesn't change regardless of what x does, thus the answer to the above equation reduces to 1-0 which is 1 (exactly).

I hope this one makes sense, 'cause I am running out of ways to explain this. 

Edit:

Ok... one last attempt:

If you look up convergence of infinite geometric series, you will find the following (for |r|<1):



We can rewrite 0.999... (repeating) as:



Anything to the 0 power is 0.  Anything to the 1 power is itself.  So one can think of the above as:



The value of r is less than one (it is 1/10), so we can apply the rule of convergence of infinite geometric series, reducing the series to:



I will leave the algebra to the reader, but a good guess as to the correct answer is 1. 

Quote:

I was trying to point out inconsistencies in viewing decimals as approximations of fractions, not argue they were approximations of fractions.

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nobhdy said:

MATHS IS WRONG

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