1=0.9999...

Avatar do usuário Henry Cunha 9900 2 17 177
I always knew I didn't really understand numbers, but I thought I had a pretty good handle on basic arithmetic. So this was a bit of a surprise to me.

It's a given that we can multiply (or add, or subtract, or divide) both sides of an equality by the same factor without violating any of the laws of arithmetic operations. So, for instance:

1/2 = .50
2x(1/2)= 2 x .50
1=1

So, if

1/3 = 0.3333....
and
3 x (1/3) = 3 x 0.3333....
therefore
1 = 0.99999...

(Extracted from Strogatz, S. The Joy of X: A Guided Tour of Math, from One to Infinity (Houghton Mifflin Harcourt 2012)

There's lots of Google stuff under "1 = .9999..." if you're interested in what that really means.
Avatar do usuário PPAULO 36030 4 32 632
The problem with Strogatz´ notion (as seen in the above thread) is that it stops somewhere, it doesn´t continues on and on to some place where it converges.

Technically, 0.9999.... equals 1. As proved in Wikipedia as follows:

x = 0.999...
10x = 9.999...
10x-x=9.999... - 0.999...

9x = 9


x = 1
http://en.wikipedia.org/wiki/0.999...


In that page there´s a proof by using limits as well. And other ways.
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