Here is a cool way to come to terms with it though it is not a proof (I'll let the experts handle that).
1/9 = .111111111111...
2/9 = .222222222222...
8/9 = .888888888888...
so this would imply that .9999999999... = 9/9 = 1
1.Originally Posted by Yankee77
Performing a devision you'll never get a decimal periodic number with an indefinite sequence of 9 or 0 digit. So the problem is only formal.
Remember how decimal periodic number are transformed in fractions?
For instance 2.(3) - where () means indefinite sequence of 3 as decimal digit
2.3 = (23-2)/9=21/9=7/3
This applies to all cases
0.(9) = 9/9 = 1
1/3 is 0.333333333333...........
3 times 1/3 is 1
but 3 times 0.333333333333........... = 0.99999999999...........
develops as the infinite sum
that is 9(1/10+1/10^2+1/10^3+...)
the infinite sum in the parenthesis is the geometric serie with ratio 1/10 and gives
9[1/(1-1/10) -1] = 9[1/(9/10)-1] = 9[10/9-1] = 9(1/9)=1
If you look at the sequence, .9,.99,.999, ... and so on, you see that each term in the sequence is a closer approximation to the number 1. Obviously, the elements of the sequence will never reach 1, but instead get closer and closer to 1. This said, the sequence converges to 1, but none of the elements are ever exactly equal to 1. But I like the idea of looking at it as an infinite geometric series better. Very clever.Originally Posted by Yankee77
Here 's a little proof, that doesn't move far into sequence territory:
Consider the difference a=|0.(9)-1|. (the parentheses mean period).
Let this number be positive. Observe that a is non-negative, and less than any number of the form 0.0(etc)01, with a random number of zeros. This contradicts a>0 (*), and so a=0.
(*) 'Cause, if a>0, there ought to be a number of the form 0.0...01 < a.
Perhaps a helpful way to think about this is to separate the concepts of 'real number' and 'decimal expansion'. We can set up the system of real numbers in various ways, some of the more high-brow being Dedekind cuts in the real numbers or the completion of the rationals via Cauchy sequences modulo null sequences. Once you have a notion of real number, you can then describe a real number by a decimal expansion. It turns out that decimal expansions have this awkward property that sometimes two different decimals express the same real number: and this happens when one of them ends in 9 recurring. Of course one way of setting up the real number system is to start with decimals expansions.
For a further discussion of this point, see the page by Tim Gowers (author of the excellent "Mathematics: a Very Short Introduction") on What is so wrong with thinking of real numbers as infinite decimals? on his website http://www.dpmms.cam.ac.uk/~wtg10/decimals.html
The answer is basic, DEFINITON: THE REPETION OF A DECIMAL EQUALS THE VALUE OF ITS CONVERGENT. Thus, what most people do not relized is this is not paradoxical (nothing in math is paradoxical) but rather it is a DEFINITION and not a theorem. It is proven by looking at it geometric sum.