How is it that 0.9 (repeating) can be equal to the value of 1?

".9 repeating equals one. In other words, .9999999... is the same number as 1."

"The standard algebra proof (which, if you modify it a little, works to convert any repeating decimal into a fraction) runs something like this. Let x = .9999999..., and then multiply both sides by 10, so you get 10x = 9.9999999... because multiplying by 10 just moves the decimal point to the right. Then stack those two equations and subtract them (this is a legal move because you're subtracting the same quantity from the left side, where it's called x, as from the right, where it's called .9999999..., but they're the same because they're equal. We said so, remember?)"

http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html

Infinity is a weird thing.
Multiplying by ten does not put a zero into the last place, the last number is still a nine.
x=0.99999...
9x=10x-x=9.99999...-0.99999...=9 ==> x=1

https://en.wikipedia.org/wiki/Infinity

Infinity is a difficult concept. You are asking: why is 0.999 . . . with an infinite number of 9's, equal to the integer 1? Well, it is just a different way of representing the same number. It even makes sense logically. Remember how you learned how to count using apples? Let us go back to that concept. Say you have 9 apples and you give one to a friend. Now what fraction of the total number of apples have you given away? Right, 1/9. How do we represent that in decimals? Right, 0.11111 (repeating). Try it by hand. You will quickly see that the 1's go on forever. Now, how many apples does your friend have? She has:
(total number of apples) times (her fraction of the total number) =
= (9) times (0.11111 (repeating)) = .99999 (repeating) = 1 apple.

newton.dep.anl.gov/newton/askasci/1995/math/MATH070.HTM webpage no longer exists

This is where philosophy and mathematics collide, or we could say Stein v. Einstein. Rose is a rose is a rose, and both common sense and Zeno tell us 0.999... can't possibly be exactly equal to one, however close it may come.

This is an example of the limitations of art and science in representing reality. 1/3 isn't exactly 33%; it's 33 1/3%. A repeating 0.3 isn't exactly one third, but ever so little shy.

We had an animated discussion about this today at lunch. I pointed out that a pie can be cut into three equal parts, but "one" cannot be divided equally by "three." The rejoinder involved that part of the pie that sticks to the knife. :-)

So, being more precise, a perfect circle can be divided into three exactly equal sectors by lines that have only one dimension. (No width.) The weakness of the decimal system is that it cannot exactly represent that. Numbers, like words, are only symbols.

These are the kinds of amusing sophistries some people think I'm nuts for enjoying. (Thanks, Quate, for presenting it.)

Two other "proofs" we discussed yesterday were:

If one divided by three equals one third, represent it as 0.3 repeating. Then multiply it by three, and you get 0.9 repeating. Therefore, 0.9 repeating equals one. Again, the trick here is in the decimal representation of a common fraction.

Try subtracting one minus 0.9 repeating. You come up with 0.0 repeating!