.9999999... Is Equal to 1.000000

And now for something completely different.  A math puzzle.  Or conundrum, if you will.

In the figure to the left, the bar above the number 9 indicates that it is to be repeated forever.  For the remainder of this post, we will represent that concept by several nines with an ellipsis (.999...).

Now, here is the conundrum.  .9 repeating is EQUAL TO ONE.   Not CLOSE to one, mind you, but EQUAL to one.

Nonsense, you reply.  It is obviously less than one.  Not by much - by an infinitely small amount, in fact.  But the simple fact (?) that it is not one is enough to demonstrate that it can't be equal to one.  It's as close as you can get to one without being one.

Wrong.  It is in fact equal to one, and that fact can be demonstrated mathematically in several ways.

The most easily understood is to revert to other familiar repeating digits.  Everyone knows that 1/3 is 0.333... and that 2/3 is 0.666...  If you add them together, you get 3/3, which is one.

But now note that the sum of the decimals on the right side of the equation is 0.999...

Therefore, one is equal to (not close to) .999...

You don't agree?  Then try this.  Subtract .999... from one.  What you have is 0.000...  An infinitely long string of zeroes, which can only be equal to zero.  And if the subtraction of .999... from one leaves zero, then the .999... must be one.  But, you say, there's a one at the end that string of zeroes.  No, there isn't, because the string of 9s doesn't end.

There are other proofs at the Polymathematics blog, along with a long series of comments, an update refuting the counterarguments, and a final refutation of the most stubborn skeptics.


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I think that our whole math system is screwed up.Maybe,somewhere in the universe,a race of aliens have devised a perfect math system which does not have any problems like what is 1/0?And how do deal with repeating decimals.
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Thanks to you Travis, and to all others echoing the proof.

Mr. Lombard: I actually thought of the concept when I was a fetus. So, as they say in Intellectual Property parlance, I "swear behind your conception date." :)
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To respond more specifically to your reasoning, the example you provide is unfair, because it happens to coincide with the limits of the decimal system. 8/2 works with the decimal system perfectly, because it divides evenly by integers. The same cannot be said for 1/3.

To explain it on its own terms, with no need for analogy. Let's say that you have a rifle with 10 horizontal aiming positions. Now, you have a target that lands at position 4 (8/2nds from your example) position. Can you hit it? Yes.

But can you hit a target at the 1/3rd horizontal position? Unfortunately, no. You need a better scope. You can change to a 100-point scope, or 1000 or 1 google plex. But you still have the same problem.

This is not a paradox - it is perfectly consistent. It is just another way of saying that you have a digital scale (10-point, 100, 1000, or whatever digital limit you are using). An infinitely-repeating digital expression will never perfectly express a fraction that does not divide evenly. That's what infinite repetition means - it is NOT FINITE.
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Thankyou, professor, this "namecalling" you have exhibited is yet another example of approximated terms to express more complex concepts. In the same way that your general, unreasoned term "Poppycock" gives insufficiently particular information concerning the concept you seek to advnace, the decimal expression .333 (repeated infinitely) gives insufficient expression to the concept of dividing into 1/3rd.

/takes his bow
//knows he is a wise-a&&
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