0.999 (9 recurring) = 1?

From wikipedia (http://en.wikipedia.org/wiki/Proof_that_0.999..._equals_1):
{
In mathematics, the repeating decimal 0.999… denotes a real number equal to one. In other words, the notations 0.999… and 1 represent the same real number. This equality has long been accepted by professional mathematicians and taught in textbooks. Proofs have been formulated with varying degrees of mathematical rigour, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.
}

One of the proofs given is:
x = 0.999 (recurring)
=> 10x = 9.999
=> 10x - x = 9.999 - 0.999
=> 9x = 9
=> x = 1

But this wrong! Whats making it wrong is that you consider the no. of digits after the decimal point remains same; or in other words, that you consider (infinity - 1) = infinity.
When n is HUGE, (n - 1) is approximately= n.

The following is the correct method. First assume that x = 0.999...... (n 9s) with n = HUGE!
x = 0.9999999999999999999999999999999999999999 (40 9s total)
10x = 9.999999999999999999999999999999999999999 (Still 40 9s; 39 after decimal point)

10x = 9.9999999999999999999999999999999999999990
- x = 0.9999999999999999999999999999999999999999
-------------------------------------------------
9x = 8.9999999999999999999999999999999999999991
-------------------------------------------------

Now make n = infinity. So it makes
9x = 8.9999999999999999999999...(infinite 9s)............99991

Now when you divide by 9, you get

x = 0.999999999999999999999999999999999999999... (infinite 9s i.e. recurring again!)

----------- x -----------

Another proof goes as:

0.333 (recurring) = 1/3
=> 3 x 0.333 = 3 x 1/3
=> 0.999 = 1

But since the problem we are dealing with (i.e. Is 0.999... = 1?) is very critical like a man standing on a knife edge on a single toe, immeasureably small factors also matter!
So we cannot say 0.333 (recurring) is equal to 1/3.
1/3 is a perfect division of a cake into 3 parts.
0.333 (recurring) is NOT a perfect division into 3 parts!!

As a corollary of this discussion, you can say:
One who can divide perfectly may be called God.

---

Share