Friday, June 03, 2005

On why 0.999... = 1

One reason why Dr. Edgar Escultura sought to trash the trichotomy axiom is because of an apparent paradox that 0.999... = 1. But 0.999... = 1 is already proven true in mathematics.

However, I agree that the equation 0.999... = 1 is not intuitive, because in the real world we know that 0.999 is less than 1.

But take note that although 0.999 is less than 1, 0.999... (with the three trailing dots) is not. Pay attention to the three dots.

The three dots in 0.999... means that the decimal point is followed by an infinite number of 9's. In the same manner, 0.333... means a number with an infinite number of 3's trailing its decimal point.

These nonterminating repeating decimals sometimes arise when dividing two counting numbers. Say, the fraction 1/3. If we want to obtain the decimal representation of 1/3, we simply divide 1 by 3. Using the manual division we learned in school, 1 divided by 3 is represented as

   
   +------   
 3 |  1

Since 1 is smaller than 3, we need to append a decimal point and a zero to 1, making the division look like this:

   +------   
 3 | 1.0

Now we can divide by treating "1.0" as "10":

       .3
   +------   
 3 |  1.0
    -   9
    ------
        1

The division leaves a remainder of 1. We can still continue to divide by appending another zero to "1.0" and "bringing down" the zero to the remainder. The result is this:

       .3
   +------   
 3 |  1.00
    -   9
    ------
        10

Treating the remainder as "10", we can divide it by 3 :

       .33
   +------   
 3 |  1.00
    -   9
    ------
        10
    -    9
    ------
         1

The remainder is again 1. We can still append zeroes and obtain the following:

       .333
   +--------
 3 |  1.000
    -   9
    ------
        10
    -    9
    -------
         10
    -     9
    -------
          1

This again leaves a remainder of 1, so the process would continue indefinitely, in fact infinitely. So we can say that the decimal representation of 1/3 is 0.333... (with the three dots). Through this process, we can also deduce that 1/9 is 0.111... and 8/9 is 0.888... .

Now let's proceed to the proof that 0.999... = 1. There are many ways to prove it, but I'll show you two ways.

One way is by stating 1 as a quotient of two identical counting numbers, say 9/9. The number 9 divided by 9 equals 1, so we can say that

1 = 9/9 

But we know that 9 = 1 + 8. So we can expand the numerator of 9/9 and obtain

1 = (1 + 8)/9 

We can split (1 + 8)/9 into two, so it now becomes

1 = 1/9 + 8/9 

We already know that 1/9 = 0.111... and 8/9 = 0.888... . So the equation now becomes

1 = 0.111... + 0.888...

We can add 0.111... and 0.888... and obtain this equation:

1 = 0.999... .

Next, I will show you another way of proving it. I learned this technique in high school.

We already know in the previous discussion that 1/3 = 0.333... . But we also have a way of obtaining the reverse, that 0.333... = 1/3.

We simply let the variable x be 0.333... .

x = 0.333...     [Equation 1]

Multiplying both sides of Equation 1 by 10, we have

10x = 3.333...   [Equation 2]

Since Equation 1 and Equation 2 are equivalent, we can "subtract" Equation 1 from Equation 2:

     10x = 3.333...   [Equation 2]
  -    x = 0.333...   [Equation 1]
  ------------------
      9x = 3.000...
  Or
      9x = 3          [Equation 3]

Dividing both sides of Equation 3 by 9, we have

9x/9 = 3/9

Or

x = 3/9 .

Reducing 3/9 into lowest terms, we obtain

x = 1/3 .

Now let us apply this technique for proving that 0.999... = 1.

Let the variable x be 0.999... .

x = 0.999...    [Equation 1]

Next, mutiply both sides of Equation 1 by 10:

10x = 9.999...  [Equation 2]

Subtract Equation 1 from Equation 2:

    10x = 9.999...    [Equation 2]
   -  x = 0.999...    [Equation 1]
   ----------------
     9x = 9           [Equation 3]

Dividing both sides of Equation 3 by 9, we have

9x/9 = 9/9

Or

x = 1 .

Therefore, since x = 0.999... and x = 1, then 0.999... = 1 .

2 Comments:

Anonymous Ceejay said...

anlupet mo p're! thanks for the info...Ceejay

11:04 AM  
Anonymous 1 said...

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10:18 PM  

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