## Wednesday, December 10, 2008

### A Different "Approach" to 0.99999999...??

An investigation for middle schoolers? Precalculus students? Calculus students? Anyone who is fascinated by patterns and an understanding of the infinite and infinite processes?? Enjoy this at any level or depth you wish...

Determine the first dozen decimal places, then the exact decimal for each of the following:

1 - 1/9

1 - 1/99

1 - 1/999

1 - 1/9999

1 - 1/99999

Continue this pattern until the denominator has a string of 9 nines.

Questions:
(1) Describe any patterns you observe. What if the denominator had a string of 100 nines? A string of N nines?
(2) What does all of this suggest (not prove) about the meaning of 0.999999... (repeating)?
(3) Oh, and by the way, you may also want to examine the decimal expansions of 1/9, 1/99, 1/999, 1/9999, ... How would you describe the exact decimal representation of 1/9999...9, where the denominator has 100 nines? N nines?

Is there anything new under the sun here? OR just another view of well-known facts about infinite repeating decimals, sums of infinite geometric series, limits and real numbers???