Math Puzzle Time!

The rational numbers are all numbers that can be represented as P/Q, where P and Q are integers; in other words, any number that can be written as a fraction.

One of the interesting facts of mathematics is that any number that can be written as a decimal that repeats forever is a rational number. For instance,
0.0909090909... = 1/11,
0.925925925... = 25/27,
0.285714285714285714... = 2/7,
and so on.

Here's the challenge:
The number 0.999... (a decimal followed by infinitely many nines) is clearly a repeating decimal.
Therefore it must be rational. Therefore it can be written as a fraction.
How can you write 0.999... as a fraction?


Have fun.

For those of you still confused...

The infinitely repeating decimal 0.9999999999999... is in fact equal to 1.

I'm not rounding it off. I'm not just saying it's close to 1. The number 0.99999.... is in fact exactly equal to 1.

Here's proof.

Let x = 0.999...
Then
10*x = 10*0.999...
so
10x = 9.999...
Subtract x from both sides.
10x-x = 9.999...-x
9x = 9.999...-x
Re-substitute the value of x on the right side of the equation
9x = 9.999...-0.999...
9x = 9
Divide both sides by 9
x = 9/9 = 1

So x = 0.999... and x = 1; therefore 0.999... = 1, QED.

There. Certain proof by algebraic manipulation that 0.999... is just a different "name" for the number 1.


If that doesn't convince you, please consider the following.

The fraction 1/3 can be written in decimal form as 0.333...
So
1/3 = 0.333...
Multiply both of those by 3.
Then
3*1/3 = 3*0.333...
3/3 = 0.999...
3/3 is clearly equal to 1, so once again
1 = 0.999...


I first started thinking about the number 0.999... while I was taking my first calculus class during my junior year of high school. Not quite sure what sparked off the chain of thought, but over the next week or two I gradually came to the conclusion that it should probably be equal to 1, because of the argument by fractions described above. Then, just a few days afterwards, I was reading a book of logic puzzles by Raymond Smullyan and came across the algebraic proof, which convinced me completely. Fascinating stuff!


Any questions?
-=-Mr. Truman