Expert: Scott A Wilson - 10/3/2010

Question
Hi.

I'm in a college course called Mathematical Thinking and Writing, and right now we're discussing rational numbers as well as prime factorization. For a paper we have assigned for Thursday, we must give an algorithm for turning a decimal into a ratio of two integers. I've researched this on the Internet (because I wasn't in class the day we learned about this), but all of the sites just give me the answer instead of helping me figure out how to do it.

The number given to me was 3.076767676...But I've found that my professor does the math differently than others. Her is an earlier example of how she does these problems if you can make sense of it:

x= 0.31707313170731...

100,000x= 31707.31707
-     1x=      .31707
----------------------
99,999x= 31,707  =3(10,569) ....
--------  ------  ----------
99,999    99,999   3(33,333).....

Until we found that x = 13/41.

Can you help me understand this at all?

Answer
In finding the fraction, take the number of 9's to be how often it repeats.  For example, with x = 0.55555..., 10x - x = 5, so 9x = 5,
so x = 5/9.

For x = 0.12121212..., since it repeats every 2, take 10^2 = 100.
From this, 100x - x = 12, so 99x = 12, so 33x = 3, so  x = 4/33.

For 0.468468468..., note repetition every 3 digits, so use 10^3.
It is known that 10^3 = 1,000, so we have 1,000x - x = 468.
That gives 999x = 468; dividing by 9 gives 111x = 52.
111 and 52 have no common factors, x = 52/111.

For 0.425342534253..., note repetition occurs every 4 digits;
this means to use 10^4, which is 10,000.  This gives us
10,000x - x = 4253.  Since 4253 is not divisible by 2 or 5
(the facotrs of 10,000 are 2*2*2*2*5*5*5*5), x = 4253/10,000.

In this case, we have a repetition every 5 digits.
That is, x = 0.317073170731707..., and that is 31707 repeated.
In this case, 100,000x - x = 31707, so that works out to be
99,999x = 31,707.

Now 31,707 has the digits 3, 1, 7, 0, and 7, and these add to 18, which is divisible by 9, so 31,707 is divisible by 9.
This says that 31,707/99,999 = 3,523/11,111.

Now most people don't know that these are both products of 271.
That is, 3523 = 13*271 and 11,111 = 41*271.  Once this is known, however, dividing the top and bottom by 271 gives 13/41.


If you are a little strange, you might know that 11 is prime; and the prime factors of repeating 1's are 111 = 3*37; 1,111 = 11*101; 11,111 = 41*271; 111,111 = 3*7*11*13*37, and 1,111,111 = 239*469.
I don't think that's common knowledge, though.

From here, 99,999 = 9*11,111 = 3*3*41*271.  Of course, maybe there are some more modern calculators that can tell you this.