Expert: Victor S. Miller - 10/15/2001

Question
How does one find the rational number representation of a repeating decimal?

I had successfully determine the answer for .3636363636... by deriving the geometric series A(n)=.36(.01)^(n-1).  I then used the values contained therein in the formula for the sum of an infinite geometric series
S(n)=A(1)/(1-r) or S(n)= (36/100)/(1-(1/100))= 4/11

I must admit, however, that this was an educated guess.  I am not sure of the correct procedure, and I have never seen it demonstrated.  I am not in a class; I am just returning to a lost love after nearly twenty years.  I was actually a political science major in college.

I still have difficulty figuring out the following

0.318181818...
1.388888888...
Is there an algorithm?

Answer
Bob, your calculation is correct.  There is an algorithm:

Find the repeating part, and calculate its value and then add on to the non-repeating part.  Taking your first example:

0.31818181818......

This is = 0.3 + (1/10) * 0.18181818181818....

But 0.18181818.... = 18/(100-1)

The 100 comes from the fact that there is a period of length
2, and that 100 is 10^2.  Now 18/99 = 2/11

So the answer is 3/10 + 2/110 = 35/110 = 7/22.

Another example

0.4123456456456....

= 0.4123 + (0.0001) * 0.456456....

and 0.456456... = 456/(1000-1).  You can complete this

The reason that this works is that

if a is a number and 0<a<1, then

1/(1-a) = 1 + a + a^2 + a^3 + ...

So we take a = 1/10^j, where j is the length of the period.

Victor