Expert: Paul Klarreich - 10/30/2008

Question
We live in France and my son got a math problem the likes of which I have never seen. I don't necessarily want the answer, just what is it called and how is it solved KEEPING THE ANSWER AS A FRACTION. I can do it if I convert it to decimals, but since the very first answer is a repeating decimal, I don't know if that invalidates the answer. Here's the problem-
1+   1
 ------
  1 +   1
     ------
      1 + 1
         ---
          2

I hope that makes sense- he did it to 2, 3, and 4 stacks and then to 7 stacks (1 plus 1 over 1 + 1 over 1 + 1 over...etc). Thanks for any help you can give!!

Answer
Questioner:   Cheryl
Category:  Advanced Math
Private:  No
 
Subject:  Fractions
Question:  We live in France and my son got a math problem the likes of which I have never seen. I don't necessarily want the answer, just what is it called and how is it solved KEEPING THE ANSWER AS A FRACTION. I can do it if I convert it to decimals, but since the very first answer is a repeating decimal, I don't know if that invalidates the answer. Here's the problem-
1+   1
------
 1 +   1
    ------
     1 + 1
        ---
         2

I hope that makes sense- he did it to 2, 3, and 4 stacks and then to 7 stacks (1 plus 1 over 1 + 1 over 1 + 1 over...etc). Thanks for any help you can give!!
..................................
Hi, Cheryl,

I understand your frustration.  It is unlikely anyone in France knows this.

This is called a CONTINUED FRACTION.  In some cases, here is what you can do:

1 + 1/(1 + 1/(1 + 1/...)  can be called x.  Now, assuming that it continues, note that:

x = 1 + 1/(1 + 1/(1 + 1/...), and
          <==  also x ==>, so

x = 1 + 1/x, which we can solve:

x^2 = x + 1, which is just a quadratic.

x^2 - x - 1 = 0

   1 + sqrt(5)
x = ------------  (not +-, because x is obviously positive)
       2
   1 + 2.23       3.23
x = --------- ~~  ----- = 1.62 or so,
       2           2

which happens to be a famous number.

I'll leave it to you to look that up -- why should I have all the fun?