CONTINUED FRACTIONS 101

These fractions within fractions let mathematicians study irrational numbers, such as Pi and , and find patterns in their infinite decimals. Finding patterns in complex, irrational numbers is crucial to fields such as cryptography and encryption.

The easiest way to jump into a continued fraction is to start with a ‘simple and finite’ continued fraction, where you’ll have nothing but 1s in the numerator position of the continued fractions, and it will resolve in just a few steps. Continued fractions for an irrational number, such as Pi, go on ad infinitum.

Take 37/13. That’s one of those awkward improper fractions. You can also write 37/13 as 2 plus 11/13. That’s step one in converting to a continued fraction.

To get to the next nested layer, you again need an improper fraction to divide. So, change your 2 plus 11/13 to 2 plus 1 over 13/11. Now divide 13 by 11, which, of course, gives you 1 plus 2/11. That’s your next layer. This process goes on until the number you’d be dividing by is 1. That’s the end of a finite, rational continued fraction, since any number divided by 1 is equal to itself.