Question

write \(\frac{19}{51}\) as continued fraction

http://upload.wikimedia.org/math/8/a/7/8a73a8728febba2c2ecb9abc2f4e0d03.png

Answer 1

\[\frac ab=\frac1{b/a}\]

Answer 2

what is 51/19

Answer 3

do i get \[\begin{align}\frac{19}{51} &= 0 + \dfrac{1}{\dfrac{51}{19}}\\~\\&= 0 + \dfrac{1}{2+\dfrac{13}{19}} \end{align}\]

Answer 4

does it have to be an infinite continued fraction

Answer 5

i think continued fraction of every rational number ends because we will eventually get 0 after dividing enough times

Answer 6

\[\frac{19}{51}\]
\[\frac{1}{51/19}\]
\[\frac{1}{2+13/19}\]
\[\frac{1}{2+\frac{1}{19/13}}\]
\[\frac{1}{2+\frac{1}{1 +\frac{6}{13}}}\]
\[\large \frac{1}{2+\frac{1}{1 +\frac{1}{13/6}}}\]
etc ...

Answer 7

oh ok

Answer 8

wow! wish there is some direct formula for getting these coefficients w/o doing so many divisions

Answer 9

what? its the divisions that makes it so fun ....

Answer 10

it is fun for sure xD a generating function or an explicit formula for like :

\(a_n = f(n)\)

Answer 11

sometimes its written as the .... addend series?

[0:2,1,2,....]