How to make continued fractions of any number?I recently found an continued fraction representation of π, and I wondered how can I make an continued fraction that converges into a number?The MAIN question is: how do you make a continued fraction for any number and can every number be represented as continued fraction?Some SPECIFIC questions:1.How is an continued fraction for any number x generated? Is there an algorithm and what is it?2.Give an example of the algorithm on some irrational number like 15 3 and on some rational number like 0.87132413.Can every number be represented as a continued fraction?4.Do continued fractions for complex numbers exist?Don't vote down for no reason. I just learned about continued fractions and I don't really know anything about them.

Question

How to make continued fractions of any number?I recently found an continued fraction representation of   π, and I wondered how can I make an continued fraction that converges into a number?The MAIN question is: how do you make a continued fraction for any number and can every number be represented as continued fraction?Some SPECIFIC questions:1.How is an continued fraction for any number x generated? Is there an algorithm and what is it?2.Give an example of the algorithm on some irrational number like       15    3   and on some rational number like   0.87132413.Can every number be represented as a continued fraction?4.Do continued fractions for complex numbers exist?Don&#039;t vote down for no reason. I just learned about continued fractions and I don&#039;t really know anything about them.

Caiden Barrett · Accepted Answer

Let the number whose continued fraction you want to find be   x.Let   [  x  ]  =  aLet the fractional part of   x i.e   f  r  a  c  (  x  )  =  bSo,   x  =  a  +  bLet c =       1    b    →  x  =  a  +  bNow, let   [  c  ]  =  pLet the fractional part of   c i.e   f  r  a  c  (  b  )  =  qHence,   c  =  p  +  qLet r =       1    q    →   c  =  p  +      1    r    x  =  a  +      1    c    →   x  =  a  +      1          p      +              1        r            Repeat the process for   rKeep repeating this process till you arrive with a rational number.But if you start off with an irrational number, you&#039;ll never arrive with a rational number.This is why irrational numbers are represented using an infinite loop of continued fractions.For complicated decimals, you could just write a computer program using the above logic.So, to answer your question, yes, every number can be represented as a continued fraction.