I recently found an continued fraction representation of $\pi$, 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 $\sqrt[3]{15}$ and on some rational number like $0.8713241$.
  3. 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.

  • 2
    $\begingroup$ You might want to begin with the Wikipedia article, which answers some of your questions. For example, it will tell you that every rational number can be expressed as a finite continued fraction, and every irrational number as an infinite continued fraction. $\endgroup$ – Brian M. Scott Mar 31 '16 at 0:00
  • $\begingroup$ take a look at math.stackexchange.com/questions/1721050/… $\endgroup$ – Will Jagy Mar 31 '16 at 0:07
  • $\begingroup$ for that matter, math.stackexchange.com/search?q=continued+fraction as well as math.stackexchange.com/questions/tagged/continued-fractions $\endgroup$ – Will Jagy Mar 31 '16 at 0:09
  • $\begingroup$ @WillJagy A coincidence that my question and question in the link you posted was asked at the same time? o.O They even have question IDs next to each other 1721050, 1721051. $\endgroup$ – KKZiomek Mar 31 '16 at 0:10
  • $\begingroup$ yes; however, there are thousands of such questions on this site, you can alternate between reading about how to do continued fractions yourself, and reading answers here that show specific problems worked out. $\endgroup$ – Will Jagy Mar 31 '16 at 0:13

Let the number whose continued fraction you want to find be $x$.

Let $[x] = a$

Let the fractional part of $x$ i.e $frac(x) = b$

So, $x = a + b$

Let c = $\frac{1}{b}$

$\rightarrow$ $x = a + \frac{1}{c} $

Now, let $[c] = p$

Let the fractional part of $c$ i.e $frac(b) = q$

Hence, $c = p + q$

Let r = $\frac{1}{q}$

$\rightarrow$ $c = p + \frac{1}{r} $

$x = a + \frac{1}{c} $

$\rightarrow$ $x = a + \frac{1}{ p + \frac{1}{r}} $

Repeat the process for $r$.

Keep repeating this process till you arrive with a rational number.

But if you start off with an irrational number, you'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.

  • $\begingroup$ Wow you answered my question two years later. You should be an archeologist-mathematician :) $\endgroup$ – KKZiomek Apr 6 '18 at 21:01
  • $\begingroup$ I became an archaeologist solely for the purpose of gaining reputation lol $\endgroup$ – Free Radical Apr 7 '18 at 1:39

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