How to make continued fractions of any number?

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.

1 Answer

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.