Wronsonia8g

2022-07-07

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.

Caiden Barrett

Expert

Let the number whose continued fraction you want to find be $x$.
Let $\left[x\right]=a$
Let the fractional part of $x$ i.e $frac\left(x\right)=b$
So, $x=a+b$
Let c = $\frac{1}{b}$
$\to$$x=a+b$
Now, let $\left[c\right]=p$
Let the fractional part of $c$ i.e $frac\left(b\right)=q$
Hence, $c=p+q$
Let r = $\frac{1}{q}$
$\to$ $c=p+\frac{1}{r}$
$x=a+\frac{1}{c}$
$\to$ $x=a+\frac{1}{p+\frac{1}{r}}$
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.

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