Wronsonia8g

Answered

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.

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.

Answer & Explanation

Caiden Barrett

Expert

2022-07-08Added 20 answers

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 = $\frac{1}{b}$

$\to $$x=a+b$

Now, let $[c]=p$

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

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.

Let $[x]=a$

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

So, $x=a+b$

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

$\to $$x=a+b$

Now, let $[c]=p$

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

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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