How to make continued fractions of any number?I recently found an continued fraction representation of...

Wronsonia8g

Wronsonia8g

Answered

2022-07-07

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 0.8713241
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

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 f r a c ( x ) = b
So, x = a + b
Let c = 1 b
x = a + b
Now, let [ c ] = p
Let the fractional part of c i.e f r a c ( b ) = q
Hence, c = p + q
Let r = 1 q
c = p + 1 r
x = a + 1 c
x = a + 1 p + 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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