Continued Fraction Expansion

The process

• Separate the integer part
• Find the inverse of the remaining part. Result will be greated than 1.
• Repeat the process for the remaining part.

Finite expansion

Take $\frac{420}{69}$ for example. $$

$\frac{420}{69} = 6 + \frac{6}{69}$

$\frac{420}{69} = 6 + \cfrac{1}{\frac{69}{6}}$

$\frac{420}{69} = 6 + \cfrac{1}{11 + \cfrac{3}{6}}$

$\frac{420}{69} = 6 + \cfrac{1}{11 + \cfrac{1}{2}}$

As $\frac{420}{69}$ is finite, its continued fraction expansion is also finite. And it can be written as $\frac{420}{69} = [6; 11, 2]$.

Infinite expansion

For irrational numbers, the expansion will be infinite.

For example $\pi$: $$

$\pi = 3 + \cfrac{1}{7 + \cfrac{1}{15 + \cfrac{1}{1 + \cfrac{1}{292 + \dotsb}}}}$

Conintued fraction expansion of $\pi$ is $[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, \ldots]$.

Convergence

In the case of infinite continued fraction expansion, on each ”+” part, the expansion can be separated. Each separated part will generate a sequence of numbers, which is converging to the original number.

For example, for $\pi$, the sequence will be: $$

$$3,\frac{22}{7},\frac{303}{106},\frac{355}{113}\; \dots$$

Here:

• Elements with the odd index are lesser than the converging value.
• Elements with the even index are greater than the converging value.