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Sahithyan's S1
Sahithyan's S1 — Mathematics

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 42069\frac{420}{69} for example.

42069=6+669\frac{420}{69} = 6 + \frac{6}{69}

42069=6+1696\frac{420}{69} = 6 + \cfrac{1}{\frac{69}{6}}

42069=6+111+36\frac{420}{69} = 6 + \cfrac{1}{11 + \cfrac{3}{6}}

42069=6+111+12\frac{420}{69} = 6 + \cfrac{1}{11 + \cfrac{1}{2}}

As 42069\frac{420}{69} is finite, its continued fraction expansion is also finite. And it can be written as 42069=[6;11,2]\frac{420}{69} = [6; 11, 2].

Infinite expansion

For irrational numbers, the expansion will be infinite.

For example π\pi:

π=3+17+115+11+1292+⋯\pi = 3 + \cfrac{1}{7 + \cfrac{1}{15 + \cfrac{1}{1 + \cfrac{1}{292 + \dotsb}}}}

Conintued fraction expansion of π\pi is [3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,…][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 π\pi, the sequence will be:

3,227,303106,355113  …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.