Continued Fraction Approximation Formula
Continued Fraction Approximation Formula
\[x = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3}}}\]
Variables
x = real number approximation
a0 = integer part
a1 = first continued fraction term
a2 = second continued fraction term
a3 = third continued fraction term
Description
What is this formula?
This formula represents a number using a continued fraction expansion for better rational approximation.
When to use it
Use it when high precision approximation of irrational numbers is required.
Example
Data:
a0 = 1, a1 = 2, a2 = 2, a3 = 2
x = 1 + 1/(2 + 1/(2 + 1/2))
Result:
x ≈ 1.4167
Applications
Used in numerical analysis, approximation theory, and computer arithmetic.
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