Mathematics
Diophantine Approximation and Continued Fractions
Quick fact
The golden ratio φ ≈ 1.618 is, in a precise sense, the most difficult number to approximate by rationals: its continued fraction is the slowest to converge, and its convergents are the ratios of consecutive Fibonacci numbers.
Why this is interesting
You've probably been taught that π is approximately 22/7. But why is that a good approximation, and could there be an even better one with a smaller denominator?
Read the full explanation
Understanding Diophantine Approximation and Continued Fractions
Diophantine approximation is about finding rational numbers p/q that are close to a given real number α. For example, π is approximately 22/7, and |π – 22/7| ≈ 0.00126. How close can we get with a given denominator? Continued fractions give a systematic way to generate a sequence of fractions that are, in a precise sense, the best possible. To build a continued fraction, you take a number like 2.5, write down its integer part (2), then invert the fractional part (1/0.5 = 2), and repeat. This produces a finite or infinite list of integers. Truncating this list at each step gives a convergent, a nested fraction. For π, the continued fraction starts [3; 7, 15, 1, 292, ...], and the first few convergents are 3, 22/7, 333/106, 355/113. Each convergent is the best rational approximation to π among all fractions with denominator no larger than its own.
A deeper explanation
The mechanism behind continued fractions is rooted in the Euclidean algorithm. For any real α, the process of extracting integer parts and inverting the remainder is exactly the Euclidean algorithm applied to α and 1, but with real numbers instead of integers. This yields a sequence of integers a0, a1, a2, ... that defines the continued fraction. The convergents pn/qn satisfy a simple recurrence: pn = an p{n-1} + p{n-2} and qn = an q{n-1} + q{n-2}, with p{-2}=0, p{-1}=1, q{-2}=1, q{-1}=0. These convergents have the remarkable property that |α – pn/qn| < 1/(qn q{n+1}) < 1/qn^2. Moreover, among all fractions with denominator ≤ qn, the convergent pn/qn is the closest to α. This is the 'best approximation' property. The golden ratio φ = (1+√5)/2 has the simplest continued fraction [1; 1, 1, 1, ...], all ones. Because every an is as small as possible, the approximation error decreases as slowly as possible, making φ the 'worst approximable' number. This extreme case is connected to the fact that φ is the root of a quadratic with the smallest constant, and it underlies the concept of an irrationality measure.