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Continued fractions: The infinite patterns hidden inside every number. Continued fractions: The infinite patterns hidden inside every number.

The secret language of infinite fractions

By Peter Teoh, Science Writer

Continued fractions are a fascinating and powerful way to represent real numbers—especially irrational ones—with remarkable precision using integers. Below is a list of 30 different popular or mathematically significant continued fractions, including well-known constants, algebraic numbers, and special functions. Each entry includes the number (or expression) and its simple (regular) continued fraction representation, often in compact notation.

1. Golden Ratio

φ = (1 + √5)/2 = [1; 1, 1, 1, 1, …]

2. Square Root of 2

√2 = [1; 2, 2, 2, 2, …]

3. Square Root of 3

√3 = [1; 1, 2, 1, 2, 1, 2, …] = [1; 1̅,̅ ̅2̅]

4. Square Root of 5

√5 = [2; 4, 4, 4, 4, …]

5. Square Root of 6

√6 = [2; 2, 4, 2, 4, …] = [2; 2̅,̅ ̅4̅]

6. Square Root of 7

√7 = [2; 1, 1, 1, 4, 1, 1, 1, 4, …] = [2; 1̅,̅ ̅1̅,̅ ̅1̅,̅ ̅4̅]

7. Square Root of 13

√13 = [3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, …] = [3; 1̅,̅ ̅1̅,̅ ̅1̅,̅ ̅1̅,̅ ̅6̅]

8. e (Euler’s Number)

e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, …] = [2; 1, 2k, 1] for k = 1, 2, 3, …

9. e²

e² = [7; 2, 1, 1, 3, 18, 5, 1, 1, …] (non-periodic, but known pattern exists)

10. π (Pi)

π = [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, …] (no known simple pattern)

11. π/2

π/2 = [1; 1, 1, 3, 31, 1, 145, …]

12. ln(2)

ln 2 = [0; 1, 2, 3, 1, 6, 3, 1, 1, 2, 1, …]

13. ln(10)

ln 10 = [2; 3, 4, 1, 2, 1, 1, 12, 1, 2, …]

14. Catalan’s Constant G

G = [0; 1, 10, 1, 8, 1, 88, 4, 1, 1, …]

15. Euler–Mascheroni Constant γ

γ = [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, …]

16. √2 − 1

√2 − 1 = [0; 2, 2, 2, 2, …]

17. (√5 − 1)/2 = 1/φ

(√5 − 1)/2 = [0; 1, 1, 1, 1, …]

18. tan(1) (radians)

tan(1) = [1; 1, 1, 3, 1, 5, 1, 7, 1, 9, …] = [1; 1, 2k+1] for k = 0, 1, 2, …

19. coth(1)

coth(1) = [1; 3, 5, 7, 9, 11, …]

20. sinh(1)

sinh(1) = [1; 5, 9, 13, 17, …] = [1; 4k+1] for k = 1, 2, 3, …

21. tanh(1)

tanh(1) = [0; 1, 3, 5, 7, 9, …]

22. √23

√23 = [4; 1, 3, 1, 8, 1, 3, 1, 8, …] = [4; 1̅,̅ ̅3̅,̅ ̅1̅,̅ ̅8̅]

23. Cube root of 2 (approximate CF)

∛2 = [1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, …] (not periodic)

24. Apéry’s Constant ζ(3)

ζ(3) = [1; 4, 1, 18, 1, 1, 1, 4, 1, 9, …]

25. 1/π

1/π = [0; 3, 7, 15, 1, 292, 1, 1, 1, 2, …] (reciprocal of π’s CF)

26. Gauss’s Constant G = 1/agm(1,√2)

G = [0; 1, 5, 21, 3, 4, 14, …]

27. Feigenbaum Constant δ ≈ 4.6692…

δ = [4; 1, 2, 43, 2, 13, 1, 1, 1, 4, …]

28. Plastic Constant ρ (real root of x³ = x + 1)

ρ = [1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, …]

29. √(2 + √2)

√(2 + √2) = [1; 1, 1, 4, 1, 1, 8, 1, 1, 12, …]

30. e^(1/e)

e^(1/e) = [1; 2, 12, 1, 2, 1, 4, 1, 1, 1, 10, …]

Notes on Continued Fractions

Simple continued fractions use only positive integers after the first term and have the form:

a₀ + 1/(a₁ + 1/(a₂ + 1/(a₃ + ⋯)))

denoted compactly as [a₀; a₁, a₂, a₃, …].

Key insights:

  • Quadratic irrationals (like √n for non-square n) always have periodic continued fractions (Lagrange’s theorem).
  • Transcendental numbers (e, π, etc.) have non-periodic expansions, though some (like e) show regular patterns.
  • Some entries (e.g., ζ(3), γ) are not proven to be irrational via continued fractions alone, but their CFs are computed numerically.

Side Notes

  • The Golden Ratio φ = (1 + √5)/2 has the simplest possible continued fraction—all 1s—making it the “most irrational” number.
  • Euler’s number e has a beautiful pattern: [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, …] where the pattern 1, 2k, 1 repeats.
  • Pi remains mysterious—despite centuries of study, no pattern has been found in its continued fraction.
  • The Feigenbaum constant δ appears in chaos theory and describes the rate at which bifurcations occur.
  • Golden Ratio Everywhere: From DNA helices to galaxy arms, φ appears throughout nature.
  • Why 292? The surprisingly large term in π’s continued fraction makes it exceptionally well-approximated by 355/113.
  • Chaos and Constants: How Feigenbaum’s δ connects continued fractions to chaos theory.
  • Best Rational Approximations: Continued fractions give the optimal way to approximate any irrational number.

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