Fibonacci Calculator
Calculate Fibonacci numbers and explore the golden ratio with BigInt support for large numbers.
Enter a position n (0 to 1,000) to calculate F(n)
The Fibonacci sequence is defined by the recurrence relation F(n) = F(n-1) + F(n-2), with seed values F(0) = 0 and F(1) = 1. Each number is the sum of the two preceding ones, creating a sequence that appears throughout nature, art, and mathematics.
The Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377...
Golden Ratio Connection: As n increases, the ratio F(n)/F(n-1) converges to the golden ratio φ (phi) ≈ 1.6180339887. This irrational number appears in the proportions of the Parthenon, spiral galaxies, and the arrangement of leaves on stems.
Algorithm: This calculator uses an iterative approach with O(n) time complexity and O(1) space complexity. BigInt is used to handle arbitrarily large Fibonacci numbers without overflow, allowing calculation of F(1000) which has over 200 digits.
Applications: Fibonacci numbers appear in computer algorithms (heap data structures, search algorithms), financial trading (Fibonacci retracements), biology (branching patterns, flower petals), and art (compositional proportions).
What is the Fibonacci sequence?
The Fibonacci sequence is a series where each number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... It was introduced to Western mathematics by Leonardo Fibonacci in 1202, though it was known earlier in Indian mathematics.
What is the golden ratio and how does it relate to Fibonacci?
The golden ratio (phi, approximately 1.618) is the limit of the ratio of consecutive Fibonacci numbers as they grow larger. F(n)/F(n-1) approaches phi as n increases. This ratio appears throughout nature, art, and architecture due to its aesthetically pleasing proportions.
Where do Fibonacci numbers appear in nature?
Fibonacci numbers appear in flower petals (lilies have 3, buttercups 5, daisies often 34 or 55), spiral arrangements in pinecones and sunflower seeds, branching patterns in trees, and the spiral shells of nautilus. These patterns emerge from efficient growth mechanisms.
How are Fibonacci numbers used in computer science?
Fibonacci numbers appear in algorithm analysis (Fibonacci heaps), search algorithms (Fibonacci search), data structures, and coding theory. They're used in financial trading (Fibonacci retracement levels), pseudorandom number generators, and as a classic example for teaching recursion and dynamic programming.