What is the Fibonacci Sequence?
The Fibonacci sequence starts with 0 and 1, and each subsequent number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
F(n) = F(n−1) + F(n−2) for n ≥ 2
Binet's Formula: F(n) = (φⁿ − ψⁿ) / √5
φ (Golden Ratio) = (1 + √5) / 2 ≈ 1.6180339887...
First 20 Fibonacci Numbers
| n | F(n) | Ratio F(n)/F(n-1) |
|---|---|---|
| 0 | 0 | — |
| 1 | 1 | — |
| 2 | 1 | 1.0000 |
| 3 | 2 | 2.0000 |
| 4 | 3 | 1.5000 |
| 5 | 5 | 1.6667 |
| 6 | 8 | 1.6000 |
| 7 | 13 | 1.6250 |
| 8 | 21 | 1.6154 |
| 9 | 34 | 1.6190 |
| 10 | 55 | 1.6176 |
| 15 | 610 | 1.6180 |
| 20 | 6,765 | 1.6180 |
Fibonacci in Real Life
- Nature: Flower petals (3, 5, 8, 13...), pinecone spirals, sunflower seeds, nautilus shell proportions
- Art & Architecture: The golden ratio ≈ 1.618 appears in the Parthenon, Leonardo da Vinci's works, and many design systems
- Computer Science: Fibonacci heaps, algorithm analysis, Fibonacci search technique
- Finance: Fibonacci retracement levels (23.6%, 38.2%, 61.8%) used in technical trading analysis
Frequently Asked Questions
Is 0 a Fibonacci number?
Yes — by modern convention, the sequence starts F(0) = 0, F(1) = 1. Some older texts start with F(1) = 1, F(2) = 1, which shifts every index by one. This calculator uses the modern F(0) = 0 convention.
What is the golden ratio and how does it relate to Fibonacci?
The golden ratio φ = (1+√5)/2 ≈ 1.6180339887. As you go further in the Fibonacci sequence, the ratio of consecutive terms (F(n+1)/F(n)) converges to φ with increasing accuracy. By F(20), the ratio is accurate to 4 decimal places.
What is the largest Fibonacci number this calculator can show?
JavaScript can represent integers exactly up to 2^53 ≈ 9 quadrillion, which is F(78) = 8,944,394,323,791,464. Beyond that, this calculator uses BigInt for exact large integer arithmetic up to F(200) and beyond.