1. What is Binet's Formula?

Binet's formula is a closed-form expression that calculates the nth Fibonacci number directly, without needing to compute all previous numbers in the sequence. It's named after French mathematician Jacques Philippe Marie Binet.

2. How Does the Calculator Work?

The calculator uses Binet's formula:

Where:

• \( \phi \) — Golden ratio (approximately 1.618)
• \( n \) — Position in Fibonacci sequence
• \( F(n) \) — nth Fibonacci number

Explanation: The formula uses the golden ratio and its conjugate to directly compute Fibonacci numbers, providing an exact integer result for any non-negative integer n.

3. Understanding the Fibonacci Sequence

Details: The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. It appears frequently in mathematics, nature, and computer science.

4. Using the Calculator

Tips: Enter a non-negative integer n to calculate the nth Fibonacci number. The calculator will return the exact integer value using Binet's closed-form formula.

5. Frequently Asked Questions (FAQ)

Q1: Why use Binet's formula instead of iterative calculation?
A: Binet's formula provides a direct calculation without recursion or iteration, making it computationally efficient for large values of n.

Q2: Does Binet's formula work for all integers?
A: The formula gives exact integer results for all non-negative integers n, despite involving irrational numbers in the calculation.

Q3: What is the golden ratio (φ)?
A: The golden ratio is approximately 1.6180339887, a mathematical constant that appears in many natural patterns and has unique mathematical properties.

Q4: Are there limitations to Binet's formula?
A: For very large n, floating-point precision limitations may affect the result, though our implementation handles this with appropriate rounding.

Q5: Who discovered this formula?
A: Although named after Binet, the formula was known to mathematicians like Abraham de Moivre and Daniel Bernoulli over a century earlier.