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Fibonacci Calculator

Calculate Fibonacci numbers, find any term in the sequence, and explore the golden ratio. Supports terms up to F(70).

Range: 0 – 70

What is Fibonacci Calculator?

The Fibonacci sequence is a series of numbers where each term is the sum of the two preceding ones, starting from 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, … The ratio of consecutive Fibonacci numbers converges to the Golden Ratio (φ ≈ 1.6180339887), which appears throughout mathematics, art, architecture, and nature. This calculator finds any Fibonacci term, lists the sequence up to that term, and shows the golden ratio approximation.

How to use

  1. 1 Enter a term number (n) to find the nth Fibonacci number.
  2. 2 Or enter a number to check if it is a Fibonacci number.
  3. 3 The calculator lists the full sequence from F(0) to F(n).
  4. 4 The ratio F(n)/F(n-1) is shown as a golden ratio approximation.

Formula

F(0) = 0, F(1) = 1 F(n) = F(n−1) + F(n−2) for n ≥ 2 Golden Ratio φ = (1 + √5) / 2 ≈ 1.6180339887 Binet's Formula: F(n) = (φⁿ − ψⁿ) / √5, where ψ = (1 − √5) / 2

Example calculation

F(10) = 55. Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Ratio F(10)/F(9) = 55/34 ≈ 1.6176 (approaches φ = 1.6180...).

Frequently asked questions

What is the Fibonacci sequence?

The Fibonacci sequence starts with 0 and 1, and each subsequent number is the sum of the previous two: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... It is named after the Italian mathematician Leonardo of Pisa, known as Fibonacci, who introduced it to Western mathematics in his 1202 book Liber Abaci.

What is the Golden Ratio?

The Golden Ratio φ (phi) ≈ 1.6180339887 is an irrational number that is the limit of the ratio F(n)/F(n-1) as n approaches infinity. It appears in the proportions of regular pentagons, the spiral of nautilus shells, phyllotaxis in plants, and is considered aesthetically pleasing in art and architecture.

What are Fibonacci numbers used for?

Beyond pure mathematics, Fibonacci numbers appear in computer science (Fibonacci search, Fibonacci heaps), financial trading (Fibonacci retracement levels), biology (leaf and petal arrangements), and the analysis of algorithms. They are also commonly used in beginner-to-advanced programming exercises.

Is 0 a Fibonacci number?

Yes. By the standard definition, F(0) = 0 is the first term of the Fibonacci sequence. Some older definitions start with F(1) = 1, F(2) = 1, which skips the 0, but the modern convention includes 0 as F(0).

What is the largest Fibonacci number this calculator handles?

This calculator supports terms up to F(70) = 190,392,490,709,135 using standard JavaScript numbers with full precision. Beyond F(70), 64-bit floating-point numbers lose integer precision.