Fibonacci Number Calculator
Compute the n-th Fibonacci number exactly, list the sequence up to n, and see how fast the ratio Fn+1/Fn approaches the golden ratio.
Interactive Fibonacci number tool
Enter an index \(n\) and this tool will compute the Fibonacci number \(F_n\) using an efficient fast-doubling algorithm with exact big-integer arithmetic. You can also display the sequence from \(F_0\) up to \(F_n\) and, for moderate \(n\), the ratio \(F_{n+1} / F_n\).
Non-negative integer, \(0 \le n \le 10000\). We use \(F_0 = 0, F_1 = 1\).
Sequence listing is automatically trimmed for very large \(n\).
The calculator always uses the full exact value; this only affects what is shown.
What is a Fibonacci number?
The Fibonacci numbers form a famous integer sequence defined by the recurrence \(F_0 = 0\), \(F_1 = 1\) and
\(F_n = F_{n-1} + F_{n-2} \quad \text{for } n \ge 2.\)
The first few terms are:
\(0,\ 1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ 55,\ \dots\)
Our calculator uses this zero-based indexing, which is common in mathematics and programming.
Closed form (Binet’s formula)
Although the Fibonacci sequence is defined by a recurrence, each term also has a closed-form expression. Let \(\varphi\) be the golden ratio:
\(\varphi = \dfrac{1 + \sqrt{5}}{2} \approx 1.6180339887\) and \(\psi = \dfrac{1 - \sqrt{5}}{2}.\)
Then Binet’s formula states:
\(F_n = \dfrac{\varphi^n - \psi^n}{\sqrt{5}}.\)
For small \(n\), this formula gives an exact integer. For large \(n\), it is mainly useful for approximations and understanding growth rates.
Growth rate and digit count
Fibonacci numbers grow exponentially. As \(n\) increases, \(F_n\) behaves roughly like \(\varphi^n / \sqrt{5}\). This means the number of decimal digits of \(F_n\) is approximately
\(\text{digits}(F_n) \approx \left\lfloor n \log_{10}(\varphi) - \log_{10}(\sqrt{5}) \right\rfloor + 1.\)
The calculator reports the exact digit count for each \(F_n\), which is often more informative than the full integer for large indices.
Fibonacci numbers and the golden ratio
One of the most striking properties of the sequence is that the ratio of consecutive terms converges to the golden ratio:
\(\lim_{n \to \infty} \dfrac{F_{n+1}}{F_n} = \varphi.\)
Our ratio panel shows \(F_{n+1}/F_n\) for moderate \(n\) and compares it to \(\varphi\), highlighting how quickly the convergence occurs.
Efficient computation: why we avoid naive recursion
A straightforward recursive implementation of the recurrence relation re-computes the same values many times and has exponential time complexity. Instead, the calculator uses a fast doubling algorithm, which computes \(F_n\) and \(F_{n+1}\) together in logarithmic time.
This is why we can safely allow indices up to \(n = 10000\) while still returning exact integer results using big-integer arithmetic.
Fibonacci numbers – FAQ
Which indexing convention is used here?
We use the standard mathematical convention \(F_0 = 0\), \(F_1 = 1\). Some programming exercises or educational resources start from \(F_1 = 1, F_2 = 1\); if you are used to that convention, remember that our \(F_n\) corresponds to their term with index \(n + 1\).
What happens if I ask for very large n?
As \(n\) grows, the number of digits of \(F_n\) grows roughly linearly in \(n\). For example, \(F_{1000}\) already has more than 200 digits. The calculator still computes the exact number, but it may trim the display according to the digit limit you select and show an informative summary instead.
Why limit the ratio computation to small or moderate n?
To compute \(F_{n+1}/F_n\) as a floating-point number, we must convert large integers into double-precision format. Beyond a certain size, that conversion loses precision or overflows. The optional limit \(n \le 70\) keeps the ratio numerically stable and clearly demonstrates the convergence to the golden ratio.
Where do Fibonacci numbers appear in practice?
Fibonacci numbers show up in algorithm design (divide-and-conquer recurrences, heap structures, search trees), discrete mathematics (combinatorial identities, tilings, lattice paths), and modelling (idealized population growth). They also underpin financial tools such as Fibonacci retracement, where ratios between terms are used to define percentage levels on price charts.
Can I use this tool for programming and coding interviews?
Yes. This calculator is useful for checking the outputs of your own implementations, verifying edge cases like \(n = 0\) or large \(n\), and exploring the effect of different indexing conventions. It does not replace understanding time and space complexity, but it helps you build numerical intuition.
Frequently Asked Questions
Is this Fibonacci number calculator exact?
Yes. It uses big-integer arithmetic internally, so all Fibonacci numbers up to the configured limit are computed exactly. Any truncation only affects how the result is displayed, not the underlying value.
Why does the tool report the number of digits?
For large indices, Fibonacci numbers can have thousands of digits. The digit count gives you an immediate sense of scale without needing to inspect the full integer, and it is often what you need for theoretical work.
Can I export or copy the sequence?
Yes. You can select the sequence block with your mouse or keyboard and copy it into a text editor, spreadsheet, or programming environment. For very long sequences, consider using the trimmed view and regenerating ranges in code.