Fibonacci Checker
Check if a number is part of the famous Fibonacci sequence. Discover the mathematical properties and patterns of Fibonacci numbers and the golden ratio.
Check Fibonacci Number
Enter a number to check if it's part of the Fibonacci sequence:
Each number is the sum of the two preceding numbers
The Fibonacci Sequence
The Fibonacci sequence is one of the most famous mathematical sequences in history. Named after the Italian mathematician Leonardo Fibonacci, it appears in nature, art, and has numerous mathematical properties.
What is the Fibonacci Sequence?
๐ Definition
Each number is the sum of the two preceding numbers
Starts with 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
๐ข Formula
F(n) = F(n-1) + F(n-2)
F(0) = 0
F(1) = 1
๐จ Golden Ratio
As numbers get larger, ratio approaches ฯ (golden ratio)
ฯ = (1 + โ5) / 2 โ 1.6180339887
Fibonacci Numbers Table
| Position (n) | Fibonacci Number | Ratio F(n)/F(n-1) | Properties |
|---|---|---|---|
| 1 | 0 | - | Starting number |
| 2 | 1 | - | First non-zero |
| 3 | 1 | 1.0 | Appears twice |
| 4 | 2 | 2.0 | First even number |
| 5 | 3 | 1.5 | Odd number |
| 6 | 5 | 1.666... | Odd number |
| 7 | 8 | 1.6 | Even number |
| 8 | 13 | 1.625 | Odd number |
| 9 | 21 | 1.615 | Odd number |
| 10 | 34 | 1.619 | Even number |
โจ Fun Fact: As you move further in the sequence, the ratio between consecutive Fibonacci numbers approaches the golden ratio (ฯ โ 1.618), which appears in art, architecture, and nature.
Mathematical Properties of Fibonacci Numbers
๐ Divisibility
Every 3rd number is even
Every 4th number is divisible by 3
Every 5th number is divisible by 5
โ GCD Property
GCD(F(m), F(n)) = F(GCD(m,n))
Greatest common divisor of any two Fibonacci numbers
โ Addition
F(a+b) = F(a+1)รF(b) + F(a)รF(b-1)
Addition formula for Fibonacci numbers
Fibonacci in Nature
| Natural Phenomenon | Fibonacci Pattern | Example |
|---|---|---|
| Plant Growth | Leaf arrangement | Pinecones, sunflowers |
| Animal Reproduction | Rabbit population | Fibonacci's original problem |
| Shell Spirals | Nautilus shell | Golden spiral pattern |
| Tree Branching | Branch distribution | Oak and elm trees |
| Hurricane Spirals | Spiral arms | Storm cloud patterns |
| DNA Structure | Helix turns | Genetic material |
Fibonacci in Art and Architecture
๐จ Golden Rectangle
Rectangle with sides in golden ratio
Considered most aesthetically pleasing
Used in painting and photography
๐๏ธ Parthenon
Ancient Greek temple proportions
Uses golden ratio extensively
Architectural harmony
๐ต Music
Fibonacci in musical scales
Instrument string lengths
Rhythm and timing
Testing Methods for Fibonacci Numbers
๐งฎ Mathematical Formula
Check if 5nยฒ+4 or 5nยฒ-4 is perfect square
Efficient for large numbers
O(1) time complexity
๐ข Sequence Generation
Generate Fibonacci numbers until reaching n
Simple and intuitive
O(log n) time complexity
๐ Matrix Exponentiation
Advanced mathematical method
Very efficient for large n
Used in computer algorithms
Fibonacci in Computer Science
- Dynamic Programming: Classic example of memoization and tabulation
- Algorithm Analysis: Used to demonstrate recursive vs iterative solutions
- Data Structures: Fibonacci heaps are advanced data structures
- Sorting Algorithms: Some sorting methods use Fibonacci numbers
- Cryptography: Used in some encryption algorithms
Large Fibonacci Numbers
The growth rate of Fibonacci numbers follows the golden ratio:
- F(n) โ ฯ^n / โ5 (where ฯ is the golden ratio)
- The sequence grows exponentially
- F(100) has 21 digits
- F(1000) has 209 digits
- F(10000) has 2090 digits
Fibonacci and the Golden Ratio
| Mathematical Relationship | Formula | Value |
|---|---|---|
| Golden Ratio | ฯ = (1 + โ5) / 2 | 1.6180339887498948... |
| Fibonacci Ratio | lim(nโโ) F(n+1)/F(n) | 1.6180339887498948... |
| Golden Ratio Conjugate | ฯ - 1 = 1/ฯ | 0.6180339887498948... |
| Lucas Numbers Ratio | L(n+1)/L(n) | Approaches ฯ |
Historical Significance
๐ค Leonardo Fibonacci
Italian mathematician (1170-1240)
Introduced sequence to Europe
Originally studied rabbit population
๐ Ancient Origins
Sequence known in ancient India
Appears in Sanskrit poetry
Used in Indian mathematics
๐ Global Discovery
Independently discovered by different cultures
Appears in Chinese mathematics
Known to Greek mathematicians
Practical Applications Today
- Financial Markets: Technical analysis uses Fibonacci retracements
- Computer Graphics: Fibonacci spirals in 3D modeling
- Medical Imaging: Spiral patterns in MRI and CT scans
- Sports Training: Periodization using Fibonacci cycles
- Game Development: Procedural generation using Fibonacci numbers
- Urban Planning: Traffic flow optimization using Fibonacci patterns