Fibonacci Calculator
Visualize how Fibonacci numbers are calculated using iterative and recursive approaches
Fibonacci Visualization
Iterative Method
Calculating:
F(6) = 8
Current Step
Completed
Pending
Mathematical Definition:
F(n) = F(n-1) + F(n-2)
Base cases: F(0) = 0, F(1) = 1
Input Controls
Uses a loop to calculate Fibonacci numbers step by step
Calculation Controls
Algorithm Analysis
Time Complexity
Iterative:
O(n)
Recursive:
O(2^n)
Space Complexity
Iterative:
O(1)
Recursive:
O(n)
Current Statistics
Input:
6
Method:
iterative
Total Steps:
0
Current Step:
0
Result:
Calculating...
Algorithm Properties
Iterative Approach:
- • Efficient: O(n) time complexity
- • Space Efficient: Uses constant O(1) space
- • No Stack Overflow: No risk of call stack overflow
- • Predictable: Linear execution pattern
Recursive Approach:
- • Elegant: Matches mathematical definition
- • Intuitive: Natural recursive structure
- • Exponential Time: O(2^n) time complexity
- • Stack Usage: Uses O(n) call stack space
How Fibonacci Works
Mathematical Definition: F(n) = F(n-1) + F(n-2)
Base Cases: F(0) = 0 and F(1) = 1
Iterative: Use a loop to calculate Fibonacci numbers step by step
Recursive: F(n) = F(n-1) + F(n-2) with base cases n ≤ 1
Performance Comparison
Iterative:
- • Time: O(n)
- • Space: O(1)
- • No stack overflow
- • More memory efficient
Recursive:
- • Time: O(2^n)
- • Space: O(n)
- • Risk of stack overflow
- • More elegant code
Real-World Applications
- • Nature: Fibonacci spiral in shells, flowers, and pinecones
- • Art: Golden ratio and aesthetic proportions
- • Finance: Fibonacci retracements in technical analysis
- • Algorithms: Dynamic programming and optimization
- • Mathematics: Number theory and combinatorics
Common Fibonacci Values
F(0) =0
F(1) =1
F(2) =1
F(3) =2
F(4) =3
F(5) =5
F(6) =8
F(7) =13