Tool for encoding/decoding numbers using Fibonacci encoding (binary words never having two consecutive 1 values)
Fibonacci Encoding - dCode
Tag(s) : Compression, Mathematics
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Fibonacci Encoding
Number to Fibonacci Coding Encoder
Fibonacci Code to Numbers Decoder
Answers to Questions (FAQ)
How to encode using Fibonacci encoding?
The Fibonacci code uses the Zeckendorf theorem (and Zeckendorf's representation of a number) which states that any integer can be written as the sum of non-consecutive Fibonacci numbers.
$$ n = \sum_{i=1}^{k} \beta_i F_{i} $$
The Fibonnacci coding consists in noting the coefficients $ \beta_i $ (being 0 or 1) to make a binary number.
Example: $ 123 $ is the sum of $ F_{11} = 89 $ and $ F_{9} = 34 $ or 1010000000 in binary (the two 1 are in position 8 and 10 starting from the right).
As the Zeckendorf representation never has 2 consecutive Fibonnacci numbers, the binary value will never have 2 times the number 1 consecutively.
How to decode Fibonacci encoding?
Each 1 of the binary word corresponds to a Fibonacci number, to find the decimal number, add all the Fibonacci numbers corresponding to the 1 of the binary word.
Example: 10100 corresponds to $ 1 \times F_5 + 0 \times F_4 + 0 \times F_3 + 1 \times F_2 + 0 \times F_1 = F_5 + F_3 = 8 + 3 = 11 $
What are the variants of the Fibonacci encoding?
A variant of Zeckendorf's theorem indicates that it is also possible to write any integer as the sum of non-consecutive Nega-Fibonacci (Generalization of Fibonnacci with negative indices) numbers, this encoding is called NegaFibonacci encoding.
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