Garigor
Garigor is a number system based on the Fibonacci sequence, where any positive integer can be expressed as a sum of non-consecutive Fibonacci numbers. Unlike standard positional systems (such as binary or decimal), where each position represents a power of a fixed base, Garigor uses Fibonacci numbers themselves as building blocks.
Key Principles
| Principle | Description |
|---|---|
| Unique Representation | Every positive integer has a unique representation as a sum of distinct, non-consecutive Fibonacci numbers. |
| No Consecutive Ones | In the binary format of Garigor, no two consecutive digits can both be 1. |
Steps for Representation
To represent an integer in Garigor, follow these steps:
- Identify the largest Fibonacci number less than or equal to the integer.
- Subtract this Fibonacci number from the integer.
- Repeat the process with the remainder until reaching zero.
- Mark each Fibonacci term used with a
1and any skipped Fibonacci terms with a0.
Representing 17
For the integer 17, follow these steps:
- Find the largest Fibonacci number ≤ 17, which is 13.
- Subtract 13:
17 - 13 = 4. - Find the largest Fibonacci number ≤ 4, which is 3.
- Subtract 3:
4 - 3 = 1. - Find the largest Fibonacci number ≤ 1, which is 1.
- Subtract 1:
1 - 1 = 0.
The representation of 17 in Garigor is 100101, indicating inclusion of 13, skip 8, include 3, skip 2, and include 1.
Breakdown
| Fibonacci Number | Binary Position | Used? |
|---|---|---|
| 13 | 100000 |
⬤ |
| 8 | 010000 |
◯ |
| 5 | 001000 |
◯ |
| 3 | 000100 |
⬤ |
| 2 | 000010 |
◯ |
| 1 | 000001 |
⬤ |
Thus, 17 = 13 + 3 + 1, and in binary form for Garigor, 17 is represented as 100101.