I know two ways to represent a number,
- Muliplication of primes
Any number can be represented uniquely as a product of prime numbers
- Binomial Representation
Any number can be represented uniquely as a sum of powers of 2
Are there any other interesting ways to uniquely represent a number?
The Fibonacci representation (sum of Fibonacci numbers with no repeats or consecutive Fibonacci numbers).
The factorial system eg $463=3\cdot5!+4\cdot4!+1\cdot3!+0\cdot2!+1\cdot1!$ where the "digit" corresponding to $n!$ can be any of $0,1,\dots,n$.
Balanced ternary ie ternary using the digits $\bar{1},0,1$, where $\bar{1}$ represents $-1$.