Does this mathematical technique have a formal name? And why does it work? When I split a number in the powers of 2. I am able to make any combination of any number that is less than it by taking each number of the series only once.
For example:
$7=1+2+4$ 
I can construct any number less than or equal to seven using the $3$ numbers $1,2,4$. 
Or take $10$ for instance:
$10 = 1 + 2 + 4 + 3$
I can construct any number that is less than or equal to $10$ using the 4 numbers.
How this decomposition is done is via breaking the number in powers of 2 until you can't break it anymore. More examples:
5 -> (1, 2, 2)
10 -> (1, 2, 4, 3)
15 -> (1, 2, 4, 8)
1000 -> (1, 2, 4, 8, 16, 32, 64, 128, 256, 489)

Does this method of decomposition have a name? And how does it allow us to make any number that is less than or equal to it?

Countrary to most of your feedback, this is not at all binary representation, because you split from the small end and not from the big end.
For binary representation to represent 7 you start with: 


*

*8 no it's too big

*4, ok it fits, 7-4=3 left, 

*2, ok it fits 3-2=1 left

*1, ok it fits 1-1=0 left, done!


What you are doing is splitting from the small end, so you will end up with a number $2^k-1 + b$, where $b$ is some remainder. I don't know if this has any name, but it sure is not a binary representation of the number.
