Please help me to prove that
**"Every natural number can be uniquely represented in sum of power's of 2".** Example 5=2^2 + 2^0.

**"Every natural number can be uniquely represented in sum of power's of 2".** Example 5=2^2 + 2^0.

No vote yet

2 votes

×

Problem Loading...

Note Loading...

Set Loading...

## Comments

Sort by:

TopNewestIt follows from that for any

\[ n \] we can find \[k\] such that \[2^k\le n<2^{k+1}\].We will proceed with induction.For \[n=1\] we see \[1=2^0\].This is the only way to express it.Suppose it is true for all \[i< n\].We show for \[n\].We show for \[n\].Now as before we can find \[k\] as per the equation \[2^k\le n<2^{k+1}\].So we see this \[k\] is unique.Now consider \[n-2^k\] and since this is \[<n\] it has a unique representation.So every positive integer has one such representation.btw this is actually proving that each number in decimal system has a unique representation in the binary system. – Riju Roy · 3 years, 8 months ago

Log in to reply

uniquepowers of \(2.\) – Michael Tang · 3 years, 8 months agoLog in to reply

– Ivan Koswara · 3 years, 8 months ago

I almost wanted to claim that the statement is wrong as it's currently worded (since \(5 = 2^0 + 2^0 + 2^0 + 2^0 + 2^0\) too), but you already put it here. :)Log in to reply

– X Zero · 3 years, 8 months ago

Thank you very much Riju Roy .Log in to reply