I wonder how mathematics would be changed if we were been using binary system in calculations instead of decimal ..
- Could theory of mathematics would change a little ?
- Are there known examples where the use of binary system leads to the new interesting mathematical facts ?
- Maybe in math history are known the spectacular results achieved thanks to binary notation ? (Probably in number theory or maybe somewhere else ?)
My example is about a simple theorem stating that any difference of two squares of odd numbers is divisible by 8. Of course we can prove it using simple algebra transformations, but it is also visible from the binary forms of the numbers.
Let's notice that squares of the following numbers (denote them as $k$) have a form:
(partially binary notation)
because other odd numbers can be denoted as $1000 n+k$, we see that differences of squares of these numbers give always last $3$ digits $000$ so the difference must be divisible by $8$.
- Could some other theorems can be also proved this way?