The use of binary numeral system for theoretical results

Question

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)

$001^2=0001$
$011^2=1001$
$101^2= 11001$
$111^2=110001$

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?

Answer 1

Many other theorems can be proved this way, or insight gained into them. For example in the Collatz conjecture every odd number leading directly to an odd number of the form $x01$ is a member of the set $\{x01, x0101, x010101,...\}$.

The proof of Fermat's last theorem uses p-adic number system, which are numbers arranged in prime bases, of which base 2 is an example, although they have the additional property of repeating to the left instead of the right. P-adics reveal many properties of number theory as they encode congruence information.

But ultimately as you move into more advanced mathematics it is actually the norm to write numbers in a variety of bases, not just base 10 or number 2, and sometimes some arbitrary base, as to do so is frequently useful just as you have conjectured.

Answer 2

Binary is another way of expressing numbers, but the numbers available are the same as decimal. Sometimes it can make informal proofs easier because there is a pattern in the digits or it can make patterns that interest us which we later prove. An example in base $10$ is $$7^2=49\\67^2=4489\\667^2=444889\\6667^2=44448889$$ This would not be so interesting in another base. The proof using $(\frac {2\cdot 10^n+1}3)^2$goes through in any base though you have to represent $10_{10}$ properly. If you were working in binary you might not find this interesting.

There are similar cute patterns in binary, but I don't have any that come to mind. Again, they can be proven in any base, but the motivation is not there unless you look at the numbers in binary.