I noticed an interesting pattern the other day. Let's take a look at the powers of 10 in binary:
- $10^0$ = 1 = 1 b
- $10^1$ = 10 = 10 10 b
- $10^2$ = 100 = 1100 100 b
- $10^3$ = 1000 = 111110 1000 b
Basically, it seems that $10^n$ for any non-negative integer $n$ written out in base 2 ends with its base 10 representation.
Does this pattern go on forever, and if so, can anyone provide me with a satisfactory explanation as to why this happens?
Multiplication of a number in binary by $2^{n}$ adds $n$ zeroes to the expression. $10^{n}=2^{n}5^{n}$, so as $2^{n}$ divides $10^{n}$, when expressed in binary you are adding $n$ zeroes to the number $5^{n}$ expressed in binary form. This is directly analogous to the base $10$ case.