Let $\text{binary}(a)$ denote the binary representation of a base-$10$ number $a$. Are the following statements correct? If yes, where can I find the proofs?
(1) $\text{binary}(a\times b)=\text{binary}(a)\times \text{binary}(b)$
(2) $\text{binary}(a+b)=\text{binary}(a)+\text{binary}(b)$
(3) $\text{binary}(a\times (b+c))=\text{binary}(a\times > b)+\text{binary}(a\times c)$