Davi has a very original calculator; it performs only two operations: the usual addition $(+)$ and another operation, denoted by $*$, which satisfies:

(i) $a * a=a$

(ii) $a * 0=2a$

(iii) $(a * b) + (c * d) = (a * c) + (b * d)$

What are the results of the operations $(2+3) * (0+3)$ and $1024 * 48$?
To calculate $(2 * 3)+(0 * 3)$, we use properties (i), (ii), and (iii). Then

$$
\begin{aligned}
& (2 * 3)+(0 * 3) \quad \stackrel{(\mathrm{iii})}{=} \quad(2 * 0)+(3 * 3) \\
& \text { (i) (ii) } 2 \times 2+3=7 \text {. }
\end{aligned}
$$

To calculate $1024 * 48$, observe that $1024=976+48$. We have:

$$
\begin{aligned}
1024 * 48 & =(976+48) *(0+48) \\
& =(976 * 0)+(48 * 48) \\
& =976 \times 2+48 \\
& =1952+48=2000
\end{aligned}
$$