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\end{Exercise}  All properties we have shown so far represent similarities between the operations in $\Fb$ and in $\ZZ$ or $\QQ$.  \\  An important difference between the sum in $\Fb$ and in $\ZZ$ or $\QQ$ is that summing the bit $1$ with the bit $1$, one gets $0$:  $$1+1=0.$$  We see now that this difference reflects also in the computation of \emph{multiples} of bits.  \\  Let us consider first $2\cdot 0$. It means $0+0$ then it   is $0$. \\  If we take $2 \cdot 1$ we have $2 \cdot 1=1+1=0$, so we can notice that $0$ is also the \emph{double} of $1$.  \begin{quote}  Multiplying a bit by $2$ we always obtain $0$.  \end{quote}  This has a precise meaning from a mathematical point of view, the concept of \emph{characteristic}.\\  Given a field (in our case $\Fb$), it can happen that   some positive\footnote{notice: positive means \emph{different} from $0$!!} integer numbers, multiplied to all the elements of the field, give $0 $ as   result (in our case, we saw that $2$ has this property).\\  The smallest such number $p$ - if it exists - is called \emph{characteristic} of the field (the characteristic is $p=2$ in the field $\Fb$). If there are no number with this properties, we say that the field has characteristic $p=0$.  \begin{Exercise}\label{ProofChar}  We have already observed that $2\cdot 0 = 2 \cdot 1=0$, so $2$ is one of the numbers that multiplied to all the elements of the field of bits,   give $0 $ as result. Can you prove that $2$ is actually the characteristic of $\Fb$?  \end{Exercise}  \begin{Exercise}\label{MaxiSum}  Compute the following operations in $\Fb$:  \begin{itemize}  \item $1+1+1+1+1+0+1+1+0+1=?$  \item $1+1+1+0+1+1+1+0+1+1+1=?$  \item $\underbrace{1+1+...+1}_{1235 \textrm{ times}}=?$  \item $\underbrace{1+1+...+1}_{15754 \textrm{ times}}=?$  \item $1245 \cdot 1=?$  \item $2482 \cdot 1=?$  \item $1234 \cdot 0=?$  \end{itemize}  Can you conclude, deriving a general rule for computing multiples of bits?  \end{Exercise}  Bits can also be seen as a way to represent the logical values TRUE/FALSE. The