I have a little problem to applying trachtenberg on this multiplication.
Actually i don't really care with the result (please don't suggest me a calculator), i just want to know how does it work.
We know the answer is :
$$\begin{array}{ccccccc} 0&2&5&6&4&1&3\\ & & & & &\times&5\\ \hline\\ 1&2&8&2&0&6&5 \end{array}$$
The rule said, that
If we had an odd digit in the multiplicand we would add $5$
And
If we had an even digit in the multiplicand we would use "the half" (without fraction) of the neighbor in the right side
Then why does digit $5$ become $8$ instead of $0$?
Please help. Thanks
Okay.....
It works because $5 = 10\div 2$ So you multiple by $10$ and divide in half.
So we want to do $3$ times $10$ divided by $2$. So that's $30\div 2$ so we want to write down $\color{blue}15$. We write down the $5$ but we aren't sure if we will be able to write down the $\color{blue}1$
Now we go to the next column. We want to wright down the $\color{blue}1$ from before but we don't know if we can. We have $\color{red}1$ in this column. We are going to multiply by $10$ and divide in half. But $\color{red}10$ is an odd number of $10$s and we can't just split it in half and have $0$. We are going to have a $\color{green}0\color{red}5$. So we add that $\color{red}5$ to our $\color{blue}1$ and write down $6$.
Now we want to write down the $\color{green}0$. But can we?
In the next column we have $4$. We will multiply by $10$ and divide by $2$. $4$ is even so we will have an even number of $10$s so dividing by $2$ will give us full number of $10$s and that will be $0$. So we don't have to add $5$ to our $\color{green}0$. We can just write it down.
And so on .....