Long multiplication school method: number of primitive operations to calculate first partial product?
Perhaps the following example will be useful. Suppose that $n = 3$, and we want to multiply $a_2 a_1 a_0$ by $b$. We first compute
$$
\begin{align*}
a_0 \times b &amp;= c_0 d_0 \\
a_1 \times b &amp;= c_1 d_1 \\
a_2 \times b &amp;= c_2 d_2 
\end{align*}
$$
(Here $c_0 d_0$ is a two-digit integer.)
So far we have done 3 operations (for general $n$, this will be $n$ operations).

Now we wish to perform the addition
$$
\begin{array}{cccc}
0&amp;d_2&amp;d_1&amp;d_0\\
c_2&amp;c_1&amp;c_0&amp;0 \\\hline
r_3&amp;r_2&amp;r_1&amp;r_0
\end{array}
$$
The book states that you need 4 operations for this (and in general, $n+1$), though 2 are enough (in general, $n-1$):
$$
\begin{align*}
r_0 &amp;= d_0 \\
s_1 r_1 &amp;= d_1 + c_0 \\
s_2 r_2 &amp;= d_2 + c_1 + s_1 \\
r_3 = s_2
\end{align*}
$$
Here $s_1,s_2$ are carry bits, and the first and last operations are for free (though the author charges one operation for each).

Summarizing, if you only charge addition and multiplication, only $2n-1$ operations are needed. If you also charge moves, $2n+1$ operations are needed. (The moves can be eliminated, though. Exercise.)