Long division taken apart, continued

Now that we have taken long division apart, in the last posting, and made a multiplication table for the divisor ${\displaystyle 21}$, in the posting before that, we are ready to look at the standard long division algorithm, with Jakow Trachtenberg's trick.

We are dividing ${\displaystyle 1543}$ by ${\displaystyle 21}$, using the multiplication table

${\displaystyle \left[\begin{array}{cc}0& 0\\ 1& 21\\ 2& 42\\ 3& 63\\ 4& 84\\ 5& 105\\ 6& 126\\ 7& 147\\ 8& 168\\ 9& 189\end{array}\right]}$.

Start by writing


${\displaystyle 21)\overline{1543}.}$

From the table, see that no whole ${\displaystyle 21}$s go into ${\displaystyle 1}$ or into ${\displaystyle 15}$, but that ${\displaystyle 7}$ whole ${\displaystyle 21}$s, but not ${\displaystyle 8}$, go into ${\displaystyle 154}$ :


${\displaystyle 21)\overline{1543}.}$
${\displaystyle 147.}$

Subtract ${\displaystyle 1470}$ from ${\displaystyle 1543}$ leaving ${\displaystyle 73}$.


${\displaystyle 21)\overline{1543}.}$
${\displaystyle \underline{1470}.}$
${\displaystyle 73.}$

From the table, again, ${\displaystyle 3}$ whole ${\displaystyle 21}$s, i.e. ${\displaystyle 63}$, go into ${\displaystyle 73}$, leaving ${\displaystyle 10}$.

${\displaystyle 73.}$
${\displaystyle 21)\overline{1543}.}$
${\displaystyle \underline{1470}}$
${\displaystyle 73.}$
${\displaystyle \underline{63}.}$
${\displaystyle 10.}$

This is the stopping point for divmod division, so that

For decimal division, one continues ${\displaystyle 21}$s into ${\displaystyle 100}$ go ${\displaystyle 4}$ times, i.e. ${\displaystyle 84}$ leaving ${\displaystyle 16}$.

${\displaystyle 73.}$
${\displaystyle 21)\overline{1543}.}$
${\displaystyle \underline{1470}}$
${\displaystyle 73.}$
${\displaystyle \underline{63}.}$
${\displaystyle 10.0}$
${\displaystyle \underline{8.4}}$
${\displaystyle 1.6}$

This continues until one has reached the desired degree of accuracy.