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 `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 `1543` by `21`, using the multiplication table

`\qquad\qquad\qquad[(0, 0),(1, 21),(2,42),(3,63),(4,84),(5,105),(6,126),(7,147),(8,168),(9,189)]`.

Start by writing

`\qquad\qquad\qquad\qquad\quad"****."`
`\qquad\qquad\qquad 21 )\bar1543.`

From the table, see that no whole `21`s go into `1` or into `15`, but that `7` whole `21`s, but not `8`, go into `154` :

`\qquad\qquad\qquad\qquad\qquad\quad7"*."`
`\qquad\qquad\qquad 21 )\bar1543.`
`\qquad\qquad\qquad\qquad\quad147 .`

Subtract `1470` from `1543` leaving `73`.

`\qquad\qquad\qquad\qquad\qquad\quad7"*."`
`\qquad\qquad\qquad 21 )\bar1543.`
`\qquad\qquad\qquad\qquad\quad\ul1470.`
`\qquad\qquad\qquad\qquad\qquad\quad73.`

From the table, again, `3` whole `21`s, i.e. `63`, go into `73`, leaving `10`.

`\qquad\qquad\qquad\qquad\qquad\quad73.`
`\qquad\qquad\qquad 21 )\bar1543.`
`\qquad\qquad\qquad\qquad\quad\ul1470`
`\qquad\qquad\qquad\qquad\qquad\quad73.`
`\qquad\qquad\qquad\qquad\qquad\quad\ul63.`
`\qquad\qquad\qquad\qquad\qquad\quad10.`

This is the stopping point for divmod division, so that `1543 :- 21 = 73 " remainder " 10.`

For decimal division, one continues `21`s into `100` go `4` times, i.e. `84` leaving `16`.

`\qquad\qquad\qquad\qquad\qquad\quad73.`
`\qquad\qquad\qquad 21 )\bar1543.`
`\qquad\qquad\qquad\qquad\quad\ul1470`
`\qquad\qquad\qquad\qquad\qquad\quad73.`
`\qquad\qquad\qquad\qquad\qquad\quad\ul63.`
`\qquad\qquad\qquad\qquad\qquad\quad10.0`
`\qquad\qquad\qquad\qquad\qquad\qquad\ul8.4`
`\qquad\qquad\qquad\qquad\qquad\qquad1.6`

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