Recursive division by Burnikel and Ziegler, explaining the breaking down of large numbers

Question

I am looking at Fast Recursive Division by Burnikel and Ziegler.

I understand $DivTwoDigitsByOne( ... )$ and $DivThreeHalvesByTwo( ... )$ as they break the numbers down.

So, for example, $12345678/1234$ would be $DivTwoDigitsByOne( 12, 34, 56, 78, 12, 34 )$

And the same for DivThreeHalvesByTwo( ... )

The part I don't get is the $RecursiveDivision( ... )$ part, (page 12), how is the big number broken into 2 numbers.

If I have a very large number $1234567812345678 / 12345678$, how do I break it down into 2 halves?

DivTwoDigitsByOne( 12, 34, 56, 78, 12, 34 )

and

DivTwoDigitsByOne( 12, 34, 56, 78, 56, 78 )

I would I then join the 2 quotients/remainder to form one?

Can someone explain the splitting of the numbers in the $RecursiveDivision( ... )$ case.