The following complementary multiplication method is reported to be used by various Arab and Persian writers ( [1] , p. 120). It is based on the identity
(1)
For instance,
or shortly
The method was recommended also by German reckoners like Widmann (1489), Riese (1522), Rudolff (1526), and Scheubel (1545). Widmann, Rudolff, and Grammateus (1518) also used the relation ([1], p. 121)
(2)
For instance,
For some other algebraic identities which can be used “to simplify” the multiplication consult [1], p. 122.
References
[1]
Smith, D. E. (1925). History of mathematics. Vol. II: Special topics of elementary mathematics. Boston: Ginn & Co.