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Complementary multiplication

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

a b = 10 b - (10 - a) b .(1)

For instance,

7 × 8= 10 × 8 - (10 - 7) × 8

= 10 × 8 - 3 × 8

= 80 - 24

= 56 ,

or shortly

10 × 8 - 3 × 8 = 80 - 24 = 56.

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)

a b= 10 b - 10 (10 - a) + (10 - a) (10 - b)
(2)

= 10 (b - (10 - a)) + (10 - a) (10 - b) .

For instance,

7× 8 = 10 × 8 - 10 × (10 - 7) + (10 - 7) (10 - 8)

= 80 - 30 + 6

= 56 .

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.

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Štefan Porubský: Complementary multiplication..

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