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Cauchy’s complementary multiplication

Cauchy [1]  proposed the following method to make the computation of the product typeset structure of two positive integers more comfortable.

Write the sum typeset structure in a different way again as a sum of two summands, say, typeset structure.  Then

x y = x^' y^' + (x - x^') (y - x^')(1)

and similarly

x y = x^' y^' + (x - y^') (y - y^') .(2)

For instance, to compute the product typeset structure we can write

23 + 67 = 30 + 60.(3)

Then

23 × 67 = 30 × 60 + (23 - 30) (67 - 30) = 1800 - 7 × 37 = 1800 - 259 = 1541.

To obtain really a simplification of the computation of the final product to find the suitable decomposition typeset structure is crucial. For instance, if we write

23 + 67 = 40 + 50

then we get

23 × 67 = 40 × 50 + (23 - 40) (67 - 40) = 2000 - 17 × 27 = 2000 - 459 = 1541.

Cauchy demonstrated his idea using two examples. To compute typeset structure he took the decomposition typeset structure. Then

609 × 616 = 600 × 625 + 9 × 16 = 375000 + 144 = 375144 .

Similarly, in the case of the square typeset structure he used the decomposition typeset structurewhich leads to

9987^2 = 9974 × 10000 + 13^2 = 99740000 + 169 = 99740169 .

A special case of the above method is the following one. Write typeset structure, typeset structure, and take typeset structure, typeset structure. Then

x y = 2 z(a + b) + (z - a) (z - b) .(4)

The choice typeset structure yields the rule called regula ignavi, that is the formula

(5 + a) (5 + b) = 10 (a + b) + (5 - a) (5 - b) .(5)

This formula is the base for the “gypsy multiplication”. For instance, typeset structure.  

References

[1]  Cauchy, A. (1840). Sur les moyens d'éviter les erreurs dans les calculs numériques. Comp. Rendus , 11, 431-442.

Cite this web-page as:

Štefan Porubský: Cauchy complementary multiplication.

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