Egyptian Mathematics for Multiplication

Derivation:

Consider two numbers  and , where ‘a’ is the radix. Our aim is to compute the product of two given numbers as well as execute the division operation between them using the ancient Egyptian mathematics. The product P can be expressed as,

                                                                                              (2.1)

     (2.2)

Where,  and k₁ is either ‘0’ or ‘1’. Equation (2) can be reformulated as,

                                                                                   (2.3)

                                                                      (2.4)

Where,  and k₂ is either ‘0’ or ‘1’. Similarly,  can be decomposed as,

                                                                                                                  (2.5)

Where, k₃ is either ‘0’ or ‘1’. Putting equation (5) in equation (4), we get,

                                (2.6)

     (2.7)

This will continue until ‘n’ cannot be divisible by 2^q where q is an integer. So at last it can be written as,

   (2.8)

In Binary mathematics, a=2, q=n and k_n=x_n-1. So equation (8) can be reformulated as,

        (2.9)

                                                                                                                   (2.10)