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Complex Multiplication

Two Complex Numbers $x=a+ib$ and $y=c+id$ are multiplied as follows:

$\displaystyle xy$ $\textstyle =$ $\displaystyle (a+ib)(c+id)= ac+ibc+iad-bd$  
  $\textstyle =$ $\displaystyle (ac-bd)+i(ad+bc).$  

However, the multiplication can be carried out using only three Real multiplications, $ac$, $bd$, and $(a+b)(c+d)$ as

\Re[(a+ib)(c+id)] &=& ac-bd\\
\Im[(a+ib)(c+id)] &=& (a+b)(c+d)-ac-bd.

Complex multiplication has a special meaning for Elliptic Curves.

See also Complex Number, Elliptic Curve, Imaginary Part, Multiplication, Real Part


Cox, D. A. Primes of the Form $x^2 + ny^2$: Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1997.

© 1996-9 Eric W. Weisstein