Complex Multiplication
Two complex numbers x=a+ib and y=c+id are multiplied as follows:
xy = (a+ib)(c+id)
(1)
= ac+ibc+iad-bd
(2)
= (ac-bd)+i(ad+bc).
(3)
In component form,
| (x,y)(x^',y^')=(xx^'-yy^',xy^'+yx^') |
(4)
|
(Krantz 1999, p. 1). The special case of a complex number multiplied by a scalar a is then given by
| a(x,y)=(a,0)(x,y)=(ax,ay). |
(5)
|
Surprisingly, complex multiplication can be carried out using only three real multiplications, ac, bd, and (a+b)(c+d) as
R[(a+ib)(c+id)] = ac-bd
(6)
I[(a+ib)(c+id)] = (a+b)(c+d)-ac-bd.
(7)
Complex multiplication has a special meaning for elliptic curves.
See also
Complex Addition, Complex Division, Complex Exponentiation, Complex Number, Complex Subtraction, Elliptic Curve, Imaginary Part, Multiplication, Real Part, SignExplore with Wolfram|Alpha
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References
Cox, D. A. Primes of the Form x2+ny2: Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1997.Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 1, 1999.Referenced on Wolfram|Alpha
Complex MultiplicationCite this as:
Weisstein, Eric W. "Complex Multiplication." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplexMultiplication.html