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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, Sign

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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 Multiplication

Cite this as:

Weisstein, Eric W. "Complex Multiplication." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ComplexMultiplication.html

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