Two complex numbers and are multiplied as follows:
|
(1) | |||
|
(2) | |||
|
(3) |
In component form,
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(4) |
(Krantz 1999, p. 1). The special case of a complex number multiplied by a scalar is then given by
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(5) |
Surprisingly, complex multiplication can be carried out using only three real multiplications, , , and as
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.
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Cite this as:
Weisstein, Eric W. "Complex Multiplication." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplexMultiplication.html