Real and Complex Products of Complex NumbersComplex numbers, as an ordered pair of real numbers, can be identified with either points or vectors in the plane. The Argand diagram, with its axes and the origin, explicitly associates complex numbers with points. The operations of addition and multiplication highlight the implicit vector link. This is especially obvious in the case of addition, which is illustrated graphically with the parallelogram rule. Two uncommon borrowings from the vector algebra have been demonstrated in a recent book by T. Andreescu to provide powerful tools for solving complex number problems. Real ProductThe real product is a complex number incarnation of the scalar product (and this is what it is called in Gardiner, Bradley, p. 239]):
It is important (although difficult) to distinguish between the symbols for the common symbol of multiplication Assuming z = z1 + iz2 and w = w1 + iw2,
so that
which we may call the real form of the real product, for it shows immediately that the real product is always a real number. Since
which is not as symmetric as (2), but may be computationally useful. There is also a symmetric derivation:
The real product has the following properties:
Maxwell's theorem serves a nice application for #5. Complex ProductThe complex product is the incarnation of the vector product in complex terms (in [Gardiner, Bradley, p. 239] the product is designated exterior):
We can verify that:
which says that z×w is purely imaginary and justifies the terminology. The complex product have the following properties:
... to be continued ... References
Complex Numbers
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