Complex trickery

Not particularly interesting, just something I found nice when doing some questions…

Conjugates

Anybody remotely familiar with complex numbers ought to know about conjugates and conjugate pairs, if not then what are you doing?!

In essence the bits I quite like are the formulae pertaining to how conjugates can be manipulated and the sometimes surprising applications this can have on (2D) vectors. Note, 2D vectors and complex numbers are virtually identical. I’ve sometimes imagined that there are some kind of complexer numbers out there using vector representations of greater order, but this just seems utterly insane and without practical application.

I like them just because the applications to real geometry can be genuinely useful, even if these do not immediately make intuitive sense (for complex numbers \(z\) and \(w\)):

• \(zz^\ast = |z|^2\)
• \((w+z)^\ast=w^\ast+z^\ast\)
• \((wz)^\ast=w^\ast z^\ast\)
• \(z={1\over w^\ast} \Longleftrightarrow z^\ast ={1\over w}\)

and so on.

Of course it goes without saying that the proofs are left as an exercise to the reader (but really, they are easy).

Some STEP questions where this sort of thing has been useful include:

• 2005 S3 Q8
• 2013 S3 Q6
• 2023 S2 Q7

(There are more out there, especially geometrically).

I also reckon that implications of some of this stuff on De Moivre’s Theorem will be interesting, though I am probably not intending to post my investigations on this since the typesetting would be nightmarish.