# Concept derivation guides

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Concept guide



## [Complex modulus and conjugate: proving z·z̄=|z|² and that conjugation preserves the operations](/en/derivations/derivation-mko-complex-modulus-conjugate-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-complex-modulus-conjugate)

For z=a+bi and w=c+di (a, b, c, d real) prove: (1) z·z̄=(a²+b²)+0i, the real embedding of the modulus squared; (2) conjugation preserves addition: (z+w)‾=z̄+w̄; (3) conjugation preserves multiplication: (zw)‾=z̄·w̄; (4) hence |zw|²=|z|²|w|². Then check everything exactly for z=3−4i and w=1+2i, distinguishing the modulus squared from the modulus.

Complex modulus and conjugation

derivation-mko-complex-modulus-conjugate-01





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