math

https://x.com/AlgebraFact/status/1802110136472817917

This identity simple, but shows important fact: the norm for complex numbers is multiplicative

(a+ci)*(b+di) = (ab-cd)+(ad+bc)i

norm(x+yi) = sqrt(x^2+y^2)

So

(ab-cd)^2+(ad+bc)^2 = norm((a+ci)*(b+di))

(a^2+c^2)(b^2+d^2) = norm(a+ci)*norm(b+di)

The two are equal

Fun fact: In quantum mechanics the state of a system can be described by a complex-valued wavefunction where its norm represents probabilities

The multiplicative property ensures that probabilities remain consistent under complex transformations—such as unitary operations—foundational to quantum state evolution