[Verse 1]
Numbers dance across a plane, two dimensions wide
Horizontal holds the real, vertical the imaginary side
Plot a point where they collide, z equals a plus b times i
Every complex number lives where axes cross and multiply
[Chorus]
Distance from the origin, modulus we call
Square root of a squared plus b squared tells us all
Conjugate's a mirror flip across the real line
Multiply them together, modulus squared you'll find
[Verse 2]
Polar form reveals the truth hidden in the mist
R times cosine theta plus i sine theta twist
Euler's formula shows the way, r times e to power i theta
Magnitude and angle dance, complex numbers' secret data
[Chorus]
Distance from the origin, modulus we call
Square root of a squared plus b squared tells us all
Conjugate's a mirror flip across the real line
Multiply them together, modulus squared you'll find
[Bridge]
When complex numbers multiply in polar dress
Moduli combine while angles coalesce
Rotation meets the scaling force
Geometry shows multiplication's course
[Verse 3]
Z one times z two equals r one r two times e raised high
To i times theta one plus theta two, watch the product fly
Multiply the distances, add the angles clean
Complex multiplication's geometric scene
[Outro]
On the Argand diagram they spin and grow
Rotation scaling, now you know
Complex plane geometry, beautiful and true
Mathematical elegance shining through