x = [p∠φ] = [p·→φ,p·↑φ] = [a,b]
y = [q∠ψ] = [q·→ψ,q·↑ψ] = [c,d]
Multiplication x·y = [(p·q)∠(φ+ψ)]
= [(p·q)·→(φ+ψ),(p·q)·↑(φ+ψ)]
= [p·q·(→φ·→ψ−↑φ·↑ψ), p·q·(↑φ·→ψ+→φ·↑ψ)]
= [p·→φ·q·→ψ−p·↑φ·q·↑ψ, p·↑φ·q·→ψ+p·→φ·q·↑ψ]
= [a·c−b·d, b·c+a·d]
Reciprocal 1/y = [(1/q)∠(-ψ)]
= [(1/q)·→(-ψ),(1/q)·↑(-ψ)]
= [1/q·→ψ,-1/q·↑ψ]
= [q·→ψ/q²,-q·↑ψ/q²]
= [c/(c²+d²),-d/(c²+d²)]
Division x/y = [(p/q)∠(φ−ψ)]
= [(p/q)·→(φ−ψ),(p/q)·↑(φ−ψ)]
= [(p/q)·(→φ·→ψ+↑φ·↑ψ), (p/q)·(↑φ·→ψ−→φ·↑ψ)]
= [p·q·(→φ·→ψ+↑φ·↑ψ)/q², p·q·(↑φ·→ψ−→φ·↑ψ)/q²]
= [(p·→φ·q·→ψ+p·↑φ·q·↑ψ)/q², (p·↑φ·q·→ψ−p·→φ·q·↑ψ)/q²]
= [(a·c+b·d)/q²,(b·c−a·d)/q²]
= [(a·c+b·d)/(c²+d²),(b·c−a·d)/(c²+d²)]
