Since ⟨ ⟨ f ∣ ∣ g ⟩ ⟩ is in general a complex number, we use the fact that the modulus squared of any complex number z is defined as | z | 2 = z z ∗ ∗ , where z ∗ ∗ is the complex conjugate of z . The modulus squared can also be expressed as