1. Let's clarify the given expressions about the complex number $z$: - $z$ alone is the complex number itself. - The conjugate of $z$ is denoted by $\overline{z}$ and has the same real part but the imaginary part with opposite sign. - $|z|^2$ is the square of the magnitude of $z$. If $z = a + bi$, then $|z| = \sqrt{a^2 + b^2}$ and so $|z|^2 = a^2 + b^2$. - $|z|$ is the magnitude (or modulus) of the complex number $z$. 2. In summary: - $z = a + bi$ - $\overline{z} = a - bi$ - $|z|^2 = z \overline{z} = a^2 + b^2$ - $|z| = \sqrt{a^2 + b^2}$ These relationships are fundamental in complex number algebra and are often used in proofs and geometric interpretations.