If z = x + i y, which of the following equals |z|^2?

If z = x + i y, which of the following equals |z|^2?

• A. x^2 + y^2
• B. x^2 - y^2
• C. x^2
• D. x^2 + 2xy

Answer: (A)

The quantity |z|^2 represents the square of the distance from the origin to the point (x, y) in the complex plane. For z = x + i y, that distance is sqrt(x^2 + y^2), so squaring it gives |z|^2 = x^2 + y^2. This also comes from multiplying by the complex conjugate: |z|^2 = z z̄ = (x + i y)(x − i y) = x^2 + y^2. The other expressions don’t match the modulus squared. x^2 − y^2 is the real part of z^2, not a distance measure. x^2 is missing the y contribution, and x^2 + 2xy contains a cross term that the modulus squared does not have.

The quantity |z|^2 represents the square of the distance from the origin to the point (x, y) in the complex plane. For z = x + i y, that distance is sqrt(x^2 + y^2), so squaring it gives |z|^2 = x^2 + y^2.

This also comes from multiplying by the complex conjugate: |z|^2 = z z̄ = (x + i y)(x − i y) = x^2 + y^2.

The other expressions don’t match the modulus squared. x^2 − y^2 is the real part of z^2, not a distance measure. x^2 is missing the y contribution, and x^2 + 2xy contains a cross term that the modulus squared does not have.