If z = x + iy, then |z| equals which expression?

If z = x + iy, then |z| equals which expression?

• A. x^2 + y^2
• B. sqrt(x^2 + y^2)
• C. sqrt(x^2 - y^2)
• D. x + y

Answer: (B)

The modulus is the distance from the origin to the point (x, y) representing z in the complex plane. That distance is found by the Pythagorean theorem, giving |z| = sqrt(x^2 + y^2). Another way to see it is that |z|^2 = z times its complex conjugate: (x + iy)(x - iy) = x^2 + y^2, so taking the square root yields the same result. This value is always nonnegative and matches the geometric distance from the origin to (x, y). The other expressions don’t represent a distance: x^2 + y^2 is the square of the modulus, sqrt(x^2 - y^2 isn’t meaningful for the modulus, and x + y is just a sum of coordinates.

The modulus is the distance from the origin to the point (x, y) representing z in the complex plane. That distance is found by the Pythagorean theorem, giving |z| = sqrt(x^2 + y^2). Another way to see it is that |z|^2 = z times its complex conjugate: (x + iy)(x - iy) = x^2 + y^2, so taking the square root yields the same result. This value is always nonnegative and matches the geometric distance from the origin to (x, y). The other expressions don’t represent a distance: x^2 + y^2 is the square of the modulus, sqrt(x^2 - y^2 isn’t meaningful for the modulus, and x + y is just a sum of coordinates.