For Z = R + jX, what is the magnitude |Z|?

• A. |Z| = R + X
• B. |Z| = sqrt(R^2 - X^2)
• C. |Z| = R - X
• D. |Z| = sqrt(R^2 + X^2)

Answer: (D)

The magnitude of a complex number is found from its geometric length in the complex plane, which uses the Pythagorean relationship. For a + jb, the magnitude is sqrt(a^2 + b^2). In Z = R + jX, the real part is R and the imaginary part is X, so the magnitude is |Z| = sqrt(R^2 + X^2). This works because squaring removes any sign of the imaginary part, so whether X is positive or negative doesn’t change the result. Quick checks: if R = 0, |Z| = |X|; if X = 0, |Z| = R.