If z = x + i y, which expression equals |z|?

If z = x + i y, which expression equals |z|?

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

Answer: (D)

Think of z as the point (x, y) in the complex plane. The modulus |z| is its distance from the origin, so by the Pythagorean theorem you get |z| = sqrt(x^2 + y^2). This is the quantity that measures the actual length from (0, 0) to (x, y). The other forms don’t match that distance: sqrt(x^2 - y^2) isn’t the distance and can be imaginary; sqrt((x - y)^2) equals |x - y|, not the Euclidean distance from the origin; and sqrt(x^2) + sqrt(y^2) equals |x| + |y|, which is a different measure (the sum of absolute values).

Think of z as the point (x, y) in the complex plane. The modulus |z| is its distance from the origin, so by the Pythagorean theorem you get |z| = sqrt(x^2 + y^2). This is the quantity that measures the actual length from (0, 0) to (x, y).

The other forms don’t match that distance: sqrt(x^2 - y^2) isn’t the distance and can be imaginary; sqrt((x - y)^2) equals |x - y|, not the Euclidean distance from the origin; and sqrt(x^2) + sqrt(y^2) equals |x| + |y|, which is a different measure (the sum of absolute values).