Correct Answer:
A. x = 0
The correct answer is A: x = 0. A complex number is generally expressed in the form z = x + iy, where x represents the real part of the number and y represents the imaginary part (with i being the imaginary unit, √-1). For a complex number to be considered purely imaginary, its real part must be zero. If x = 0, the number simplifies to z = 0 + iy = iy. Examples of such numbers include 3i, -5i, or simply i, all of which have a real part of zero. While for a *purely* imaginary number, y must also be non-zero to exclude the number zero itself, the fundamental condition for a complex number to be categorized as imaginary (meaning it lies on the imaginary axis) is that its real component, x, is zero.
- B: y = 0 is incorrect. If y = 0, the complex number becomes x + 0i = x, which is a purely real number. For example, 5 or -2 are real numbers, not imaginary ones.
- C: x ≠ 0 is incorrect. If x ≠ 0, it means the real part is non-zero. This implies the number is a general complex number with both a real and an imaginary component (e.g., 3 + 2i), not specifically an imaginary number.
- D: y = 1 is incorrect. This condition means the imaginary part is specifically 1 (e.g., x + i). This doesn't inherently make the number imaginary unless x = 0. An imaginary number can have any non-zero real value for its imaginary part (e.g., 2i, -7i, or 0.5i), so specifying y = 1 is too restrictive and not the defining characteristic.