The argument of a complex number is defined as what?

The argument of a complex number is defined as what?

• A. Angle between the positive real axis and the line joining that number to the origin
• B. Distance from origin
• C. Real part
• D. Imaginary part

Answer: (A)

The argument describes the direction of the complex number in the plane. It is the angle formed by the positive real axis and the line from the origin to the point representing the number, usually measured counterclockwise. This angle appears in the polar form z = r(cos theta + i sin theta) or z = r e^{i theta}, where r is the distance from the origin (the modulus) and theta is the argument. The distance from the origin is a separate quantity, so it’s not the angle. The real part and imaginary part tell you the horizontal and vertical coordinates, not the direction. For the origin itself, the direction is not defined, so the argument is undefined there.

The argument describes the direction of the complex number in the plane. It is the angle formed by the positive real axis and the line from the origin to the point representing the number, usually measured counterclockwise. This angle appears in the polar form z = r(cos theta + i sin theta) or z = r e^{i theta}, where r is the distance from the origin (the modulus) and theta is the argument. The distance from the origin is a separate quantity, so it’s not the angle. The real part and imaginary part tell you the horizontal and vertical coordinates, not the direction. For the origin itself, the direction is not defined, so the argument is undefined there.