In Exercises 11–26, plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. −4i

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. In the case of the complex number -4i, the real part is 0 and the imaginary part is -4. Understanding complex numbers is essential for visualizing them on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.

Polar Form of Complex Numbers

The polar form of a complex number expresses it in terms of its magnitude (or modulus) and angle (or argument) relative to the positive real axis. It is represented as r(cos θ + i sin θ) or re^(iθ), where r is the distance from the origin to the point in the complex plane, and θ is the angle formed with the positive real axis. Converting a complex number to polar form is crucial for operations like multiplication and division.

Argument and Magnitude

The argument of a complex number is the angle θ formed with the positive real axis, while the magnitude is the distance from the origin to the point representing the complex number. For the complex number -4i, the magnitude is 4, calculated as √(0² + (-4)²), and the argument is -90 degrees or 270 degrees, indicating its position on the negative imaginary axis. Understanding these concepts is vital for accurately plotting and converting complex numbers.