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. 2 + 2i

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 coefficient of the imaginary unit 'i'. In the given example, 2 + 2i, the real part is 2 and the imaginary part is also 2. Understanding complex numbers is essential for visualizing them on the complex plane.

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). It is represented as r(cos θ + i sin θ) or r e^(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 to polar form is crucial for operations involving complex numbers.

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 2 + 2i, the magnitude can be calculated using the formula r = √(a² + b²), and the argument can be found using the arctan function. These concepts are fundamental for converting complex numbers to polar form.