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

Verified step by step guidance

1. Step 1: Identify the real and imaginary parts of the complex number. Here, the real part is 2 and the imaginary part is 2.
2. Step 2: Plot the complex number on the complex plane. The x-axis represents the real part, and the y-axis represents the imaginary part. Plot the point (2, 2).
3. Step 3: Calculate the magnitude (or modulus) of the complex number using the formula $r=\sqrt{a^2+b^2}$, where $a$ is the real part and $b$ is the imaginary part.
4. Step 4: Determine the argument (angle) of the complex number using the formula $\theta ={\tan }^{-1}(\frac{b}{a})$.
5. Step 5: Write the complex number in polar form as $r(\cos \theta +i\sin \theta )$ or $r{e}^{i\theta }$, using the magnitude and argument calculated in the previous steps.

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.