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.−3 + 4i

Verified step by step guidance

1. Step 1: Identify the real and imaginary parts of the complex number. Here, the real part is -3 and the imaginary part is 4.
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 (-3, 4).
3. Step 3: Calculate the magnitude (or modulus) of the complex number using the formula \( r = \sqrt{a^2 + b^2} \), where \( a = -3 \) and \( b = 4 \).
4. Step 4: Determine the argument (angle) of the complex number using the formula \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). Adjust the angle based on the quadrant where the complex number is located.
5. Step 5: Write the complex number in polar form as \( r(\cos \theta + i\sin \theta) \) or \( re^{i\theta} \), 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 this case, the complex number −3 + 4i consists of a real part of -3 and an imaginary part of 4. Understanding how to represent and manipulate complex numbers is essential for converting them into polar form.

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 a complex number to polar form involves calculating these two components.

Magnitude and Argument

The magnitude of a complex number is calculated using the formula r = √(a² + b²), which gives the distance from the origin to the point represented by the complex number in the complex plane. The argument, θ, is the angle formed with the positive real axis, calculated using θ = arctan(b/a). Understanding how to compute both the magnitude and argument is crucial for accurately converting a complex number into its polar form.