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

Verified step by step guidance

1. Step 1: Identify the complex number in the form a + bi. Here, the complex number is 0 - 4i, where a = 0 and b = -4.
2. Step 2: Plot the complex number on the complex plane. The real part (a) is 0, so it lies on the imaginary axis at -4.
3. Step 3: Calculate the magnitude (or modulus) of the complex number using the formula \( r = \sqrt{a^2 + b^2} \).
4. Step 4: Determine the argument (angle) of the complex number. Since the number is on the negative imaginary axis, the argument is \( \theta = \frac{3\pi}{2} \) radians or 270 degrees.
5. Step 5: Write the complex number in polar form as \( r(\cos \theta + i\sin \theta) \) or \( re^{i\theta} \).

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.