In Exercises 11–14, plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. 1 − i

Verified step by step guidance

1. Step 1: Identify the complex number. The given complex number is $1 - i$, where the real part is 1 and the imaginary part is -1.
2. Step 2: Plot the complex number on the complex plane. The real part (1) is plotted on the x-axis, and the imaginary part (-1) is plotted on the y-axis.
3. Step 3: Calculate the modulus (magnitude) of the complex number using the formula $|z| = \sqrt{a^2 + b^2}$, where $a$ is the real part and $b$ is the imaginary part.
4. Step 4: Determine the argument (angle) of the complex number. Use the formula $\theta = \tan^{-1}(\frac{b}{a})$, where $a$ is the real part and $b$ is the imaginary part. Adjust the angle based on the quadrant in which the complex number lies.
5. Step 5: Write the complex number in polar form as $z = r(\cos \theta + i\sin \theta)$ or $z = re^{i\theta}$, where $r$ is the modulus and $\theta$ is the argument.

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', which is defined as the square root of -1. In the given question, the complex number 1 - i has a real part of 1 and an imaginary part of -1.

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. This form is particularly useful for multiplication and division of complex numbers.

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 (a, b) in the complex plane. The argument, θ, is the angle formed with the positive real axis, found using θ = arctan(b/a). This angle can be expressed in degrees or radians, depending on the context.