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

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1. Identify the complex number given: \$1 - i\(. Here, the real part is \(1\) and the imaginary part is \)-1$.
2. Plot the complex number on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. The point will be at coordinates \((1, -1)\).
3. Calculate the modulus (or magnitude) \(r\) of the complex number using the formula \(r = \sqrt{(\text{real part})^2 + (\text{imaginary part})^2}\). Substitute the values to get \(r = \sqrt{1^2 + (-1)^2}\).
4. Find the argument (or angle) \(\theta\) of the complex number using \(\theta = \tan^{-1}\left(\frac{\text{imaginary part}}{\text{real part}}\right)\). Substitute the values to get \(\theta = \tan^{-1}\left(\frac{-1}{1}\right)\).
5. Write the complex number in polar form as \(r(\cos \theta + i \sin \theta)\) or \(r \operatorname{cis} \theta\), where \(r\) is the modulus and \(\theta\) is the argument you calculated.

Key Concepts

Complex Numbers and the Complex Plane

A complex number is expressed as a + bi, where a is the real part and b is the imaginary part. It can be represented as a point or vector in the complex plane, with the x-axis as the real axis and the y-axis as the imaginary axis. Plotting involves locating the point (a, b) on this plane.

Polar Form of Complex Numbers

Polar form expresses a complex number using its magnitude (distance from the origin) and argument (angle with the positive real axis). It is written as r(cos θ + i sin θ) or r∠θ, where r = √(a² + b²) and θ = arctangent(b/a). This form highlights the number's geometric properties.

Calculating the Argument (Angle)

The argument θ of a complex number is the angle between the positive real axis and the line connecting the origin to the point (a, b). It can be found using θ = arctan(b/a), adjusted for the correct quadrant. The angle can be expressed in degrees or radians depending on the context.