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

Verified step by step guidance

1. Step 1: Identify the complex number. In this case, the complex number is \(-3\), which can be written as \(-3 + 0i\).
2. Step 2: Plot the complex number on the complex plane. The real part is \(-3\) and the imaginary part is \(0\), so the point is located on the real axis at \(-3\).
3. Step 3: Determine 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: Calculate the argument (angle) of the complex number. Since the number is on the negative real axis, the argument is \(\pi\) radians or \(180^\circ\).
5. Step 5: Write the complex number in polar form using the formula \(z = r(\cos \theta + i\sin \theta)\), 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 imaginary part. In this context, the complex number -3 can be represented as -3 + 0i, indicating that it lies on the real axis of 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. For -3, the polar form can be calculated using its modulus and argument.

Magnitude and Argument

The magnitude of a complex number is the distance from the origin to the point in the complex plane, calculated using the formula r = √(a² + b²). The argument is the angle formed with the positive real axis, which can be found using the arctangent function. For the complex number -3, the magnitude is 3, and the argument is π radians (or 180 degrees), as it lies on the negative real axis.