In Problems 13-24, plot each complex number in the complex plane and write it in polar form. Exp

Step 1: Identify the real and imaginary parts of the complex number.u000AThe complex number given is

In Problems 13-24, plot each complex number in the complex...

Key Concepts

Complex Numbers

Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined by the relation i² = -1. They extend the concept of one-dimensional real numbers to the two-dimensional complex plane, enabling solutions to equations that have no real solutions.

Complex Plane

The complex plane, also known as the Argand diagram, is a two-dimensional plane where each complex number is represented by a point. The horizontal axis represents the real part of the complex number, while the vertical axis represents the imaginary part, allowing for a geometric interpretation of complex numbers.

Polar Form

The polar form of a complex number expresses the number in terms of its magnitude and angle relative to the positive real axis. It is typically written as r(cos ? + i sin ?) or re^(i?), where r is the modulus and ? is the argument.

Modulus

The modulus of a complex number, denoted by r, is the distance from the origin to the point representing the complex number in the complex plane. It is calculated using the formula r = ?(a² + b²), providing a measure of the number's magnitude.

Argument

The argument of a complex number is the angle, typically denoted by ?, between the positive real axis and the line connecting the origin with the point representing the complex number. The argument indicates the direction of the complex number in the plane.

Conversion from Rectangular to Polar

Conversion from the rectangular (a + bi) form to the polar form involves calculating the modulus using r = ?(a² + b²) and finding the argument by using trigonometric functions, typically ? = arctan(b/a). When necessary, adjustments are made to ? to determine the correct angle in the appropriate quadrant, often expressed in degrees.

In Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. $-2+3 i$

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