Problem 14 - Algebra and Trigonometry

Question

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. $$ 2-2 i $$

Step-by-Step Solution

Answer

The polar form of the complex number 2 - 2i is \( \sqrt{8} (cos(-45°) + isin(-45°)) \).

Step 1: Plot the Complex Number

Firstly, plot the complex number 2 - 2i on the complex plane. The real part (2) corresponds to the x-axis and the imaginary part (-2) corresponds to the y-axis.

Step 2: Find the modulus of the complex number

The modulus of a complex number in the form a + bi is given by \( \sqrt{a^2 + b^2} \). For this complex number, the modulus can be calculated as follows: \( \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4 } = \sqrt{8} \).

Step 3: Calculate the argument of the complex number

The argument of a complex number, often denoted by theta (θ), is the angle it makes with the positive x-axis, and can be calculated by using the formula atan2(b,a). For this complex number, the argument can be calculated as follows: atan2(-2, 2), which results in -45 degrees or -π/4 rad.

Step 4: Write the complex number in polar form

The polar form of a complex number is written as \( r(cos(θ) + isin(θ)) \). This results in \( \sqrt{8} (cos(-45°) + isin(-45°)) \) for this complex number.

Key Concepts

Polar Form of Complex Numbers

Complex numbers, the sum of a real part and an imaginary part, can also be expressed in a different form known as polar form. This is particularly useful when performing multiplications, divisions, and finding powers and roots of complex numbers. Instead of using the rectangular coordinates (the 'a' and 'b' in 'a + bi'), the polar form uses r and θ, where r is the modulus and θ is the argument of the complex number.

In polar form, a complex number is represented as \( r(\cos(\theta) + i\sin(\theta)) \), where r corresponds to the distance of the number from the origin in the complex plane, and \theta indicates the counterclockwise angle it makes with the positive real axis. This shift from cartesian to polar coordinates provides a more geometric perspective on the relationship between complex numbers and their operations.

Modulus of a Complex Number

The modulus of a complex number, often seen as the 'absolute value' in the real number context, signifies the distance of the complex number from the origin in the complex plane. It is denoted by \( |z| \) and is calculated using the Pythagorean theorem, given the real part (a) and the imaginary part (b) of the complex number \( a + bi \).

The formula to find the modulus is \( |z| = \sqrt{a^2 + b^2} \). For our example, \( |2 - 2i| = \sqrt{2^2 + (-2)^2} = \sqrt{8} \). Understanding the modulus is vital when converting a complex number to polar form, as it represents the radius r in the polar coordinate system.

Visualizing Modulus on the Complex Plane

Imagine drawing a right triangle whose hypotenuse connects the origin to the point representing the complex number. The modulus is the length of this hypotenuse. In our case, the modulus is the distance from the origin to the point (2, -2).

Argument of a Complex Number

The argument of a complex number is the angle formed by the line joining the complex number to the origin with the positive direction of the x-axis. It provides a way of describing the direction in which the complex number is located from the origin of the complex plane.

To find the argument, denoted by \theta, one typically uses trigonometric functions. The common formula involves the atan2 function: \( \theta = atan2(b, a) \) which accounts for the signs of both the real and the imaginary parts of the complex number, ensuring the correct quadrant for the angle is selected.

For the example 2 - 2i, the argument calculated as \( atan2(-2, 2) \) yields -45 degrees or -π/4 radians. An important aspect of the argument is that it is not unique; adding multiples of \( 2π \) to an angle gives another equivalent argument because of the circular nature of the polar coordinate system.

Interpreting Angles in Different Quadrants

Since complex numbers can be located in any of the four quadrants of the complex plane, the angle must be adjusted accordingly to reflect its actual position. In our example, the complex number is in the fourth quadrant, hence the negative angle.