Problem 24 - 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. $$ -2+3 i $$

Step-by-Step Solution

Answer

The polar form of the complex number -2 + 3i is \(\sqrt{13} (cos-0.9828 + i sin -0.9828)\).

Step 1: Identify the Real and Imaginary Parts

First, identify the real and imaginary parts of the complex number. In this case, \(a = -2\) is the real part and \(b = 3\) is the imaginary part.

Step 2: Compute the Magnitude

Next, compute the magnitude of the complex number using the formula \r = \sqrt{a^2 + b^2}\. So, \(r = \sqrt{(-2)^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}\).

Step 3: Compute the Angle

The argument (angle) \(\theta\) of the complex number is defined by the formula \(\theta = atan2(b, a)\), where the function atan2 is the arctangent function, but takes into account the quadrant of the complex number in the complex plane. Therefore, \(\theta = atan2(3, -2)\), which we can decode as -56.31 degrees or -0.9828 in radians.

Step 4: Write in Polar Form

The polar form of complex number is given by \( r(cos \theta + i sin \theta) \). Thus, replacing the values we got from the previous steps, we get \(\sqrt{13} (cos-0.9828 + i sin -0.9828)\).

Key Concepts

Polar Form

The polar form of a complex number is an alternative way to express complex numbers. Instead of the usual rectangular form \(a + bi\), it utilizes the magnitude (distance from origin) and argument (angle) of the complex number.
In polar form, a complex number is written as \(r(\cos \theta + i \sin \theta)\), where:

• \(r\) is the magnitude or modulus.
• \(\theta\) is the argument, given either in degrees or radians.

The conversion from rectangular to polar form helps in visualizing and making calculations easier, especially multiplication and division.
When converting, always remember that the angle \(\theta\) is crucial, as it indicates the direction in which the complex number lies with respect to the real axis.

Magnitude

The magnitude of a complex number, denoted by \(r\), measures how far the point representing the complex number is from the origin in the complex plane.
To find the magnitude, use the formula \(r = \sqrt{a^2 + b^2}\), where \(a\) is the real part, and \(b\) is the imaginary part.
In our case for \(-2 + 3i\):

• Compute \((-2)^2 + 3^2 = 4 + 9 = 13\)
• Then \(r = \sqrt{13}\)

The magnitude is always a non-negative number and helps in the transformation into polar form by acting as the radius in the expression \(r(\cos \theta + i \sin \theta)\).

Argument (Angle)

The argument, or angle \(\theta\), is a measure of direction from the positive real axis to the complex number's vector in the complex plane.
This angle can be found using the arctangent function \(\theta = \text{atan2}(b, a)\), which safely handles any quadrant placement of the complex number.
For \(-2 + 3i\), the computation is:

• \(\theta = \text{atan2}(3, -2)\)
• The result is approximately -0.9828 radians or -56.31 degrees.

Understanding the argument is essential, especially when plotting complex numbers and converting into polar form, as it gives the exact direction of the complex number.

Imaginary Part

The imaginary part of a complex number is the component paired with the imaginary unit \(i\). It allows complex numbers to extend beyond the real numbers.
In the context of our example \(-2 + 3i\), the imaginary part is 3.

• The imaginary part is denoted as \(b\) in the rectangular form \(a + bi\).
• It impacts both the magnitude and the angle of the complex number when converting to polar form.

Thus, 3 represents the vertical component on the complex plane, and its value plays a key role in finding the argument and the overall relationship with the real part.

Real Part

The real part of a complex number is straightforward. It is the scalar part that exists on the real number line
In our expression \(-2 + 3i\), \(-2\) is the real part.
The real part can be;

• Denoted as \(a\) in a complex number written as \(a + bi\).
• It affects how we measure both the magnitude and argument of the number when expressed in polar form.

The real part \(-2\) here is critical in determining the direction in which the complex number lies, providing the horizontal measure that combines with the imaginary part.