Problem 16 - Advanced Engineering Mathematics

Question

Find \(z_{1} z_{2}\) and \(z_{1} / z_{2} .\) Write the number in the form \(a+i b\). $$ \begin{aligned} &z_{1}=\sqrt{2}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right) \\ &z_{2}=\sqrt{3}\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right) \end{aligned} $$

Step-by-Step Solution

Answer

The product is \(1.225 + 2.121i\); the quotient is \(0.707 + 0.075i\).

Step 1: Identify Given Complex Numbers

We are given two complex numbers in their polar forms: \( z_1 = \sqrt{2} ( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} ) \) and \( z_2 = \sqrt{3} ( \cos \frac{\pi}{12} + i \sin \frac{\pi}{12} ) \). We will convert these into their standard forms for multiplication and division.

Step 2: Convert Polar to Rectangular Form

The rectangular form of a complex number is given by \( a + ib \). For \( z_1 \), use \( a = r \cos \theta \) and \( b = r \sin \theta \): \( z_1 = \sqrt{2} \left( \frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} \right) = 1 + i \). For \( z_2 \), calculate \( \cos \frac{\pi}{12} \approx 0.9659 \) and \( \sin \frac{\pi}{12} \approx 0.2588 \), giving us \( z_2 = \sqrt{3}(0.9659 + 0.2588i) \approx 1.673 + 0.448i \).

Step 3: Multiply Complex Numbers

To find \( z_1z_2 \), multiply \( (1 + i)(1.673 + 0.448i) \). Use the distributive property: \( 1.673 + 0.448i + 1.673i + 0.448i^2 \). Recall, \( i^2 = -1 \), so the expression becomes \( 1.673 + 0.448i + 1.673i - 0.448 \), which simplifies to \( (1.673 - 0.448) + (0.448 + 1.673)i = 1.225 + 2.121i \).

Step 4: Divide Complex Numbers

To find \( \frac{z_1}{z_2} \), use the formula \( \frac{a+ib}{c+id} = \frac{(ac+bd) + i(bc-ad)}{c^2+d^2} \). Using \( z_1 = 1 + i \) and \( z_2 = 1.673 + 0.448i \), calculate the denominator: \( 1.673^2 + 0.448^2 = 2.999 \). The numerator is \( (1 \times 1.673 + 1 \times 0.448) + i(1 \times 0.448 - 1 \times 1.673) = 2.121 + 0.224i \). Therefore, \( \frac{z_1}{z_2} = \frac{2.121 + 0.224i}{2.999} \approx 0.707 + 0.075i \).

Key Concepts

Polar to Rectangular Conversion

Complex numbers can be represented in various ways. One of the most common representations is the polar form, which uses a radius (magnitude) and angle to define a point in the complex plane. The polar form of a complex number is often written as \( r(\cos(\theta) + i\sin(\theta)) \), where \( r \) is the magnitude and \( \theta \) is the angle.
To use complex numbers in calculations, like multiplication or division, it is often easier to convert them to rectangular form. This form is expressed as \( a + ib \), where \( a \) and \( b \) are real numbers.
Converting from polar to rectangular form involves using the trigonometric functions, cosine and sine. Given \( r \) and \( \theta \):

• Calculate \( a \) using \( a = r \cos(\theta) \)
• Calculate \( b \) using \( b = r \sin(\theta) \)

For example, converting \( z_1 = \sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}) \) involves calculating \( a = \sqrt{2} \times \frac{1}{\sqrt{2}} = 1 \) and \( b = \sqrt{2} \times \frac{1}{\sqrt{2}} = 1 \), giving us \( z_1 = 1 + i \).
This simplification makes subsequent operations like multiplication and division more straightforward.

Multiplication of Complex Numbers

Multiplying complex numbers is straightforward in their rectangular forms. The operation resembles multiplying two binomials in algebra, using the distributive property.
Consider two complex numbers \( (a + ib) \) and \( (c + id) \). The multiplication results in:

• Real part: \( ac - bd \)
• Imaginary part: \( ad + bc \)

This arises because \( i^2 = -1 \), which changes the sign of the product \( bd \).
In our example, multiplying \( z_1 = 1 + i \) and \( z_2 = 1.673 + 0.448i \), we calculate:

• Real part: \( 1 \times 1.673 - 1 \times 0.448 = 1.225 \)
• Imaginary part: \( 1 \times 0.448 + 1 \times 1.673 = 2.121 \)

So, \( z_1z_2 = 1.225 + 2.121i \).
This method simplifies calculations and is essential for working with complex numbers in engineering, physics, and mathematics.

Division of Complex Numbers

Dividing complex numbers involves multiplying the numerator and denominator by the complex conjugate of the denominator. This process simplifies the denominator to a real number and makes the division manageable.
For two complex numbers \( z_1 = a + ib \) and \( z_2 = c + id \), to find \( \frac{z_1}{z_2} \):

• Multiply both numerator and denominator by \( c - id \) (the conjugate of \( z_2 \)).
• The new denominator becomes \( c^2 + d^2 \).
• The new numerator is \( (ac + bd) + i(bc - ad) \).

In our example, dividing \( z_1 = 1 + i \) by \( z_2 = 1.673 + 0.448i \):

• Calculate \( 2.121 + 0.224i \), the result of the numerator \((1\times1.673 + 1\times0.448) + i(1\times0.448 - 1\times1.673) \).
• Divide by the real number \( 2.999 \) (\( 1.673^2 + 0.448^2 \)).

Resulting in \( 0.707 + 0.075i \).
Using the conjugate simplifies division, turning it into a series of arithmetic operations.

Rectangular Form of Complex Numbers

The rectangular form of a complex number most literally connects to the Cartesian coordinate system. It's expressed as \( a + ib \), where \( a \) represents the real part and \( b \) represents the imaginary part.
In applications, this form allows for straightforward addition, subtraction, and comparisons between complex numbers. Each complex number can be visualized as a point where \( a \) is the x-coordinate and \( b \) is the y-coordinate on the complex plane.
This practical representation sets the stage for operating on and visualizing complex numbers, making it one of the most usable forms in mathematics.
As seen in our examples, \( z_1 = 1 + i \) and \( z_2 = 1.673 + 0.448i \) are easy to handle, offering a direct path to multiplication and division, reflecting their origin from the polar form to this direct \( a + ib \) expression.

• This allows for easy plotting and understanding of operations which are crucial in fields like electrical engineering and quantum physics.