Problem 35 - Precalculus with Limits

Question

Writing a Complex Number in Standard Form In Exercises \(31-40\) , write the standard form of the complex number. Then represent the complex number graphically. $$\frac{9}{4}\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right)$$

Step-by-Step Solution

Answer

Complex number in standard form is \( \frac{-9\sqrt{2}}{8} + i\frac{9\sqrt{2}}{8} \). The point on the graph representing the complex number is at (\( \frac{-9\sqrt{2}}{8}, \frac{9\sqrt{2}}{8} \)).

Step 1: Transform the Polar Form to Standard Form

In standard form, the complex number is written as \( a + bi \). Where \( a \) is the real part, and \( b \) is the imaginary part. The polar form is written as \( r(\cos\theta + i\sin\theta)\), where \( r \) is the modulus of the complex number and \( \theta \) is the argument or angle made by the complex number with the positive real axis. In the given problem, \( r = \frac{9}{4} \), and \( \theta = \frac{3\pi}{4} \). To convert from polar form to standard form, the real part \( a \) will be \( r\cos\theta = \frac{9}{4}\cos \frac{3\pi}{4} = \frac{-9\sqrt{2}}{8} \), and the imaginary part \( b \) will be \( r\sin\theta = \frac{9}{4}\sin \frac{3\pi}{4} = \frac{9\sqrt{2}}{8} \). So, the standard form is \( a + bi = \frac{-9\sqrt{2}}{8} + i\frac{9\sqrt{2}}{8} \)

Step 2: Plot the Complex Number Graphically

To graphically represent the complex number, plot the point on the complex plane where the x-coordinate corresponds to the real part (\( a \)) and the y-coordinate corresponds to the imaginary part (\( b \)). So, the point for this complex number will be at (\( \frac{-9\sqrt{2}}{8}, \frac{9\sqrt{2}}{8} \)). This point represents the given complex number graphically.

Key Concepts

Polar to Rectangular Conversion

When dealing with complex numbers, it is often necessary to switch between different forms. The polar form and rectangular (or standard) form are two popular representations.In polar form, a complex number is expressed as \( r(\cos \theta + i \sin \theta) \), where \( r \) represents the modulus (or magnitude) and \( \theta \) is the angle (or argument) measured from the positive real axis.To convert from polar to rectangular form, use the following steps:

• Find the real part: \( a = r \cos \theta \)
• Find the imaginary part: \( b = r \sin \theta \)

These correspond to the coordinates on the complex plane where the number will be plotted.As an example, consider a complex number with \( r = \frac{9}{4} \) and \( \theta = \frac{3\pi}{4} \). Calculating the real and imaginary components gives us \( a = \frac{-9\sqrt{2}}{8} \) and \( b = \frac{9\sqrt{2}}{8} \), making the rectangular form \( \frac{-9\sqrt{2}}{8} + i\frac{9\sqrt{2}}{8} \). This process is essential for visualizing and working with complex numbers in standard algebraic form.

Standard Form of Complex Number

The standard form of a complex number is quite straightforward: it should be expressed as \( a + bi \). Here, \( a \) is the real part, and \( b \) is the imaginary part, with \( i \) representing the imaginary unit defined as \( i^2 = -1 \).This form allows us to easily perform algebraic operations such as addition, subtraction, multiplication, and division with complex numbers, similar to real numbers.Take for instance the standard form of our previous example: \( \frac{-9\sqrt{2}}{8} + i\frac{9\sqrt{2}}{8} \). This form makes it easy to identify the real and imaginary components and manipulate them accordingly.Key points to remember:

• The real part \( a \) is independent of \( i \).
• The imaginary part \( b \) is multiplied by \( i \).

Recognizing these components in standard form can simplify various mathematical processes involving complex numbers.

Graphical Representation of Complex Numbers

Complex numbers can also be represented graphically on a complex plane, which is similar to a coordinate plane. The horizontal axis (x-axis) corresponds to the real part of the complex number, and the vertical axis (y-axis) corresponds to the imaginary part.To plot a complex number, identify its standard form \( a + bi \) and locate the point \( (a, b) \) on the complex plane. For instance, for the complex number \( \frac{-9\sqrt{2}}{8} + i\frac{9\sqrt{2}}{8} \), you plot the point at \( \left( \frac{-9\sqrt{2}}{8}, \frac{9\sqrt{2}}{8} \right) \).Steps to graph:

• Plot the real part along the x-axis.
• Plot the imaginary part along the y-axis.
• Mark the intersection of these values on the plane as the location of the complex number.

Graphical representation provides a visual way to understand the magnitude and direction of complex numbers, making it easier to grasp concepts such as addition or subtraction geometrically. This is useful for intuitively understanding operations performed on complex numbers.