Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

Question

Represent the complex number graphically, and find the standard form of the number. $$\sqrt{12}\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right)$$

Step-by-Step Solution

Answer

The standard form of the complex number is \( 0 - \sqrt{12}i \). The graphical representation of the number is a point at \( (0, - \sqrt{12}) \) on the complex plane.

Step 1: Identify the polar form of complex number

The given complex number is in polar form \( r(\cos \theta + i \sin \theta) \), where \( r = \sqrt{12} \) is the magnitude, and \( \theta = \frac{3 \pi}{2} \) is the angle.

Step 2: Convert to Cartesian coordinates

Complex numbers in standard form are represented as \( a + bi \), where \( a = r \cos \theta \) and \( b = r \sin \theta \). So x-coordinate \( a = r \cos \theta = \sqrt{12} \cos \frac{3 \pi}{2} = 0 \) and y-coordinate \( b = r \sin \theta = \sqrt{12} \sin \frac{3 \pi}{2} = -\sqrt{12} \).

Step 3: Write the standard form

Plug \( a \) and \( b \) values into \( a + bi \) to get standard form. Hence, the standard form of the given complex number is \( 0 - \sqrt{12}i \).

Step 4: Represent graphically

To plot this complex number on the complex plane, mark the point at \( (0, -\sqrt{12}) \). This point represents the complex number in its standard form.

Key Concepts

Polar Form of Complex Numbers

Complex numbers can be represented in various forms, and one of the most illuminative is the polar form. Instead of using the traditional Cartesian coordinates, the polar form expresses a complex number in terms of its magnitude and the angle it makes with the positive real axis. This is written as \( r(\text{cos} \theta + i \text{sin} \theta) \), where \( r \) is the modulus or the absolute value of the complex number, and \( \theta \) represents the argument or angle.

Understanding the polar form is essential when dealing with complex numbers in fields such as electrical engineering and physics, where oscillatory behaviors are represented. The polar form simplifies multiplication and division of complex numbers because it's easier to multiply and divide their magnitudes and add and subtract their angles, compared to dealing with individual real and imaginary parts. The process of finding a complex number's polar form is best mastered through practice and by verifying the results with graphical representations.

Conversion to Cartesian Coordinates

The journey from polar to Cartesian coordinates is a crucial skill for anyone diving into the field of complex numbers. This conversion is performed by using the equations \( a = r \cos \theta \) and \( b = r \sin \theta \) to find the real part \( a \) and the imaginary part \( b \) of the complex number, respectively.

This process can be compared to translating a language with its own unique expressions into a more universally understood form. By converting a complex number into Cartesian coordinates, it's possible to perform a variety of operations like addition, subtraction, and understanding complex roots. When dealing with complex numbers, it's beneficial to become comfortable with both forms and to be able to switch between them effectively.

Representing Complex Numbers Graphically

Visualizing complex numbers on a graph can greatly enhance comprehension of their behavior and relationships. When we chart a complex number, we place it on the complex plane, which is much like the familiar Cartesian coordinate system with a horizontal real axis and a vertical imaginary axis.

Representing a complex number graphically involves plotting a point where the horizontal axis value corresponds to the real part of the number, and the vertical axis value corresponds to the imaginary part. This pictorial representation can be particularly enlightening when comparing magnitudes and arguments of different complex numbers, which can be thought of much like the way we visualize vectors in physics. Indeed, the graphic representation is not just a theoretical exercise—it equips students with an intuitive grasp of complex operations and relationships, laying the groundwork for more advanced study in complex analysis and related subjects.

Standard Form of Complex Numbers

The standard form of a complex number is perhaps the most direct and common way of expressing these numbers. It's given in the form of \( a + bi \), where \( a \) represents the real part and \( b \) embodies the imaginary part, with \( i \) being the imaginary unit. This form is straightforward and enables easy execution of basic algebraic operations such as addition and subtraction.

Mastering the conversion from polar or graphical representation to the standard form is a fundamental skill. The standard form is not just about notation—by understanding and utilizing it effectively, students can solve complex equations, understand the fundamentals of complex functions, and begin to explore the rich tapestry of analysis that hinges on these intriguing mathematical entities.