Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-20: Represent the complex number geometrically. $$ 4(-1+2 i) $$

Step-by-Step Solution

Answer

The complex number \(4(-1 + 2i)\) is geometrically represented by the point \((-4, 8)\) on the complex plane.

Step 1: Understand the Complex Number in Rectangular Form

The given complex number is in the form \(a + bi\). The equation \(4(-1 + 2i)\) represents a complex number where \( a = -1 \) and \( b = 2 \). This is the rectangular representation of the number.

Step 2: Simplify the Complex Number

We need to simplify the expression \(4(-1 + 2i)\). Distribute the 4 into the terms inside the parentheses: \(4 \times -1 = -4\) and \(4 \times 2i = 8i\). Thus, the complex number is \(-4 + 8i\).

Step 3: Plot the Complex Number on the Complex Plane

In a complex plane, the horizontal axis (real axis) represents the real part, and the vertical axis (imaginary axis) represents the imaginary part. Here, the real part is \(-4\) and the imaginary part is \(8\). Plot the point corresponding to \(-4\) on the horizontal axis and \(8\) on the vertical axis. The point \((-4, 8)\) is the geometric representation.

Key Concepts

Rectangular Form

The rectangular form of a complex number is its representation in the format \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary coefficient. This format is akin to representing a point on a two-dimensional plane using Cartesian coordinates. Here, \(a\) and \(b\) delineate the position on the real and imaginary axes, respectively.

• For a complex number like \(-1 + 2i\), \(a = -1\) is the real component, and \(b = 2\) is the imaginary component.
• In our scenario with \(4(-1 + 2i)\), simplifying the expression gives us \(-4 + 8i\), still in rectangular form.

The rectangular form is essential in understanding the structure of complex numbers and serves as a foundation for their manipulation in various mathematical operations.

Complex Plane

The complex plane is a two-dimensional plane used to visually represent complex numbers. It is essential because it provides a geometric interpretation of complex numbers. On this plane:

• The horizontal axis, known as the real axis, represents the real part of the complex number.
• The vertical axis, known as the imaginary axis, represents the imaginary part.

Each complex number corresponds to a point on this plane. The real part determines the position along the horizontal axis, while the imaginary part determines the position along the vertical axis. For instance, the complex number \(-4 + 8i\) would be plotted at the point \((-4, 8)\). This graphical approach helps in visualizing operations on complex numbers, such as addition, subtraction, and understanding concepts like magnitude and argument.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in the world of complex numbers. It is defined by the property \(i^2 = -1\). This discovery breaks the boundaries of real numbers, enabling the expansion into complex numbers:

• The number \(i\) allows for the square roots of negative numbers, a task impossible with just real numbers.
• In the expression \(-1 + 2i\), the term \(2i\) signifies that the imaginary unit multiplies with 2, showcasing its role in defining the imaginary component.

Understanding the imaginary unit is crucial as it unlocks advanced mathematical concepts and applications, paving the way for solving equations that encompass square roots of negative numbers.