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Chapter 8: Problem 20

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

Short Answer

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

Step by step solution

01

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.
02

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 \(-4 + 8i\).
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -4-4 i $$

Use Euler's formula to prove De Moivre's theorem.

Find the two square roots of \(1+\sqrt{3} i\).

Exer. 1-10: Find the absolute value. $$ |5+8 i| $$

Exer. 23-30: Find the solutions of the equation. $$ x^{5}+1=0 $$
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