Problem 27 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ 3-(4-6 i) $$

Step-by-Step Solution

Answer

The standard form is \(-1 + 6i\).

Step 1: Remove Parentheses

Distribute the negative sign across the terms inside the parentheses. The expression becomes:\[3 - (4 - 6i) = 3 - 4 + 6i.\]

Step 2: Combine Real Parts

Identify and combine the real numbers in the expression from Step 1. The expression simplifies to:\[3 - 4 = -1.\]

Step 3: Write in Standard Form

Combine the results from Steps 1 and 2 to get the expression in standard form. Standard form is typically written as a complex number with the real part first and the imaginary part second. Thus, the final expression is:\[-1 + 6i.\]

Key Concepts

Standard Form

Complex numbers appear frequently in mathematics, often represented using a specific structure called the standard form. Standard form of a complex number is \(a + bi\), where \(a\) represents the real part, and \(bi\) is the imaginary part. In this form, \(a\) and \(b\) are real numbers, while \(i\) is the imaginary unit satisfying the equation \(i^2 = -1\).The key aspect of standard form is that the real and imaginary components are distinct. This is crucial for performing arithmetic operations and making comparisons.

• Order matters: The real part \(a\) is always placed before the imaginary part \(bi\).
• Simplification: Expressions should be simplified as much as possible before being written in standard form.

Writing in standard form allows for an easier grasp of the fundamental nature of the complex number, facilitating operations like addition, subtraction, and more.

Real Part

Understanding the real part of a complex number is essential. Within the standard form \(a + bi\), \(a\) is referred to as the real part. Even though complex numbers include an imaginary component, the real part functions just like an ordinary real number you work with in mathematics.For instance, in the expression \(-1 + 6i\), the real part is \(-1\). Finding and working with the real part involves several operations:

• Identification: It's the constant without the imaginary unit \(i\).
• Manipulation: Can be used in all arithmetic operations like any other real number.

This familiarity with the real part makes complex numbers less daunting and more approachable, especially when performing operations like addition and subtraction.

Imaginary Part

The imaginary part of a complex number brings out its unique characteristics. In standard form, \(a + bi\), the term \(bi\) signifies this component, where \(b\) is a real number and \(i\), the imaginary unit.In our solved expression \(-1 + 6i\), the imaginary part is \(6i\). Here's a closer look at how to identify and handle the imaginary part:

• Recognition: It is the term coupled with \(i\).
• Contributions: Represents quantities that exist in a perpendicular direction to the real number line.

A good grasp of the imaginary part not only helps in comprehending complex numbers but also broadens your mathematical understanding. This knowledge is practical when dealing with more advanced topics, such as polar coordinates and frequency analysis.