Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation and write the result in standard form. $$(-4+3 i)+(6-2 i)$$

Step-by-Step Solution

Answer

The result of the operation \(-4+3 i)+(6-2 i)\) is \(2+i\)

Step 1: Identify the real and imaginary components

For the first complex number \(-4+3i\), \(-4\) is the real part and \(3i\) is the imaginary part. For the second complex number \(6-2i\), \(6\) is the real part and \(-2i\) is the imaginary part.

Step 2: Add the real components together

When adding the real parts together, -4 + 6 = 2

Step 3: Add the imaginary components together

When adding the imaginary parts together, \(3i - 2i = i\)

Step 4: Write the result in standard form

Combining the real and imaginary parts together, the result is \(2+i\)

Key Concepts

Standard Form of a Complex Number

Understanding the standard form of a complex number is fundamental in complex number arithmetic. A complex number is expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \). In this standard form, the \( a \) represents the real part and \( bi \) represents the imaginary part of the complex number.

For instance, in the provided exercise, we worked with two complex numbers \( -4+3i \) and \( 6-2i \). When these are written in the standard form, it becomes easy to perform operations such as addition, subtraction, and multiplication. The beauty of the standard form lies in its simplicity and structure, which allows for straightforward calculation rules that resemble those of real numbers—yet with a twist due to the presence of the imaginary unit \( i \).

Real and Imaginary Components

Distinguishing between the real and imaginary components of complex numbers is crucial for performing arithmetic operations. The real part of a complex number is simply the coefficient of the real unit, while the imaginary part is the coefficient of the imaginary unit \( i \).

In our textbook example, \( -4+3i \) has a real component of \( -4 \) and an imaginary component of \( 3 \) (remember the \( i \) is part of the imaginary unit, not the imaginary component itself). Similarly, the complex number \( 6-2i \) has a real component of \( 6 \) and an imaginary component of \( -2 \). Recognizing these components is the first step in doing any complex arithmetic, as it allows us to separate the calculation into smaller, more digestible parts—real calculations with real numbers, and imaginary calculations with imaginary numbers.

Arithmetic of Complex Numbers

When it comes to the arithmetic of complex numbers, the rules are elegantly similar to those for real numbers, with careful attention to the unique behavior of the imaginary unit \( i \). To add complex numbers, as done in our exercise, we simply add their corresponding components—real with real, and imaginary with imaginary.

Following this approach, we combined the real parts \( -4 \) and \( 6 \), and the imaginary parts \( 3i \) and \( -2i \) of our complex numbers. This resulted in \( 2 + i \) after simplifying each addition separately. Subtraction would follow the same pattern but with taking differences, and multiplication and division have their special rules due to the \( i^2 = -1 \) property. Particularly, multiplication demands distributing each part and simplifying, and division involves a process known as 'complex conjugation' to eliminate the imaginary unit from the denominator.

By breaking down these operations into their real and imaginary components, and then simplifying, we are able to maneuver through complex arithmetic with ease and accuracy.