Problem 42 - Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i.\) $$ (5-4 i)-(3+2 i) $$

Step-by-Step Solution

Answer

The result is \(2 - 6i\).

Step 1: Identify the components of each complex number

The first complex number is \(5 - 4i\), where \(5\) is the real part and \(-4i\) is the imaginary part. The second complex number is \(3 + 2i\), where \(3\) is the real part and \(+2i\) is the imaginary part.

Step 2: Subtract the real parts

Subtract the real part of the second complex number from the real part of the first complex number. \[5 - 3 = 2\]

Step 3: Subtract the imaginary parts

Subtract the imaginary part of the second complex number from the imaginary part of the first complex number. \[-4i - 2i = -6i\]

Step 4: Combine the results

Combine the results from Step 2 and Step 3 to form the final expression in the form \(a + bi\).\[2 - 6i\]

Key Concepts

Real Part of a Complex Number

Complex numbers are numbers that consist of two parts: the real part and the imaginary part. The real part of a complex number is important as it represents the magnitude in the "real world" sense, akin to regular numbers on the number line. When dealing with a complex number like \( a + bi \), the "\( a \)" represents the real part.
For example, in the complex number \( 5 - 4i \), the real part is \( 5 \). It can be thought of as the projection of the complex number on the horizontal axis of the complex plane.
Understanding the real part helps us handle operations such as addition, subtraction, and eventually multiplication of complex numbers. When performing operations, we treat these real parts just like any other regular numbers, keeping them separate from the imaginary components, which we'll discuss next.

Imaginary Part of a Complex Number

The imaginary part of a complex number sheds light on the component that cannot be placed on the traditional number line. It introduces an "imaginary" dimension needed to fully understand and visualize complex numbers. In a complex number \( a + bi \), the "bi" represents the imaginary part.
For instance, in the complex number \( 5 - 4i \), \(-4i\) is the imaginary part. Imaginary parts involve the imaginary unit \( i \), where \( i^2 = -1 \). In the complex plane, the imaginary part corresponds to the vertical position, at a right angle to the real axis.
When subtracting the imaginary parts of two complex numbers like \(-4i - 2i\), we think of it as combining terms separately from the real parts, leading to a result of \(-6i\). This highlights the importance of treating the imaginary and real parts distinctly.

Subtracting Complex Numbers

Subtracting complex numbers involves handling the real and imaginary parts separately. It can be seen as a two-step process because each component requires individual attention. Imagine we have two complex numbers: \( 5 - 4i \) and \( 3 + 2i \).
Here's how it works:

• First, subtract the real parts: \( 5 - 3 = 2 \).
• Next, subtract the imaginary parts: \(-4i - 2i = -6i \).

These steps ensure clarity and precision in operations, as adding or subtracting real numbers is done independently from combining imaginary numbers.
Once both operations are conducted, you combine them into the familiar form \( a + bi \). For the above example, it results in \( 2 - 6i \). This structured approach makes working with complex numbers as intuitive as working with vectors in geometry.