Problem 29 - Intermediate Algebra

Question

Add or subtract. Write the sum or difference in the form \(a+b i .\) See Example 3 $$ (6+5 i)-(8-i) $$

Step-by-Step Solution

Answer

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

Step 1: Identify Real Parts

The expression given is \((6 + 5i) - (8 - i)\). To start, let's identify the real parts of each complex number. In \((6 + 5i)\), the real part is 6. In \((8 - i)\), the real part is 8.

Step 2: Identify Imaginary Parts

Now, identify the imaginary parts of each complex number. For \((6 + 5i)\), the imaginary part is 5 (since it is 5i). For \((8 - i)\), the imaginary part is -1 (since it is -i).

Step 3: Subtract Real Parts

Now subtract the real parts: \[ 6 - 8 = -2 \]

Step 4: Subtract Imaginary Parts

Next, subtract the imaginary parts: \[ 5 - (-1) = 5 + 1 = 6 \]

Step 5: Combine Results

Combine the results from the subtraction of the real and imaginary parts to get the final answer: \[ -2 + 6i \]

Key Concepts

Real and Imaginary Parts

Complex numbers have a unique structure composed of two parts. The **real part** and the **imaginary part**. Understanding these two components is crucial.

• The real part of a complex number is the number that stands alone without any "i" attached. For example, in the complex number \(6 + 5i\), the real part is \(6\).
• The imaginary part is the number which is multiplied by "i". In \(6 + 5i\), this part is \(5\), specifically \(5i\) where \(i\) is the imaginary unit defined as \(i^2 = -1\).

A complex number is usually expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Recognizing and separating these parts allows for easier operations, such as addition and subtraction.

Addition and Subtraction of Complex Numbers

The process of adding or subtracting complex numbers involves handling both parts separately. This means:

• Real parts of complex numbers are added or subtracted from other real parts.
• Imaginary parts are added or subtracted from other imaginary parts.

For example, when subtracting the complex numbers \(6 + 5i\) and \(8 - i\), we handle them like this:

• Subtract the real parts: \(6 - 8 = -2\)
• Subtract the imaginary parts: \(5 - (-1) = 5 + 1 = 6\)

Thus, the resulting complex number is \(-2 + 6i\). This separation of real and imaginary components simplifies calculations and helps in better understanding of how complex numbers interact.

Algebraic Expressions

Complex numbers can be thought of as algebraic expressions. These expressions include both real and imaginary parts. In any complex expression, it is necessary to:

• Ensure each complex number is reduced to the form \(a + bi\).
• Manipulate algebraic expressions to combine like terms.

When dealing with addition and subtraction, the principles of algebraic expressions help streamline the operation. Terms in algebra involving "i" behave like variables; you combine similar terms.
Understanding how to manipulate these expressions is key to solving arithmetic operations involving complex numbers. The same rules used in managing regular algebraic expressions apply but with an added layer for imaginary numbers. By equating real parts to real and imaginary to imaginary, it's easier to envision solving these complex algebraic tasks.