Problem 76 - Intermediate Algebra

Question

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (5-3 i)+(7-8 i) $$

Step-by-Step Solution

Answer

The result is \(12 - 11i\).

Step 1: Identify Real and Imaginary Parts

First, let's identify the real and imaginary parts of each complex number separately in the expression \((5-3i)+(7-8i)\). The real parts are 5 and 7, and the imaginary parts are \(-3i\) and \(-8i\).

Step 2: Combine the Real Parts

Add the real parts of the complex numbers together. This means we add 5 and 7: \[ 5 + 7 = 12 \]

Step 3: Combine the Imaginary Parts

Add the imaginary parts of the complex numbers together. This means we add \(-3i\) and \(-8i\): \[ (-3i) + (-8i) = -11i \]

Step 4: Form the Complex Number

Combine the results from Steps 2 and 3 to form a complex number in the format \(a+bi\): \[ 12 + (-11i) = 12 - 11i \]

Key Concepts

Real and Imaginary Parts

When working with complex numbers, it's crucial to understand their structure. A complex number is expressed in the form \(a + bi\). Here, \(a\) represents the real part, while \(bi\) is the imaginary part, where \(b\) is a real number and \(i\) is the imaginary unit, defined as \(i = \sqrt{-1}\). This implies that the imaginary part of the complex number has a value of zero except for the presence of the imaginary unit.

• Real Part: Represents the non-imaginary component of the number. For example, in \(5 - 3i\), the real part is 5.
• Imaginary Part: Represents the portion with the imaginary unit. In \(5 - 3i\), the imaginary part is \(-3i\).

Recognizing these components separately helps in performing operations like addition or subtraction with complex numbers.

Addition of Complex Numbers

Adding complex numbers involves adding their respective real and imaginary parts. Consider the expression \((5-3i) + (7-8i)\) as an example.

• Step 1: Identify the real parts and the imaginary parts of the numbers. Here, 5 and 7 are the real parts, and \(-3i\) and \(-8i\) are the imaginary parts.
• Step 2: Add the real parts together and separately add the imaginary parts. The real part calculation is \(5 + 7 = 12\), while the imaginary part calculation is \((-3i) + (-8i) = -11i\).

After performing these operations, combine these results to get the final sum as a new complex number: \(12 - 11i\). This process demonstrates how the addition of complex numbers operates under similar principles to regular number addition, but keeps the real and imaginary components distinct.

Algebraic Operations with Complex Numbers

Algebraic operations with complex numbers, such as addition, subtraction, multiplication, and division, follow specific rules due to the unique nature of complex numbers.

• Addition and Subtraction: Direct by combining the corresponding real and imaginary parts separately, as shown in our earlier example.
• Multiplication: Utilizes the distributive property with the rule \(i^2 = -1\). For example, multiplying \((a + bi)\) with \((c + di)\) yields \((ac - bd) + (ad + bc)i\).
• Division: Involves multiplying the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary unit from the denominator while simplifying the expression.

These operations allow complex numbers to be manipulated algebraically and are essential for solving equations where complex numbers naturally arise. Mastery of these concepts is vital for deeper mathematical studies and practical applications in engineering and physics.