Problem 41 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) See Example 3 $$ (7-3 i)-(4+2 i) $$

Step-by-Step Solution

Answer

The result is \(3 - 5i\).

Step 1: Rewrite the Expression

Start by rewriting the given expression \[(7 - 3i) - (4 + 2i)\]in a more manageable form. Since subtraction is the same as adding the opposite, it can be rewritten as:\[7 - 3i + (-4 - 2i).\]

Step 2: Separate Real and Imaginary Parts

Separate the real and imaginary components of the expression. The real parts are:\[7 + (-4) = 3.\]The imaginary parts are:\[-3i + (-2i) = -5i.\]

Step 3: Combine the Components

Now, combine the results from the real and imaginary parts to get the final expression:\[3 - 5i.\]

Step 4: Write in Standard Form

Ensure the result is written in the form \(a + bi\) by confirming that 3 is the real part and -5 is the imaginary coefficient:\[a + bi = 3 - 5i.\]

Key Concepts

Algebraic Operations with Complex Numbers

Algebraic operations with complex numbers can be similar to operations with real numbers once you understand the structure. Complex numbers are expressed in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.

When performing operations like addition or subtraction, it’s essential to treat the real and imaginary parts separately.

• For addition: Combine the real parts from each number, and then do the same for the imaginary parts.
• For subtraction: Subtract the real parts and subtract the imaginary parts accordingly.

This approach simplifies the process and avoids confusion by focusing on one component at a time.

To illustrate, consider the expression \((7 - 3i) - (4 + 2i)\). Rewriting it as \(7 - 3i + (-4 - 2i)\) reveals that it is equivalent to adding opposites, making it straightforward to handle each type separately.

Understanding Imaginary Numbers

Imaginary numbers are an essential part of complex numbers. The foundation of imaginary numbers lies in the unit \( i \), defined as \( i^2 = -1 \). This imaginary unit is what allows us to extend real numbers into the complex plane.

An imaginary number is typically expressed as \( bi \), where \( b \) is a real number and \( i \) is the imaginary unit. Various operations involving imaginary numbers, such as multiplication or exponentiation, rely on the property \( i^2 = -1 \). For instance, multiplying two imaginary numbers \( bi \) and \( ci \) involves:

• Direct multiplication of the coefficients, \( b \times c \).
• Multiplying the imaginary units and using the fact \( i^2 = -1 \), leading to \( -bc \).

Imaginary numbers can be abstract at first, yet they have practical applications in electrical engineering and wave analysis, making them incredibly useful in real-world scenarios.

Their interaction with real numbers forms the basis of complex numbers, creating a robust mathematical system that facilitates solving equations that have no real solutions alone.

Real and Imaginary Parts in Complex Numbers

In any complex number written as \( a + bi \), the real part is \( a \) and the imaginary part is \( bi \). Understanding how to separate and manipulate these parts is crucial for successfully handling complex numbers.

When subtracting two complex numbers, like in the original exercise, you will:

• Identify the real components and perform the operation on these numbers.
• Do the same with the imaginary components.

In the example of \( (7 - 3i) - (4 + 2i) \), separate the numbers:

• Real parts: \( 7 - 4 = 3 \).
• Imaginary parts: \( -3i - 2i = -5i \).

The final expression \( 3 - 5i \) clearly distinguishes the real and imaginary parts, keeping the standard form of \( a + bi \). This method not only helps in basic arithmetic but also prepares you for more complex operations like multiplication, division, and conjugate calculations.

Ensuring the result is in standard form confirms your understanding and provides clarity, essential for communicating mathematical results effectively.