Problem 40 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) See Example 3 $$ (5+3 i)-(6-9 i) $$

Step-by-Step Solution

Answer

The answer is \(-1 + 12i\).

Step 1: Identify the Expression

You are given the expression \[ (5+3i) - (6-9i) \].This expression involves subtraction of two complex numbers.

Step 2: Distribute the Negative Sign

Apply the distributive property by multiplying the negative sign into the second complex number.\[ 5 + 3i - 6 + 9i \].

Step 3: Combine Like Terms

Group the real parts and the imaginary parts together:\[ (5 - 6) + (3i + 9i) \].

Step 4: Simultaneous Simplification

Simplify both parts separately:Real part: \( 5 - 6 = -1 \)Imaginary part: \( 3i + 9i = 12i \)

Step 5: Write the Simplified Expression

Combine the simplified real and imaginary parts to form the final answer:\[ -1 + 12i \].

Key Concepts

Subtraction of Complex Numbers

Complex numbers are composed of a real part and an imaginary part, typically denoted as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. When subtracting complex numbers, the main goal is to treat the real and imaginary parts separately.

In our original exercise, we had two complex numbers: \((5 + 3i)\) and \((6 - 9i)\). To subtract the second from the first, you need to subtract each corresponding component:

• Subtract the real parts: 5 (from the first number) and 6 (from the second number).
• Subtract the imaginary parts: 3i (from the first number) and -9i (negative because of subtraction).

This separate treatment ensures that each part is handled correctly, allowing you to further simplify the expression.

Distributive Property

The distributive property is a crucial step in the subtraction of complex numbers, especially when dealing with negative signs. The property helps to simplify expressions by distributing the operations across each term involved.

In the expression \( (5 + 3i) - (6 - 9i) \), the negative sign before \((6 - 9i)\) affects both terms within the parentheses, converting the expression to \(5 + 3i - 6 + 9i\). This happens because:

• -1 times 6 gives -6
• -1 times -9i gives +9i

The distributive property, in essence, helps break apart and distribute the subtraction operation, making it easier to later combine like terms.

Combining Like Terms

Once we apply the distributive property, the next important step is to combine like terms. This involves bringing together similar parts of the expanded expression.

After applying the distributive step, we have the expression \(5 + 3i - 6 + 9i\). The like terms consist of:

• Real numbers: 5 and -6
• Imaginary numbers: 3i and 9i

Combining these like terms:
Real part: \(5 - 6 = -1\)
Imaginary part: \(3i + 9i = 12i\)
By organizing and simplifying these similar terms, we get the final form of the expression, which is easier to read and understand.

Imaginary Numbers

Imaginary numbers are an essential part of complex numbers, represented by \(i\), which stands for the square root of -1. In the realm of complex numbers, the imaginary unit \(i\) is paired with real numbers to express values that cannot be found on the traditional number line.

For our problem, the presence of imaginary parts 3i and -9i required careful handling during subtraction and simplification. Imaginary numbers add an additional layer to arithmetic operations due to their unique property: \(i^2 = -1\).

Understanding how to work with imaginary numbers, like combining them with real numbers and performing operations on them, is crucial when solving complex number problems. They allow us to perform mathematical operations and solve equations that are otherwise impossible using just real numbers.