Problem 17 - College Algebra

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ (-5+3 i)-(6-i) $$

Step-by-Step Solution

Answer

The result is \(-11 + 4i\).

Step 1: Identify Components

The problem gives us two complex numbers: 1. \(-5 + 3i\) 2. \(6 - i\)We need to perform the operation indicated, which is subtraction.

Step 2: Distribute and Change Signs

Distribute the negative sign across the second complex number: \[(-5 + 3i) - 6 + i\] This changes the signs of the real and imaginary components of \(6 - i\), giving us \(-6 + i\).

Step 3: Combine Like Terms

Add the real parts and the imaginary parts separately:Real parts: \(-5 - 6 = -11\)Imaginary parts: \(3i + i = 4i\)

Step 4: Write the Simplified Form

Combine the results from real and imaginary parts: The simplified form of the complex number is \(-11 + 4i\).

Key Concepts

Complex Number Subtraction

When you subtract complex numbers, you essentially handle real and imaginary components separately. Imagine complex numbers as pairs of numbers: one real and one imaginary. When performing operations like subtraction, treat these components independently.

• Real parts are subtracted with real parts.
• Imaginary parts are subtracted with imaginary parts.

This makes complex number subtraction straightforward because it adheres to common rules of arithmetic, just applied to two separate categories of numbers—real and imaginary. For example, if we subtract \(-5 + 3i\) from \(6 - i\), distribute the subtraction to both components of the second number. Adjust the signs to subtract effectively.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a unique number in mathematics defined by the property \(i^2 = -1\). This distinct characteristic is what differentiates complex numbers from real numbers. When dealing with complex numbers, the imaginary unit is your tool for managing square roots of negative numbers.

• An imaginary number is represented as \(bi\), where \(b\) is a real number.
• The imaginary part of \(-5 + 3i\) is \(+3i\), while in \(6 - i\), it's \(-i\).

Understanding the imaginary unit is crucial, as it forms the building block of complex numbers and allows you to perform operations like addition, subtraction, multiplication, and division within this numeric system.

Combining Like Terms

Combining like terms is a method used to simplify mathematical expressions. It involves grouping similar parts of the expression to form a single term. In complex numbers, this process applies by:

• Adding or subtracting the real parts.
• Adding or subtracting the imaginary parts independently.

For example, in the subtraction \((-5 + 3i) - (6 - i)\), you would combine the real numbers \(-5\) and \(-6\), resulting in \(-11\). Similarly, you'd combine the imaginary parts \(+3i\) and \(+i\), resulting in \(+4i\). This step ensures you're simplifying complex expressions appropriately by managing components according to their nature.

Simplifying Complex Expressions

Simplifying complex expressions involves consolidating the expression into a simpler form without changing its value. After combining like terms, you reach a final expression that represents your original problem in its simplest term.

• Ensure all similar terms are combined fully.
• Express the result in the format \(a + bi\), where \(a\) and \(b\) are real numbers.

In this exercise, after subtracting and combining terms, you arrive at the simplified form \(-11 + 4i\). This standardized expression allows you to clearly communicate and utilize the value of the complex number. Mastering the skill of simplification is crucial for further operations involving complex numbers, such as division or solving equations.