Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Multiply or divide as indicated. Simplify each answer. $$(-6+5 i)+(4-4 i)+(2-i)$$

Step-by-Step Solution

Answer

The simplified expression is 0.

Step 1: Identify and Group Like Terms

The expression given is \((-6 + 5i) + (4 - 4i) + (2 - i)\). First, identify the real numbers and imaginary parts. Group together the real parts (-6, 4, 2) and the imaginary parts (5i, -4i, -i).

Step 2: Sum the Real Parts

Add the real parts together: \(-6 + 4 + 2\). Calculate this to get \(0\).

Step 3: Sum the Imaginary Parts

Add the imaginary parts together: \(5i - 4i - i\). Calculate this to get \(0i\), which simplifies just to \(0\).

Step 4: Combine Real and Imaginary Parts

Combine the sum of the real parts and the sum of the imaginary parts to form the simplified expression: \(0 + 0i\), which simplifies to \(0\).

Key Concepts

real numbers

Real numbers form the foundation of many aspects of mathematics. They include all the numbers that can be found on the number line. These are also the numbers we usually use in everyday life. Real numbers include:

• Integers: like -1, 0, and 1
• Fractions: like 1/2
• Decimals: both finite, like 0.5, and repeating, like 0.333...
• Positive and negative numbers

In the context of complex numbers, the real part is the component without any imaginary unit 'i'. For the given expression \((-6 + 5i) + (4 - 4i) + (2 - i)\), the real numbers are -6, 4, and 2. By summing these, you perform a basic arithmetic process to simplify the expression. Real numbers are crucial because they help us handle quantities seen in the real world, like distance and temperature. By adding them separately from imaginary numbers, clarity is maintained in the mathematical processes.

imaginary numbers

Imaginary numbers were introduced to tackle equations that don't have real solutions. A classic example is the square root of negative numbers. The core of an imaginary number is 'i', the square root of -1. Imaginary numbers can be combined with real numbers to form complex numbers of the form a + bi. Here:

• 'a' represents the real part
• 'b' represents the imaginary part

In our exercise \((-6 + 5i) + (4 - 4i) + (2 - i)\), the imaginary parts are 5i, -4i, and -i. Adding them separately from the real numbers allows us to focus on each part of the expression clearly. This distinction between real and imaginary components helps simplify processes, especially when dealing with more complex algebraic manipulations. The sum of these imaginary parts results in a value of 0i, which can just be simplified to 0.

simplification

Simplification in algebra refers to reducing an expression to its most basic form. The goal is to make mathematical expressions as straightforward and efficient as possible. When working with complex numbers like our example \((-6 + 5i) + (4 - 4i) + (2 - i)\), simplification involves:

• Identifying and grouping like terms: real numbers with real, and imaginary with imaginary.
• Performing the necessary arithmetic operations on these groups.

Simplifying the given complex sum requires separately adding real parts to get 0, and imaginary parts to get 0i. Together, these form the extremely simplified result of 0. Simplification ultimately provides clarity and understanding, allowing us to work with the essential components of mathematical expressions and easily see their underlying structure.