Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ (-2-4 i)+(1+6 i) $$

Step-by-Step Solution

Answer

The simplified complex number is \(-1 + 2i\).

Step 1: Add the real parts

To simplify the expression \((-2-4i)+(1+6i)\), start by adding the real parts of the complex numbers together. The real parts are -2 and 1. Calculate: \[-2 + 1 = -1\]

Step 2: Add the imaginary parts

Next, add the imaginary parts of the complex numbers. The imaginary parts are -4i and 6i. Calculate:\[-4i + 6i = 2i\]

Step 3: Combine the results

Combine the results of the real part and the imaginary part to form the simplified complex number:\[-1 + 2i\]

Key Concepts

Algebraic Operations

Algebraic operations with complex numbers work similarly to operations with real numbers. However, complex numbers involve both a real part and an imaginary part, denoted as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. In the exercise given, we are asked to perform addition of complex numbers.

• Addition involves combining like terms. In complex numbers, this specifically means combining the real parts separately from the imaginary parts.
• It is important to maintain the structure \(a + bi\) throughout the operation to ensure accuracy and proper simplification.

When performing algebraic operations, always remember to treat the real and imaginary parts individually. This keeps the operations straightforward and prevents mistakes that can arise from mixing parts.

Imaginary Numbers

Imaginary numbers form the cornerstone of complex numbers. They arise from the need to handle square roots of negative numbers. The imaginary unit is denoted by \(i\), where \(i^2 = -1\). This crucial relationship allows us to work with expressions that include square roots of negative values, such as \(\sqrt{-4} = 2i\).

• Understanding imaginary numbers is essential because they extend the real number system into the complex plane, where every number is a combination of real and imaginary parts.
• Although imaginary numbers initially seem abstract, they have practical applications in engineering, physics, and complex problem-solving.

In the context of the exercise, the imaginary parts \(-4i\) and \(6i\) are summed just like real numbers, resulting in \(2i\). This demonstrates how imaginary units essentially behave like regular algebraic terms during operations.

Simplification of Expressions

Simplifying complex number expressions involves reducing them to their most basic form \(a + bi\) by performing both real and imaginary calculations. The goal is to streamline the expression into an easily understandable form while preserving its value and mathematical integrity.

• Start by addressing each component separately, as seen in the original exercise where the real and imaginary parts are calculated independently.
• By simplifying both parts separately, it's easier to combine them into a concise complex expression.

In this exercise, after performing the addition \((-2) + 1\) and \((-4i) + 6i\), the result \(-1 + 2i\) is the simplified form of the original complex expression. Simplification not only helps in clear communication but also in further mathematical manipulation, be it solving equations or transforming into polar form.