Problem 16 - Algebra 2

Question

Simplify. $$ (-2+7 i)+(-4-5 i) $$

Step-by-Step Solution

Answer

The simplified form is \(-6 + 2i\).

Step 1: Understand the problem

The expression we have is \[(-2 + 7i) + (-4 - 5i)\]which consists of adding two complex numbers together. Each complex number is of the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

Step 2: Group real and imaginary components

Start by grouping the real components and the imaginary components separately. The real components are \(-2\) and \(-4\), and the imaginary components are \(7i\) and \(-5i\). This gives us:\[-2 + (-4) + (7i) + (-5i)\]

Step 3: Add real components

Add the real numbers from Step 2 together:\[-2 + (-4) = -6\]

Step 4: Add imaginary components

Add the imaginary parts from Step 2 together:\[7i + (-5i) = 2i\]

Step 5: Combine results

Combine the results from Step 3 and Step 4 to form the simplified complex number:\[-6 + 2i\]

Key Concepts

Real and Imaginary Components

In complex numbers, understanding the real and imaginary components is crucial. Complex numbers are often expressed in the form \(a + bi\), where \(a\) represents the real part, and \(bi\) is the imaginary component. Here, \(i\) is the imaginary unit, which satisfies the equation \(i^2 = -1\).
In the expression \((-2 + 7i) + (-4 - 5i)\), the real components are \(-2\) and \(-4\). The imaginary components, which involve the imaginary unit, are \(7i\) and \(-5i\).

• When identifying these parts, remember: the real component does not involve \(i\), whereas the imaginary part does.
• Separating these components lays the groundwork for easy manipulation of complex numbers.

Recognizing these components helps in basic operations like addition and simplification, which we explore next.

Addition of Complex Numbers

Adding complex numbers involves handling each part—real and imaginary—separately. In the expression \((-2 + 7i) + (-4 - 5i)\), start by focusing on each component separately.
Follow these simple steps:

• **Add the real parts**: Combine the real numbers \(-2\) and \(-4\) to get \(-6\).
• **Add the imaginary parts**: Combine the imaginary parts \(7i\) and \(-5i\), resulting in \(2i\).

By dealing with each part separately, you simplify addition into basic arithmetic. This separate treatment ensures accuracy and ease of computation. Once calculated, the new complex number is assembled by simply combining the newly calculated real and imaginary parts, giving us \(-6 + 2i\).
Remember, addition in complex numbers retains your initial format—\(a + bi\)—for the final simplified expression.

Simplifying Complex Expressions

Simplifying complex expressions is often the culmination of understanding and applying a few basic operations correctly. In our example, the expression \((-2 + 7i) + (-4 - 5i)\) leads us to the final, simplified form \(-6 + 2i\). This smaller problem of adding corresponds to simplification.
By combining like terms, we ensure the expression is in its most streamlined format. Simplification involves:

• **Grouping like components**, as we analyzed earlier, the real and imaginary parts.
• **Performing basic operations** like addition or subtraction for each component.
• **Reconstructing** the result into the standard form of a complex number, \(a + bi\).

This process not only makes complex numbers easier to manage but also sets a foundation for tackling more intricate problems involving multiplication or division.
The key to simplification is practice and ensuring each part is correctly handled and recombined to form an easy-to-read final answer.