Problem 34 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(9 + 2i\).

Step 1: Write the Given Expression

The given mathematical expression to simplify is \((5-2i)+(4+4i)\). This expression involves adding two complex numbers.

Step 2: Separate Real and Imaginary Parts

Identify the real and imaginary parts of each complex number. For \((5-2i)\), the real part is 5, and the imaginary part is \(-2i\). Similarly, for \((4+4i)\), the real part is 4, and the imaginary part is \(4i\).

Step 3: Add the Real Parts Together

Add the real parts of the complex numbers: \(5 + 4 = 9\). This sum is the real part of the simplified expression.

Step 4: Add the Imaginary Parts Together

Add the imaginary parts of the complex numbers: \(-2i + 4i = 2i\). This sum is the imaginary part of the simplified expression.

Step 5: Combine the Real and Imaginary Parts

Combine the results from Step 3 and Step 4 to form the simplified complex number: \(9 + 2i\).

Key Concepts

Simplifying Complex Expressions

When working with complex numbers, simplifying expressions is an essential skill. Complex numbers are often written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. Simplifying complex expressions involves combining like terms to make the expression more concise.
In the original exercise, we are given the expression \((5 - 2i) + (4 + 4i)\). To simplify, it's important to:

• Identify and group the real parts of the complex numbers.
• Identify and group the imaginary parts of the complex numbers.

After grouping, we add these parts separately, which makes it easier to see which numbers can be combined. This process not only helps simplify calculations but also reinforces an understanding of how to manipulate complex numbers in algebraic expressions.

Addition of Complex Numbers

Adding complex numbers is quite similar to adding real numbers, but with an additional step to consider the imaginary parts. Each complex number has two components, a real part and an imaginary part. When we add complex numbers, we add their corresponding parts separately.
In our expression \((5 - 2i) + (4 + 4i)\), follow these simple steps:

• Firstly, add the real numbers: \(5 + 4 = 9\).
• Then, add the imaginary numbers: \(-2i + 4i = 2i\).

The result of adding the components separately is \(9 + 2i\). This is the simplified result of the original complex expression. Always remember to track both parts—real and imaginary—to maintain accuracy throughout the process.

Real and Imaginary Parts

Understanding the difference between the real and imaginary parts of a complex number is crucial in working with them. Each complex number \(a + bi\) contains a real part \(a\) and an imaginary part \(bi\).
In the given exercise, for the complex number \((5 - 2i)\):

• The real part is \(5\).
• The imaginary part is \(-2i\).

Similarly, for \((4 + 4i)\):

• The real part is \(4\).
• The imaginary part is \(4i\).

Distinguishing these parts allows us to manipulate and perform operations correctly. Always ensure to separate and process real and imaginary parts individually, whether adding, subtracting, or simplifying expressions with complex numbers.