Problem 11 - Algebra for College Students

Question

Add or subtract as indicated. $$(-8+4 i)+(2+6 i)$$

Step-by-Step Solution

Answer

The sum is \(-6 + 10i\).

Step 1: Identify Real and Imaginary Parts

In the expression \[(-8 + 4i) + (2 + 6i)\]both complex numbers are of the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. The first complex number, \(-8 + 4i\), has a real part of \(-8\) and an imaginary part of \(4i\). The second complex number, \(2 + 6i\), has a real part of \(2\) and an imaginary part of \(6i\).

Step 2: Add Real Parts

Add the real parts of both complex numbers. That is,\[-8 + 2 = -6\].

Step 3: Add Imaginary Parts

Add the imaginary parts of both complex numbers. That is,\[4i + 6i = 10i\].

Step 4: Form the Resulting Complex Number

Combine the results of the real and imaginary parts to form the resulting complex number. This gives us,\[-6 + 10i\].

Key Concepts

Understanding Real and Imaginary Parts

Complex numbers are composed of two components: the real part and the imaginary part. In a standard form, a complex number is written as \( a + bi \), where \( a \) represents the real part, and \( bi \) represents the imaginary part. The letter \( i \) is the imaginary unit, which satisfies the equation \( i^2 = -1 \). For example, in the complex number \(-8 + 4i\), \(-8\) is the real part and \(4i\) is the imaginary part.
Understanding these components is crucial as they help in performing arithmetic operations like addition or subtraction. To identify them:

• Look at the term without the \( i \); this is your real part.
• The term with \( i \) is your imaginary part.

Recognizing and separating these parts is the first step in working with complex numbers effectively.

Addition of Complex Numbers Simplified

When adding complex numbers, the process is quite straightforward. You need to combine the real parts together and the imaginary parts together to form a new complex number. Let's consider the addition of \((-8 + 4i)\) and \((2 + 6i)\).
Firstly, identify the real and imaginary parts:

• For \(-8 + 4i\), real: \(-8\), imaginary: \(4i\)
• For \(2 + 6i\), real: \(2\), imaginary: \(6i\)

Now, add the real parts:
\(-8 + 2 = -6\)
And add the imaginary parts:
\(4i + 6i = 10i\)
The new complex number formed by combining these sums is \(-6 + 10i\).
By following these simple steps, you can easily add any two complex numbers.

Working with Algebraic Expressions Involved in Complex Numbers

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. When dealing with complex numbers, these expressions often incorporate both real and imaginary numbers, making the manipulation a bit unique.
To handle expressions like \((-8 + 4i) + (2 + 6i)\), treat each part separately:

• Real numbers behave like regular numbers under addition or subtraction.
• Imaginary parts also follow standard arithmetic rules, but keep the \( i \) factor in mind, which does not multiply with the real coefficient.

Furthermore, realize the simplification process mirrors that of simple algebra: gathering like terms. Here, liken terms to separating real from imaginary components.
Understanding these concepts allows you to simplify and solve more complex mathematical tasks involving complex numbers effortlessly.