Problem 62 - Intermediate Algebra

Question

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (7+4 i)+(4-4 i) $$

Step-by-Step Solution

Answer

The result is \(11 + 0i\).

Step 1: Identify the Real Components

In the expression \((7+4i)+(4-4i)\), identify the real parts of each complex number. They are 7 and 4.

Step 2: Add the Real Components

Add the real components together: \(7 + 4 = 11\).

Step 3: Identify the Imaginary Components

Next, identify the imaginary parts of each complex number, which are \(4i\) and \(-4i\).

Step 4: Add the Imaginary Components

Add the imaginary components together: \(4i + (-4i) = 0i\).

Step 5: Combine the Real and Imaginary Results

Combine the results from Step 2 and Step 4 to get the final result. The sum is \(11 + 0i\).

Key Concepts

Adding Complex Numbers

When working with complex numbers, it's helpful to know how to add them. Complex numbers have two parts: a real part and an imaginary part. They are in the form \(a + bi\), where \(a\) is the real component and \(b\) is the coefficient of the imaginary part. Typically, when adding complex numbers, you handle each part separately.

• Begin by identifying both the real and imaginary parts of each complex number.
• Next, add the real parts together.
• Then add the imaginary parts together.

These separate sums give you the real and imaginary components of the resulting complex number. It's straightforward, once you break it down into these steps.

Real Components

The real components of complex numbers are simply the non-imaginary parts, represented by \(a\) in \(a + bi\). When adding complex numbers, it's crucial to focus on the real components at the start. For example, in the expression \((7+4i)+(4-4i)\):

• Identify the real parts which are 7 and 4.
• Add them together to get \(7 + 4 = 11\).

This real part of the sum becomes the real component of your resultant complex number. The process remains consistent regardless of the numbers involved.

Imaginary Components

Imaginary components involve the components that are multiplied by \(i\), the imaginary unit, standing for \(\sqrt{-1}\). In our example, these were \(4i\) and \(-4i\).

• Start by identifying these imaginary parts in each complex number.
• Add them, taking into account the sign. Here, \(4i + (-4i) = 0i\).

Remember, the sum of imaginary parts will always be an imaginary number, like \(bi\) or possibly \(0i\) (like in this example). When the imaginary components cancel each other out, your final result may not have any imaginary part, only a real one.