Problem 75 - Intermediate Algebra

Question

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (2+4 i)+(6-5 i) $$

Step-by-Step Solution

Answer

The result is \(8 - i\).

Step 1: Identify Real and Imaginary Parts

The expression \((2+4i) + (6-5i)\) is given. Identify the real parts and the imaginary parts of the complex numbers. Here, the real parts are 2 and 6, and the imaginary parts are \(4i\) and \(-5i\).

Step 2: Add the Real Parts

Add the real parts of the complex numbers. The real part of the first complex number is 2, and the real part of the second one is 6. So, \(2 + 6 = 8\).

Step 3: Add the Imaginary Parts

Add the imaginary parts of the complex numbers. The imaginary part of the first complex number is \(4i\), and the imaginary part of the second one is \(-5i\). So, \(4i + (-5i) = -1i\).

Step 4: Write the Result in Standard Form

Combine the results from the previous steps. The sum of the real parts is 8, and the sum of the imaginary parts is \(-1i\). Therefore, the result is \(8 - i\).

Key Concepts

Addition of Complex Numbers

When we talk about the addition of complex numbers, we are referring to combining two numbers that each have a real part and an imaginary part. Complex numbers take the form of \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.
To add two complex numbers, simply add the real parts together and the imaginary parts together.

• If you have \((2+4i)+(6-5i)\), add 2 (from the first number) to 6 (from the second number) to get the real part.
• Similarly, add the imaginary parts: \(4i + (-5i)\) becomes \(-1i\).

Once both parts are added, you would combine these results to express the final answer in the standard form \(a+bi\). In our example, the final result is \(8 - i\). This simplicity makes the addition of complex numbers quite straightforward.

Imaginary Unit

The imaginary unit, commonly denoted as \(i\), is a special number defined by the property \(i^2 = -1\). This concept is introduced to allow us to handle square roots of negative numbers, which aren't possible in the set of real numbers alone.
For example, \(\sqrt{-1} = i\).
Because of the properties of the imaginary unit, it offers great flexibility when working with numbers in mathematical computations, particularly with complex numbers.

• It allows us to extend the number system beyond purely real numbers.
• When combined with a real number, \(i\) creates a complex number (e.g., \(4i\)).

In the exercise we worked on, \(i\) helped us clearly denote imaginary parts like \(4i\) and \(-5i\). It's important to remember that \(i\) operates within specific mathematical rules, particularly that any power of \(i\) can be reduced using its fundamental property.

Real and Imaginary Parts

Every complex number is built from two essential components: a real part and an imaginary part. Understanding these is crucial to mastering complex numbers.

• The **real part** is the portion of the number without the imaginary unit \(i\). For instance, in \(2 + 4i\), the real part is 2.
• The **imaginary part** is the term including the imaginary unit \(i\). In the example \(2 + 4i\), the imaginary part is \(4i\).

In operations involving complex numbers, these parts often need to be treated separately. For the addition task \((2+4i) + (6-5i)\), you treat 2 and 6 as like terms, leading to a new real part, and \(4i\) and \(-5i\) as like terms, leading to a new imaginary part.
Recognizing real vs. imaginary parts ensures proper application of arithmetic rules and helps keep complex number tasks simple.