Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation and write the result in standard form. $$(13-2 i)+(-5+6 i)$$

Step-by-Step Solution

Answer

The result of the operation is \(8 + 4i\).

Step 1: Identify the Real and Imaginary Parts

Identify the real part and the imaginary part in both complex numbers.\nThe first complex number, \(13-2i\), has 13 as its real part and -2 as its imaginary part.\nThe second complex number, \(-5+6i\), has -5 as its real part and 6 as its imaginary part.

Step 2: Perform Addition Separately on Real and Imaginary Parts

The real parts get added together: \(13 + (-5) = 8\).\nThe imaginary parts get added together (note that the value 'i' is kept with the imaginary part): \(-2i + 6i = 4i\).

Step 3: Write the Result in Standard Form

Now, combine the added real part and the added imaginary part to write the result in standard form which is \(8 + 4i\).

Key Concepts

Addition of Complex Numbers

Adding complex numbers is quite similar to adding polynomials. Each complex number has two parts: real and imaginary.
When adding two complex numbers, we add the real parts together and the imaginary parts together separately.

• Take the two complex numbers, for example, \(13-2i\) and \(-5+6i\).
• The real parts are \(13\) and \(-5\).
• The imaginary parts are \(-2i\) and \(6i\).

So, by performing the additions separately, you take \(13+ (-5) = 8\) for the real parts and \(-2i + 6i = 4i\) for the imaginary parts.

The result is the combination of these additions, presented as \(8 + 4i\). It is important to keep the imaginary unit \('i'\) with its associated number during the addition of imaginary parts.

Real and Imaginary Parts

Complex numbers are composed of two distinct parts: a real part and an imaginary part. Understanding these parts is essential for working with complex numbers effectively.

The real part is similar to any ordinary number you deal with in everyday math: a simple integer or a decimal without any imaginary unit attached. For example, in the complex number \(13-2i\), the real part is \(13\).
The imaginary part, on the other hand, involves the imaginary unit \('i'\), satisfying the equation \(i^2 = -1\). In \(13-2i\), the imaginary part is \(-2i\).

Identifying these parts correctly is the first step in performing operations with complex numbers. It's what allows us to add and manipulate them as seen in the example, where recognizing the real \(-5\) and imaginary \(6i\) parts of \(-5+6i\) was crucial.

Standard Form of Complex Numbers

Writing the result of a complex number operation in standard form is crucial for clarity and ensures everyone understands the notation. The standard form of a complex number is expressed as \(a + bi\).
Here, \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

• \(8 + 4i\) indicates \(8\) as the real part and \(4\) as the coefficient of the imaginary part \(i\).
• Make sure to express the complete number together, without breaking the union of \(a\) and \(bi\).

After performing operations like addition, reassembling the result in this form helps present the answer correctly.
For example, once you've added the real and imaginary parts as shown in the exercise \((8 + 4i)\), make sure the complex number is in standard form before considering it your final result. This practice simplifies not just addition, but any other operations involving complex numbers. Similar rules apply for subtraction, multiplication, and division.