Problem 46 - A Graphical Approach to College Algebra

Question

Add or subtract as indicated. Write each sum or difference in standard form. $$(4-i)+(2+5 i)$$

Step-by-Step Solution

Answer

The sum is \(6 + 4i\).

Step 1: Identify Real and Imaginary Parts of Each Complex Number

The given expression is \((4-i)+(2+5i)\). Here, \(4-i\) has the real part 4 and the imaginary part \(-1i\). In \(2+5i\), the real part is 2 and the imaginary part is \(5i\).

Step 2: Add Real Parts Together

Add the real components from both complex numbers: \(4 + 2 = 6\).

Step 3: Add Imaginary Parts Together

Add the imaginary components of both complex numbers: \(-1i + 5i = 4i\).

Step 4: Write the Result in Standard Form

Combine the results from Step 2 and Step 3 to write the sum in standard form: \(6 + 4i\).

Key Concepts

Addition of Complex Numbers

To add complex numbers, you simply need to add their respective real and imaginary parts. A complex number has two components: the real part and the imaginary part. For example, in the expression \((4-i)+(2+5i)\), you will add the real parts, 4 and 2, and the imaginary parts, \-1i and 5i, separately.
Addition is pretty straightforward when you break it into these basic steps:

• Add the real parts: 4 and 2, which gives 6.
• Add the imaginary parts: \-1i and 5i, which gives 4i.

After calculating these sums independently, the solution is a new complex number. So, the sum of \((4-i)+(2+5i)\) becomes \(6 + 4i\). Notice how each part, real and imaginary, is calculated separately and then combined to give the final result. This method is logical and consistent, ensuring you correctly handle complex number addition.

Real and Imaginary Parts

Understanding the real and imaginary parts of a complex number is essential when working with them. A complex number is typically written in the form \(a + bi\), where \(a\) is the real part and \(b\) represents the imaginary part, multiplied by the imaginary unit \(i\).

Consider the complex number \(4-i\):

• The real part is 4.
• The imaginary part is \-1i (or just -1 if you disregard the i).

For the number \(2+5i\):

• The real part is 2.
• The imaginary part is 5i.

When solving problems involving complex numbers, it is crucial to distinctly identify each part. This understanding will not only aid in addition but also in other operations involving complex numbers like subtraction, multiplication, and division.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part. This format is convenient and universally recognized for expressing complex numbers.

Let's take the result from our previous problem: \(6 + 4i\). Here:

• 6 is the real part.
• 4i is the imaginary part.

Expressing complex numbers in standard form instantly clarifies which components belong to the real and imaginary dimensions. It facilitates ease in computation and consistency across mathematical problems. Remembering this format will also help you perform operations such as addition, subtraction, and even understanding more complex manipulations like finding conjugates or modulus of complex numbers.