Problem 72 - Intermediate Algebra

Question

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (6-2 i)+7 i $$

Step-by-Step Solution

Answer

The result is \(6 + 5i\).

Step 1: Identify Like Terms to Combine

In the expression \((6 - 2i) + 7i\), identify the like terms. The real number is 6, and the imaginary terms are \(-2i\) and \(7i\).

Step 2: Combine the Real Part

Since the expression \((6 - 2i) + 7i\) has only one real term, you simply carry it over without changes. Thus, the real part remains 6.

Step 3: Combine the Imaginary Parts

Add the imaginary coefficients: \(-2i\) and \(7i\). This gives \((-2 + 7)i = 5i\).

Step 4: Write the Result in Standard Form

Combine your results from the previous steps to express the solution in the standard form \(a + bi\). You get \(6 + 5i\).

Key Concepts

Imaginary Numbers

Imaginary numbers are one of the two components of complex numbers. They involve the square root of negative numbers, which is not possible within the realm of real numbers. In mathematics, we represent imaginary numbers using the letter 'i'. Here, 'i' is defined by the property that

• \( i^2 = -1 \).

Using this concept, we can handle what was once thought impossible: taking square roots of negative numbers. For instance, the square root of

• -4 is \(2i\),

which simplifies to

• \(2 \times (i \text{ or } \sqrt{-1})\).

The expression involving imaginary numbers in our example

• \(-2i + 7i\),

represents the imaginary part of the complex number in question.

Real Numbers

Real numbers serve as the foundation in understanding complex numbers. They include both positive and negative numbers, as well as zero, without involving any imaginary component. Each real number can be represented on a number line, making them very familiar to us.

In the expression we evaluated, the real part is the number 6. This is straightforward since it does not involve 'i'. When solving complex numbers, it's important to treat real numbers separately from imaginary numbers.

You should always remember to keep the process of identifying real numbers simple:

• Real numbers remain unchanged when no other real numbers are present to combine with.

In our case, since 6 is the only real number term, combining it involves simply writing it as is.

In the complete result, the real number is a basic component:

• 6 in our example.

Addition of Complex Numbers

Adding complex numbers can initially seem intimidating. However, it becomes much simpler when you break it down into its real and imaginary parts. A complex number is often expressed in the form

• \(a + bi\),

where

• 'a' is the real part,
• 'b' is the coefficient of the imaginary part.

When adding complex numbers, concentrate on combining like terms:

• Real numbers with real numbers.
• Imaginary numbers with imaginary numbers.

In our exercise, we evaluated

• \((6 - 2i) + 7i\).

First, combine the real number component, which was simply 6, as it was the only real-term present.

Next, handle the imaginary parts

• \(-2i + 7i\)

by adding the coefficients of 'i'. This gave us

• \( ( -2 + 7 ) i = 5i\).

Finally, place the individual results into the standard

• \(a + bi\)

form, resulting in

• \(6 + 5i\).

Breakdown and understanding of these processes make addition of complex numbers clear and manageable.