Problem 37 - Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i.\) $$ (3+4 i)+(5-6 i) $$

Step-by-Step Solution

Answer

The answer is \(8 - 2i\).

Step 1: Identify the Real Components

The given expression is \((3+4i) + (5-6i)\). Begin by identifying the real parts in both complex numbers. Here, the real numbers are 3 from the first expression and 5 from the second expression.

Step 2: Identify the Imaginary Components

Next, identify the imaginary parts of each complex number. The imaginary part in the first complex number is 4i, and in the second, it is -6i.

Step 3: Add the Real Components

Add the real parts from each complex number: 3 + 5. This equals 8, which will be the real part of the result.

Step 4: Add the Imaginary Components

Add the imaginary parts from each complex number: \(4i + (-6i)\). This simplifies to \(-2i\), which will be the imaginary part of the result.

Step 5: Write the Final Answer

Combine the results of Step 3 and Step 4 to express the answer in the form \(a + bi\). Thus, the final answer is \(8 - 2i\).

Key Concepts

Addition of Complex Numbers

Addition of complex numbers involves combining the real and imaginary parts separately. Imagine it as a two-step process where you deal with real numbers first, then imaginary numbers. Consider the example:

• Given: \[ (3 + 4i) + (5 - 6i) \]

To add these complex numbers:

• First, add the real components: 3 (from the first complex number) and 5 (from the second complex number).
• Second, add the imaginary components: 4i and -6i.

Bringing these two results together, the final sum in this case is 8 for the real part and -2i for the imaginary part. Thus, \[ (3 + 4i) + (5 - 6i) = 8 - 2i \]Remember, when performing addition, always keep the components separate until the final step where they join back together in one standard notation.

Real and Imaginary Components

Every complex number is made up of two parts: the real component and the imaginary component. It looks like this: \[ a + bi \]

• 'a' is the real component: a regular number we use every day.
• 'bi' is the imaginary component: a bit different as it involves 'i', the imaginary unit, where \[ i^2 = -1 \].

Consider \[ (3 + 4i) \]: here 3 is the real part, and 4i is the imaginary part. When examining complex numbers, always locate these two components. They will guide you in operations such as addition and subtraction, allowing you to work with these numbers just like regular two-dimensional vectors on a graph.

Expressing Complex Numbers in Standard Form

Standard form for complex numbers is like keeping everything neat and tidy. This form is expressed as \[ a + bi \], where 'a' is the real part, and 'bi' is the imaginary part. Think of it as a simple address where the real number lives normally, and the 'bi' part has its own section.

• When you finish any complex arithmetic task, like addition, make sure your result reflects this form.
• Use this form to clearly see both the real and imagined parts at once.

For example, when you add \[ (3 + 4i) + (5 - 6i) \], you reorganize and finish with \[ 8 - 2i \]. This keeps the format consistent and the math easy to follow.