Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Find the sum of each pair of complex numbers. Express your answer in rectangular form. Do not use a calculator. $$2+6 i,-2 i$$

Step-by-Step Solution

Answer

The sum is \(2 + 4i\).

Step 1: Identify the complex numbers

We are given the complex numbers \(2 + 6i\) and \(-2i\). We'll add these two complex numbers together.

Step 2: Add the real parts

Identify the real part of each complex number: \(2\) from \(2 + 6i\) and \(0\) from \(-2i\). Add the real parts: \(2 + 0 = 2\).

Step 3: Add the imaginary parts

Identify the imaginary part of each complex number: \(6i\) from \(2 + 6i\) and \(-2i\) from \(-2i\). Add these together: \(6i - 2i = 4i\).

Step 4: Combine the results

Combine the results from Steps 2 and 3. The sum of the complex numbers \(2 + 6i\) and \(-2i\) is \(2 + 4i\).

Key Concepts

Rectangular Form

Complex numbers often appear in different formats, but one of the most common forms is the rectangular form. This is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The value of \(i\) is equal to the square root of \(-1\). In the rectangular form, the number \(a\) represents the real part and \(bi\) represents the imaginary part. Together, they define the position of the complex number in a two-dimensional plane, functioning similarly to coordinates in geometry.

Being comfortable with the rectangular form makes it much easier to perform operations like addition, subtraction, and even multiplication of complex numbers. It’s specifically handy when you're working without a calculator. While working with the rectangular form, always be mindful of separating the real and imaginary parts, as it plays a crucial role when performing arithmetic operations.

Real and Imaginary Parts

In the realm of complex numbers, understanding what makes up the real and imaginary parts is key to gripping the concept. Real and imaginary parts are the two components that form a complex number. The real part is simply the number without any \(i\) next to it. In the complex number \(2 + 6i\), the real part is \(2\). In contrast, the number \(-2i\) includes no real part, so it’s effectively \(0\).

On the flip side, the imaginary part includes the coefficient of \(i\). In our example, the imaginary part of \(2 + 6i\) is \(6i\) and for \(-2i\), it's \(-2i\). Always treat \(i\) separately to ensure you’re handling the components correctly, especially during addition. The imaginary unit \(i\) allows complex numbers to exist in a form distinct from regular real numbers. Understanding this distinction is fundamental when solving problems involving complex numbers.

Addition of Complex Numbers

To add complex numbers, you need to focus on their components separately: first adding the real parts and then the imaginary parts. Let's break it down with an example of the complex numbers \(2 + 6i\) and \(-2i\).

• Start with the real parts: Take \(2\) from \(2 + 6i\) and \(0\) from \(-2i\). Add them together, and you'll get a sum of \(2\).
• Next, handle the imaginary parts: Combine \(6i\) from \(2 + 6i\) and \(-2i\) from \(-2i\). Adding these gives you \(4i\).

You piece the results back together to get the resulting complex number \(2 + 4i\). This makes the process straightforward, by keeping each part distinct and following the rule of treating real and imaginary numbers separately. It also ensures you're capturing the full essence of these multi-dimensional numbers, and expressing them correctly back in rectangular form.