Problem 11 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ (0+3 i)+(0-3 i) $$

Step-by-Step Solution

Answer

The sum is \(0\).

Step 1: Identify the Components

We have the complex numbers \(0 + 3i\) and \(0 - 3i\). These are in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

Step 2: Add Real Parts

The real parts of both complex numbers are both 0. When we add these, we get: \[0 + 0 = 0\]

Step 3: Add Imaginary Parts

Next, we add the imaginary parts of the two complex numbers. This is done as follows: \[3i + (-3i) = 0\]

Step 4: Combine Results

Combine the results from the real and imaginary parts to form the final answer: \[0 + 0i = 0\]

Key Concepts

Addition of Complex Numbers

Complex numbers consist of both a real part and an imaginary part and are generally expressed in the form \(a + bi\). Here, \(a\) is the real part, and \(bi\) is the imaginary part. Adding two complex numbers is a straightforward process, as demonstrated in the exercise.

• First, add the real parts of the numbers.
• Then, add the imaginary parts.
• Combine these sums to get your final result.

Adding \((0 + 3i)\) and \((0 - 3i)\), we take the real parts (0 and 0), which add up to 0. Next, the imaginary parts \(3i\) and \(-3i\) add up to 0. Hence, the total sum is 0.
This is because imaginary units cancel each other out, illustrating a unique characteristic of complex addition.

Imaginary Numbers

Imaginary numbers are numbers that involve the imaginary unit, represented by \(i\). The fundamental property of this number is \(i^2 = -1\). Situations arise in mathematics, especially involving square roots of negative numbers, where real numbers are not sufficient, necessitating the use of imaginary numbers.
For example, when you encounter \(\sqrt{-1}\), it is represented as \(i\). In our exercise, \(3i\) is an imaginary number. Imaginary numbers are crucial in complex number calculations even when they cancel each other, like in our exercise solution \(3i + (-3i) = 0\).