Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ \left(\frac{1}{2}+\frac{1}{2} i\right)+\left(\frac{1}{4}-\frac{3}{4} i\right) $$

Step-by-Step Solution

Answer

The sum is \( \frac{3}{4} - \frac{1}{4} i \).

Step 1: Identify the real and imaginary parts

In the given complex numbers, we have the first number as \( \frac{1}{2} + \frac{1}{2} i \) and the second number as \( \frac{1}{4} - \frac{3}{4} i \). The real parts are \( \frac{1}{2} \) and \( \frac{1}{4} \). The imaginary parts are \( \frac{1}{2} i \) and \( -\frac{3}{4} i \).

Step 2: Add the real parts

Add the real parts of the complex numbers: \( \frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \).

Step 3: Add the imaginary parts

Add the imaginary parts of the complex numbers: \( \frac{1}{2} i + (-\frac{3}{4} i) = \frac{2}{4} i - \frac{3}{4} i = -\frac{1}{4} i \).

Step 4: Combine the results

Combine the sum of the real parts and the sum of the imaginary parts to form the complex number: \( \frac{3}{4} - \frac{1}{4} i \).

Key Concepts

Addition of Complex Numbers

Adding complex numbers is a straightforward process, once you understand how to handle their unique components. A complex number consists of a real part and an imaginary part. When you're adding complex numbers, you should treat these parts separately. Let's consider the exercise: we have two complex numbers, \( \frac{1}{2} + \frac{1}{2} i \) and \( \frac{1}{4} - \frac{3}{4} i \).

• Start by adding the real parts: here, that would be \( \frac{1}{2} \) from the first number and \( \frac{1}{4} \) from the second number. When you add \( \frac{1}{2} + \frac{1}{4} \), you get \( \frac{3}{4} \).
• Next, add the imaginary parts: from the first number, \( \frac{1}{2} i \), and from the second, \( -\frac{3}{4} i \). Adding \( \frac{1}{2} i + (-\frac{3}{4} i) \) simplifies to \( -\frac{1}{4} i \).

Combine these results to create a new complex number, which in this case is \( \frac{3}{4} - \frac{1}{4} i \). Remember, the key to adding complex numbers correctly is handling the real and imaginary components separately.

Real and Imaginary Parts

Understanding the real and imaginary parts of a complex number is crucial for performing operations such as addition. A complex number is typically written as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. In our particular exercise, we dealt with the numbers \( \frac{1}{2} + \frac{1}{2} i \) and \( \frac{1}{4} - \frac{3}{4} i \).

• The real part of the first number is \( \frac{1}{2} \), and its imaginary part is \( \frac{1}{2} i \).
• Similarly, the second number has a real part \( \frac{1}{4} \) and an imaginary part \( -\frac{3}{4} i \).

Identifying these components allows for easier manipulation, such as addition. The real components \( \frac{1}{2} \) and \( \frac{1}{4} \) are added separately from the imaginary components \( \frac{1}{2} i \) and \( -\frac{3}{4} i \). Recognizing and remembering the structure of complex numbers is a huge step toward mastering them in mathematical operations.

Complex Number Operations

Complex number operations extend beyond basic arithmetic; however, addition is a fundamental building block for more complex calculations. Performing operations with complex numbers involves treating the real and imaginary parts independently. For instance, when multiplying two complex numbers \( (a + bi) \) and \( (c + di) \), you use the distributive property:1. Multiply the real parts: \( ac \).2. Add the cross-products of real and imaginary: \( ad + bc \).3. Multiply the imaginary parts: \( bd \) into \( i^2 \), remembering \( i^2 = -1 \) which converts to a real number.Division also follows a pattern, generally involving the multiplication by the complex conjugate to simplify the denominator to a real number, enabling standard division.Through these methods, complex number operations open the door to a variety of applications, including physics and engineering, where they simplify the handling of oscillations and waves. Beginning with fundamental operations like addition helps build the necessary skills to tackle more complex challenges in mathematics involving complex numbers.