Problem 22 - Algebra 2 and Trigonometry

Question

In \(18-25,\) write the complex conjugate of each number. $$ \frac{1}{2}-3 i $$

Step-by-Step Solution

Answer

The complex conjugate of \( \frac{1}{2} - 3i \) is \( \frac{1}{2} + 3i \).

Step 1: Identify the Conjugate Definition

The complex conjugate of a complex number \( a + bi \) is \( a - bi \). For a number expressed as \( a - bi \), its conjugate is \( a + bi \).

Step 2: Identify Real and Imaginary Parts

The given number is \( \frac{1}{2} - 3i \). Here, \( a = \frac{1}{2} \) and \( b = 3 \).

Step 3: Apply the Conjugate Formula

Using the conjugate formula \( a + bi \) for the given number \( \frac{1}{2} - 3i \), the conjugate is \( \frac{1}{2} + 3i \).

Step 4: Write the Result

The complex conjugate of \( \frac{1}{2} - 3i \) is \( \frac{1}{2} + 3i \).

Key Concepts

Complex Conjugate

The concept of a complex conjugate is key when dealing with complex numbers. A complex number is composed of a real part and an imaginary part in the form \( a + bi \), where \( a \) is the real component, and \( bi \) is the imaginary component. The complex conjugate of this number simply changes the sign of the imaginary part, making it \( a - bi \).
This reversal of sign is quite helpful in various mathematical operations, especially in simplifying divisions involving complex numbers. By turning the imaginary component's sign, we effectively create a mirror image of the original number on the complex plane. Thus, if we have \( \frac{1}{2} - 3i \), its complex conjugate becomes \( \frac{1}{2} + 3i \).

• Identifying the conjugate makes it easier to rationalize denominators when dividing complex numbers.
• It can also be used to find the magnitude of a complex number by multiplying the number by its conjugate.

Understanding conjugates is therefore a fundamental step in mastering the arithmetic of complex numbers.

Real and Imaginary Parts

Complex numbers, written in the form \( a + bi \), stand out from regular numbers due to their two components: the real part \( a \) and the imaginary part \( b \). These parts function together to expand our understanding of numbers beyond the traditional number line into a plane called the complex plane.
For the given exercise, we have \( \frac{1}{2} - 3i \), where \( \frac{1}{2} \) represents the real part and \(-3i \) is the imaginary part. The imaginary unit \( i \) is defined as \( \sqrt{-1} \). This definition allows us to handle square roots of negative numbers, which was previously not possible with just real numbers.

• The real part controls horizontal movement on the complex plane.
• The imaginary part allows for vertical movement.

By recognizing and working with these parts separately, we can perform operations such as addition, subtraction, and finding complex conjugates.

Algebraic Expressions

Algebraic expressions involving complex numbers follow many of the same rules as real-number expressions, with some unique quirks. Each complex number can be manipulated and expressed in terms of its real and imaginary parts, allowing for a wide variety of algebraic operations.
In the provided exercise with \( \frac{1}{2} - 3i \), we manipulate the expression by applying the rules of complex conjugation. Recognizing the separate real and imaginary parts lets us change \( a - bi \) to its conjugate \( a + bi \). This becomes especially useful when dealing with division.

• When multiplying or dividing complex numbers, using conjugates can simplify expressions.
• Complex numbers can be added and subtracted by simply combining like terms — real parts with real parts, imaginary parts with imaginary parts.

Understanding these algebraic principles is important when you enter more advanced study areas like engineering or physics, where complex numbers are frequently used.