Problem 19 - Algebra 2 and Trigonometry

Question

In \(18-25,\) write the complex conjugate of each number. $$ 2-5 i $$

Step-by-Step Solution

Answer

The complex conjugate of \(2 - 5i\) is \(2 + 5i\).

Step 1: Understanding Complex Numbers

A complex number is written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. For the complex number \(2 - 5i\), \(a = 2\) and \(b = -5\).

Step 2: Defining the Complex Conjugate

The complex conjugate of a complex number \(a + bi\) is obtained by changing the sign of the imaginary part, resulting in \(a - bi\).

Step 3: Applying the Conjugate Formula

For the given complex number \(2 - 5i\), the complex conjugate is obtained by changing the sign of the imaginary part \(-5i\) to \(+5i\).

Step 4: Writing the Final Answer

Thus, the complex conjugate of \(2 - 5i\) is \(2 + 5i\).

Key Concepts

Complex Conjugate

In the world of complex numbers, the concept of the complex conjugate is quite fundamental. A complex conjugate switches the sign of the imaginary part while leaving the real part unchanged. Consider a generic complex number expressed in the form \(a + bi\), where \(a\) represents the real part and \(bi\) the imaginary part. The complex conjugate of this number is \(a - bi\).

This concept plays an important role especially when it comes to simplifying expressions and solving equations involving complex numbers. For instance, multiplying a complex number by its conjugate will always result in a real number. This is because the imaginary parts cancel out, leaving behind only real terms. Hence, it is not just a mathematical gimmick but a powerful tool.

To see this in action, let's explore the complex number \(2 - 5i\). According to our rule, its complex conjugate would be \(2 + 5i\), effectively reflecting the original number across the real axis in the complex plane.

Real Part

The real part of a complex number is the component that does not involve the imaginary unit \(i\). In any complex number \(a + bi\), \(a\) is known as the real part. It acts much like any other real number, given its independence from the imaginary companion.

This means that even though we deal with complex numbers, the real part is easy to handle if you have already grasped standard arithmetic operations. For the number \(2 - 5i\), the real part is \(2\). This concept seems straightforward, but its implications are profound when we dive into more complex operations like addition, subtraction, multiplication, and division of complex numbers.

To add or subtract complex numbers, you merely combine the real parts with each other, treating them as regular numbers. Thus, thinking about the real part separately can simplify your work tremendously, as it keeps operations transparent and computations error-free.

Imaginary Part

The imaginary part of a complex number involves the coefficient of the imaginary unit \(i\). If we consider a complex number \(a + bi\), then \(b\) is the imaginary part. Despite its name, the imaginary part is quite real in mathematical computations.

Imaginary numbers arise in scenarios where solutions cannot be expressed along the real number line, allowing for the extension into another dimension of thought. For the complex number \(2 - 5i\), the imaginary part is \(-5\). Remember, the imaginary unit \(i\) itself is defined by the property \(i^2 = -1\), which makes it a unique entity in mathematics.

When adding or subtracting complex numbers, you will handle the imaginary parts distinctly from the real ones. It's quite similar to how you deal with like terms in algebra. This separation keeps mathematical procedures clean and manageable, whether you are solving equations or simplifying expressions.