Problem 80 - Precalculus : Building Concepts and Connections

Question

Use the following definition. A complex number \(a+b i\) is often denoted by the letter \(z .\) Its conjugate, \(a-b i\) is denoted by \(\bar{z}\). Show that the imaginary part of \(z\) is equal to \(\frac{z-\bar{z}}{2 i}\)

Step-by-Step Solution

Answer

The imaginary part of the complex number \(z\) is \(\frac{z-\bar{z}}{2i}\)

Step 1: Understand the complex number and its conjugate

The complex number \(z\) is given as \(z = a + bi\), where \(a\) and \(b\) are real numbers. The conjugate of \(z, \bar{z}\) is \(a - bi\). The imaginary part of \(z\) is represented as \(b\). The goal is to demonstrate that this \(b\) can be obtained from the formula \(\frac{z-\bar{z}}{2i}\)

Step 2: Substitute values of \(z\), \(\bar{z}\) into the formula

Substitute the values of \(z = a + bi\) and \(\bar{z} = a - bi\) into the formula \(\frac{z-\bar{z}}{2i}\) to get \(\frac{(a + bi) - (a - bi)}{2i}\)

Step 3: Simplify the Equation

Simplify the equation from step 2 to obtain \(\frac{2b}{2i}\), which can be further simplified to give \(b = \frac{z-\bar{z}}{2i}\). This indicates that the imaginary part of the complex number \(z\) is indeed \(\frac{z-\bar{z}}{2i}\)

Key Concepts

Imaginary Part of Complex Number

When we dive into complex numbers, understanding their structure is essential. A complex number is like a two-part recipe, comprising a real number and an imaginary part. It is represented in the form of \(a + bi\), where \(a\) stands for the real part, while \(b\) stands for the imaginary part, and \(i\) is the square root of -1.

For instance, if we have the complex number \(3 + 4i\), the real part is 3, and the imaginary part is 4. The magic of the imaginary part comes into play because it helps us to solve equations that we can't with real numbers alone, like the square root of negative numbers.

Understanding how to extract this imaginary part is crucial for algebraic manipulation and further applications in mathematics and engineering. By using the formula \(\frac{z-\bar{z}}{2i}\), you can neatly isolate the imaginary part \(b\) of the complex number \(z\), making calculations tidy and manageable.

Complex Conjugate

The concept of the complex conjugate is simple yet powerful in the realm of complex numbers. If you have a complex number \(z = a + bi\), its complex conjugate, represented as \(\bar{z}\), is \(a - bi\). It's like looking in a mirror but across the real number line, swapping the sign in front of the imaginary part.

Why is this useful?

The complex conjugate comes in handy when performing operations like division or when finding the magnitude of a complex number. Really, it's all about balance and symmetry. The conjugate pairs up with the original number to eliminate the imaginary part when multiplied, leaving you with a real number. It's like a math dance where two complex numbers come together and end up with their feet firmly on the ground of real numbers.

Algebraic Manipulation

Algebraic manipulation in the context of complex numbers involves a variety of skills, including adding, subtracting, and multiplying these intriguing number pairs. One of the neat tricks you'll often use is related to the imaginary part of a complex number and its conjugate.

The process involves manipulating expressions to simplify them or to express them in a desired form. For example, if you want to avoid division by a complex number, you multiply by the conjugate. This strategic move transforms the denominator into a real number, which is much more manageable.

Practice makes perfect

Algebraic manipulation is like a puzzle, where understanding the relationships and properties of complex numbers allows you to reconfigure parts until you reveal the most elegant form. The more problems you solve, the better you'll get at spotting patterns and techniques, thereby deepening your comprehension of complex numbers and their behaviors.