Problem 25 - Precalculus: Functions and Graphs

Question

Write the expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$\frac{4-2 i}{-5 i}$$

Step-by-Step Solution

Answer

\(\frac{2}{5} + \frac{4}{5}i\)

Step 1: Multiply by the Conjugate of the Denominator

To eliminate the imaginary number from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The denominator is \(-5i\), and its conjugate is \(5i\). Thus, multiply: \[ \frac{(4 - 2i) \cdot 5i}{(-5i) \cdot 5i} \]

Step 2: Perform the Multiplication

First, multiply the numerator: \((4 - 2i) \cdot 5i = 20i - 10i^2\). Since \(i^2 = -1\), we have: \( -10i^2 = 10\). Therefore, the expanded numerator is \(10 + 20i\). Now, multiply the denominator: \((-5i) \cdot 5i = -25i^2 = 25\).

Step 3: Simplify the Expression

Now that both the numerator and denominator are simplified, divide them by 25: \[ \frac{10 + 20i}{25} = \frac{10}{25} + \frac{20i}{25} \]Simplify both terms: \(\frac{10}{25} = \frac{2}{5}\) and \(\frac{20}{25} = \frac{4}{5}\).

Step 4: Write in the Form \(a + bi\)

Combine the simplified terms to write the final expression in the form \(a + bi\): \[ \frac{2}{5} + \frac{4}{5}i\]where \(a = \frac{2}{5}\) and \(b = \frac{4}{5}\).

Key Concepts

Complex Conjugate

In the world of complex numbers, the term "complex conjugate" sounds fancy, but it simplifies a lot of math equations. Imagine you have a complex number in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The complex conjugate of \(a + bi\) is \(a - bi\). A simple rule is to change the sign of the imaginary part. This operation is incredibly useful, especially when looking to simplify division of complex numbers.
By multiplying a complex number by its conjugate, you eliminate the imaginary parts in certain calculations, thus simplifying expressions significantly.

• If you have a complex number like \(-5i\), its conjugate is \(5i\).
• Multiplying a complex number by its conjugate makes the imaginary parts cancel out. For example, \((-5i) \times (5i) = -25i^2 = 25\). The imaginary parts "disappear" because \(i^2 = -1\).
• This technique is essential in dividing complex numbers to retain real number denominators.

Understanding the concept of complex conjugates is key to manipulating complex numbers effectively.

Division of Complex Numbers

Dividing complex numbers might seem intimidating at first, but the process becomes manageable when employing the complex conjugate. The main goal when dividing complex numbers is to "rationalize" the denominator, making it a real number. Here is the step-by-step guide:

• Identify the complex denominator, like \(-5i\).
• Multiply both the numerator and the denominator by the conjugate of the denominator. For example, if your denominator is \(-5i\), use \(5i\) as the conjugate.
• Perform the multiplication in both the numerator and denominator. This turns the denominator into a real number, thanks to the property \(i^2 = -1\).
• In the exercise, multiplying the numerator \((4-2i)\) by \(5i\) results in \(20i - 10i^2 = 10 + 20i\).
• The denominator \((-5i) \cdot (5i) = 25\) because \(-25i^2 = 25\).

Subsequently, divide the resulting real and imaginary parts by the real denominator value to simplify your expression.
When divided, you achieve the final form \(a + bi\), which in the example becomes \(\frac{2}{5} + \frac{4}{5}i\).

Practicing this method makes handling complex numbers much more straightforward and routine.

Imaginary Unit

The imaginary unit, represented by \(i\), is a curious mathematical concept often encountered when dealing with square roots of negative numbers. In math, \(i\) is defined such that \(i^2 = -1\). This property allows for operations involving complex numbers, which integrate both real numbers and imaginary parts.
Here's how it works:

• When you multiply \(i\) by itself, you get \(-1\). This is unique to the imaginary unit, setting it apart from regular numbers.
• In expressions like \((-5i) \cdot 5i\), the outcome of \(i^2\) changes an imaginary product into a real number: here, \(25\).
• The imaginary unit makes it possible to consider numbers beyond the normal number line. It allows solutions to equations that previously had no real solution, like \(x^2 + 1 = 0\).

In the realm of complex numbers, \(i\) plays a pivotal role in expanding our understanding of mathematical equations, providing structures and solutions where none seemed possible before.