Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: 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

The expression is \(\frac{2}{5} + \frac{4}{5}i\).

Step 1: Identify the Expression

The given expression to rewrite in the form \(a + bi\) is \(\frac{4-2i}{-5i}\). This is a fraction with a complex number in the numerator and purely imaginary number in the denominator.

Step 2: Multiply by the Conjugate

To remove the imaginary number from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(-5i\) is \(5i\). Thus, multiply the fraction by \(\frac{5i}{5i}\).

Step 3: Distribute Multiplication in the Numerator

Now we perform the multiplication in the numerator: \((4 - 2i) \cdot 5i = 20i - 10i^2\). Recall that \(i^2 = -1\).

Step 4: Simplify the Numerator

Substitute \(i^2 = -1\): \(20i - 10(-1) = 20i + 10\). Therefore, the numerator becomes \(10 + 20i\).

Step 5: Simplify the Denominator

Now calculate the denominator: \(-5i \cdot 5i = -25i^2 = 25\), since \(i^2 = -1\).

Step 6: Write the Expression in Standard Form

Combine and simplify the expression: \(\frac{10 + 20i}{25}\). Separate into real and imaginary parts: \(\frac{10}{25} + \frac{20}{25}i\).

Step 7: Simplify the Fractions

Reduce the fractions: \(\frac{2}{5} + \frac{4}{5}i\). Therefore, the expression in the form \(a + bi\) is \(\frac{2}{5} + \frac{4}{5}i\).

Key Concepts

Complex Fraction

A complex fraction is a fraction where the numerator, the denominator, or both, contain complex numbers. In the provided exercise, the expression \(\frac{4-2i}{-5i}\) is an example of a complex fraction. Here's how you deal with such expressions:

• Identify which part of the fraction has complex numbers.
• Find the conjugate of the complex number in the denominator to eliminate the imaginary unit \(i\) from it.
• Multiply both the numerator and the denominator by the conjugate – this operation simplifies the complex fraction.

Understanding these steps helps in converting complex fractions into a more understandable form of \(a + bi\), where \(a\) and \(b\) are real numbers.

Imaginary Number

Imaginary numbers extend the real number system by adding solutions to equations like \(x^2 + 1 = 0\). The unit imaginary number is denoted as \(i\), where \(i^2 = -1\). In our exercise, \(-5i\) in the denominator represents a purely imaginary number.

• Imaginary numbers are used in engineering, electromagnetism, and quantum mechanics, where operations with these values make complex computations possible.
• A purely imaginary number is one where the real part is zero, such as \(-5i\).
• The presence of \(i\) allows us to perform operations beyond the standard number line.

Grasping the concept of imaginary numbers is essential to handle fractions or expressions involving \(i\), making them a critical component of complex numbers.

Conjugate

The conjugate of a complex number is formed by changing the sign of its imaginary part. For instance, the conjugate of \(-5i\) is \(5i\). Conjugates play a critical role in simplifying complex fractions.

• Use conjugates to eliminate the imaginary unit from denominators of fractions.
• Multiplying a complex number by its conjugate results in a real number, as shown in the exercise where \(-5i \cdot 5i = 25\).
• This process, called rationalizing, is crucial because it brings the complex number to a manageable form.

The application of conjugates is not limited to fractions; it is an essential tool in many areas of mathematics and physics.

Standard Form of Complex Numbers

The standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers. In the exercise solution, the final expression \(\frac{2}{5} + \frac{4}{5}i\) is the standard form.

• "\(a\)" represents the real part, while "\(bi\)" represents the imaginary part.
• Writing in standard form is essential for identifying the components of a complex number easily.
• This form is used extensively because it's simple and straightforward, making calculations and understanding results more intuitive.
• Ensure fractions involved in \(a\) or \(b\) are simplified, as shown when \(\frac{10}{25}\) is reduced to \(\frac{2}{5}\).

Mastering the conversion to standard form simplifies the process of working with complex expressions in mathematics and related fields.