Problem 55 - Intermediate Algebra

Question

Perform each indicated operation. Write the result in the form \(a+b i\). $$ \frac{4-5 i}{2 i} $$

Step-by-Step Solution

Answer

The result is \(-\frac{5}{2} - 2i\).

Step 1: Multiply Numerator and Denominator by the Conjugate of the Denominator

To simplify the division of complex numbers, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of the denominator \(2i\) is \(-2i\). The expression becomes: \[ \frac{(4 - 5i) imes (-2i)}{(2i) imes (-2i)} \]

Step 2: Perform the Multiplications

First, calculate the denominator: \[ (2i) imes (-2i) = -4i^2 = 4 \] since \(i^2 = -1\).Next, calculate the numerator: \((4 - 5i) imes (-2i) = 4 imes (-2i) + (-5i) imes (-2i) = -8i + 10i^2 = -8i + 10(-1) = -8i - 10\).

Step 3: Combine and Simplify

Now, put the results together:\[ \frac{-8i - 10}{4} = \frac{-10}{4} + \frac{-8i}{4} \]Simplify both parts:\[ -\frac{5}{2} - 2i \]

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

The expression is now in the form \(a + bi\), where\[ a = -\frac{5}{2} \] and \[ b = -2 \].This gives us the final form: \(-\frac{5}{2} - 2i\).

Key Concepts

Imaginary Unit

In the world of complex numbers, the concept of the imaginary unit is essential. The imaginary unit, denoted as \( i \), is defined by the property \( i^2 = -1 \). This definition is the foundation for working with complex numbers and allows mathematicians to extend the real number system to include solutions to certain equations that don't have solutions within the real numbers alone.
Real numbers are all numbers we can find on the number line like 0, 1, 2, or even fractions like 1/2. Imaginary numbers, on the other hand, involve the term \( i \) and appear when we deal with square roots of negative numbers. For instance, the square root of \(-1\) isn’t a number we can place on a traditional number line, hence, we use \( i \) to represent it.

• Basic property: \( i^2 = -1 \)
• Expression example: \( 2i + 3 \) where \( 3 \) is real and \( 2i \) is imaginary.

Understanding \( i \) is crucial as it helps represent complex numbers in the form \( a + bi \), where \( a \) and \( b \) are real numbers.

Complex Conjugate

The idea of the complex conjugate is pivotal in simplifying complex expressions, especially during division. If you have a complex number in the form \( a + bi \), its conjugate is \( a - bi \). It’s simply obtained by changing the sign of the imaginary part.
Complex conjugates are useful because when you multiply a complex number by its conjugate, the result is a real number. This is particularly helpful when you need to rationalize the denominator of a complex fraction.

Consider a complex number \( z = 6 + 4i \):

• Conjugate: \( \bar{z} = 6 - 4i \)
• Multiplying \( z \) by \( \bar{z} \) gives \((6+4i)(6-4i) = 36 - 16i^2 = 36 + 16 = 52 \)

Using conjugates simplifies the division of complex numbers, as we can turn a fraction with complex terms into a simpler real expression.

Division of Complex Numbers

Dividing complex numbers involves a straightforward method when you understand the conjugate technique. In any division task involving complex numbers, the aim is to eliminate the imaginary unit \( i \) from the denominator. This is done by multiplying both the numerator and denominator by the complex conjugate of the denominator.
For instance, with the problem \( \frac{4-5i}{2i} \), multiply numerator and denominator by the conjugate of \( 2i \), which is \( -2i \):

• Numerator becomes \( (4-5i) \times (-2i) = -8i - 10 \).
• Denominator becomes \( (2i) \times (-2i) = 4 \).

Ultimately, the division results in a real number plus an imaginary number: \( - \frac{5}{2} - 2i \). This conversion from a complex division to a straightforward expression provides clarity and resolves the initially cumbersome fraction.

Simplification of Complex Expressions

Simplifying complex expressions often involves a few key steps: multiplication, combination, and reduction. When terms involve the imaginary unit \( i \), keeping track of the imaginary unit’s properties is vital. Remember, the primary property is that \( i^2 = -1 \).
When dealing with complex expressions, especially those involving division, the conjugate’s multiplication simplifies them, turning them into expressions with no imaginary component in the denominator.

Consider a division like \( \frac{4-5i}{2i} \):

• Multiply by the conjugate: numerator and denominator resolve to real terms.
• Combine like terms: ensures clarity of result, split into real and imaginary.
• Simplify: breaking down coefficients, as in reducing \( \frac{-10}{4} \) to \( -\frac{5}{2} \).

This process renders even the complex expressions into understandable and practical forms, often making manipulation and further calculations easier and more accurate.