Problem 88 - Intermediate Algebra

Question

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{-5 i}{2-4 i} $$

Step-by-Step Solution

Answer

The answer is \(1 - 0.5i\).

Step 1: Identify the components

Write the complex fraction \( \frac{-5i}{2-4i} \). The numerator is \(-5i\), and the denominator is \(2-4i\).

Step 2: Multiply by the conjugate

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

Step 3: Simplify the denominator

Calculate the denominator \((2-4i)(2+4i)\) using the difference of squares formula: \((a-b)(a+b) = a^2 - b^2\). Here, \(a = 2\), \(b = 4i\), so:\[ 2^2 - (4i)^2 = 4 - 16(-1) = 4 + 16 = 20 \].

Step 4: Simplify the numerator

Distribute \(-5i\) in the numerator: \(-5i(2+4i) = -5i \times 2 + (-5i) \times 4i\) which simplifies to:\[ -10i - 20i^2 \]. Knowing \(i^2 = -1\), substitute to get \[-10i + 20 = 20 - 10i\].

Step 5: Combine results

Now we have the fraction: \[ \frac{20-10i}{20} \]. This simplifies by dividing both terms of the numerator by 20:\[ \frac{20}{20} + \frac{-10i}{20} = 1 - 0.5i \].

Step 6: Express in standard form

The expression \(1 - 0.5i\) is already in the standard form \(a + bi\), where \(a = 1\) and \(b = -0.5\).

Key Concepts

Complex Conjugate

The concept of a complex conjugate is crucial when dealing with complex numbers. A complex conjugate is formed by changing the sign of the imaginary part of a complex number. For a complex number expressed in the form \(a + bi\), its conjugate is \(a - bi\). This simple change of sign is highly useful in operations involving complex numbers, especially division.

When dividing complex numbers, you use the complex conjugate of the denominator to eliminate the imaginary unit from the denominator. By multiplying the numerator and the denominator by this conjugate, you simplify the expression into a real number. This step is key to expressing complex numbers neatly in their standard form, \(a + bi\).

• The original denominator: \(2 - 4i\)
• Its conjugate: \(2 + 4i\)

Using the conjugate helps convert the denominator into a real number and is a very efficient strategy when simplifying fractions involving complex numbers.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental unit in the complex number system. It is defined by the property that \(i^2 = -1\). This definition might sound abstract, but it serves a crucial role in mathematics by helping to handle the square roots of negative numbers.

When working with imaginary numbers, it's crucial to understand how they operate with real numbers. Imaginary units have the following traits in arithmetic operations:

• \(i^2 = -1\)
• \(i^3 = i^2 \times i = -i\)
• \(i^4 = i^2 \times i^2 = 1\)

Knowing these cycles helps in simplifying expressions involving powers of \(i\). In division problems like the one in the exercise, substituting \(i^2 = -1\) simplifies expressions, as seen when the term \(-20i^2\) turns into \(+20\) due to this substitution.

Simplifying Fractions

Simplifying fractions in the context of complex numbers is slightly different from regular fractions. The main goal is to express the quotient in a standard complex form \(a + bi\), where \(a\) and \(b\) are real numbers.

To simplify a complex fraction, you often multiply both the numerator and the denominator by the complex conjugate of the denominator. This step transforms the denominator into a real number via the difference of squares formula: \((a-b)(a+b) = a^2 - b^2\). In our example, this turns \((2-4i)(2+4i)\) into 20, a real number.

• The denominator, \((2 - 4i)(2 + 4i)\), simplifies to 20.
• The numerator expands to \(-10i + 20\).

The fraction \(\frac{20 - 10i}{20}\) is ultimately simplified by dividing each part of the complex number by the real denominator, resulting in the standard form \(1 - 0.5i\). This process highlights the utility of the complex conjugate and careful algebraic manipulation to achieve a solution that is clear and usable.