Problem 88 - Algebra for College Students

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 quotient is \(1 - \frac{1}{2}i\).

Step 1: Identify the Problem

We need to divide two complex numbers: \(-5i\) and \(2 - 4i\). Our goal is to express the quotient in the standard form of a complex number, which is \(a + bi\), where \(a\) and \(b\) are real numbers.

Step 2: Multiply by the Conjugate

To eliminate the imaginary part in the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. Here, the conjugate of \(2 - 4i\) is \(2 + 4i\). Multiply the expression: \[\frac{-5i}{2-4i} \times \frac{2+4i}{2+4i}\]

Step 3: Perform the Multiplication

Calculate the multiplication in the numerator and the denominator: - For the numerator: \(-5i \times (2 + 4i) = -10i - 20i^2\)- Simplify using \(i^2 = -1\): \(-10i + 20\)- For the denominator: \((2 - 4i)(2 + 4i) = 4 + 8i - 8i - 16i^2\) Simplify using \(i^2 = -1\): \(4 + 16 = 20\)

Step 4: Simplify the Quotient

Recall the equation from the previous step: \[\frac{-10i + 20}{20}\]Divide each term by 20 to simplify:\[-\frac{10}{20}i + \frac{20}{20}\]Finally, simplify:\[-\frac{1}{2}i + 1\]

Step 5: Write the Final Result

The simplified form of the complex number is \[1 - \frac{1}{2}i\]This is the quotient in the standard form \(a + bi\).

Key Concepts

Quotients of Complex Numbers

When you divide complex numbers, the process isn't as straightforward as dividing simple numerical fractions. Complex numbers have both a real and an imaginary part, requiring us to use specific methods for division. A complex number typically takes the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The goal when finding the quotient of two complex numbers is to rewrite the division in this standard form.Here's the basic procedure:

• Form a fraction with the given complex numbers.
• Multiply the numerator and the denominator by the conjugate of the denominator to get rid of the imaginary part in the denominator.
• Simplify the result to express it in the standard form \(a + bi\).

By transforming the denominator to a real number using the conjugate, we handle the division like normal arithmetic.

Standard Form of Complex Numbers

The standard form of a complex number is represented as \(a + bi\). This format is useful because it clearly separates the real part \(a\) and the imaginary part \(bi\) of the complex number. Expressing complex numbers in this form simplifies operations like addition, subtraction, and multiplication.A few points to remember:

• \(a\) and \(b\) must be real numbers.
• \(i\) stands for the imaginary unit, where \(i^2 = -1\).
• Every complex number has a conjugate \(a - bi\), which changes the sign of the imaginary part.

By adhering to this standard format, one can easily perform calculations and ensure consistency in expressions. This is especially critical when simplifying results to the standard form after operations like division.

Multiplying by the Conjugate

Multiplying by the conjugate is a crucial step when you need to simplify the division of complex numbers. The conjugate of a complex number \(a + bi\) is \(a - bi\). When you multiply a complex number by its conjugate, it eliminates the imaginary component, yielding a real number. This process helps to simplify expressions involving complex numbers.Here's why this technique works:

• The product of a complex number and its conjugate is always real: \((a+bi)(a-bi) = a^2 + b^2\).
• This technique is especially useful when simplifying the denominator of a fraction involving complex numbers.

For example, in the division problem we solved, the complex number \(2 - 4i\) has the conjugate \(2 + 4i\). By multiplying both the numerator and denominator by this conjugate, the denominator becomes a real number (20), making the final quotient easier to simplify and express in standard form.