Problem 87 - Algebra for College Students

Question

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{-2 i}{3-5 i}$$

Step-by-Step Solution

Answer

The quotient in standard form is \( \frac{5}{17} - \frac{3}{17}i \).

Step 1: Identify the Complex Numbers

We are given the complex fraction \( \frac{-2i}{3-5i} \). Here, the numerator is \(-2i\) and the denominator is \(3-5i\).

Step 2: Multiply the Numerator and Denominator by the Conjugate of the Denominator

To simplify the fraction and write it in standard form, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(3-5i\) is \(3+5i\). Thus, multiply by \( \frac{3+5i}{3+5i} \):\[ \frac{-2i(3+5i)}{(3-5i)(3+5i)} \]

Step 3: Simplify the Denominator

The denominator \( (3-5i)(3+5i) \) uses the difference of squares formula: \((a-b)(a+b) = a^2 - b^2\). Calculate: \(3^2 - (5i)^2 = 9 - (-25) = 9 + 25 = 34\). So, the denominator simplifies to 34.

Step 4: Expand and Simplify the Numerator

Expand the numerator \(-2i(3+5i) \): \(-2i \times 3 - 2i \times 5i = -6i - 10i^2\). Recall \(i^2 = -1\), so \(-10i^2 = 10\). Thus, the numerator becomes \(10 - 6i\).

Step 5: Combine and Simplify to Standard Form

Now, the fraction is \( \frac{10 - 6i}{34} \). Split the fraction: \( \frac{10}{34} - \frac{6i}{34} \). Simplifying each part, we get \( \frac{5}{17} - \frac{3}{17}i \).

Step 6: Write in Standard Form

The complex number in standard form is \( \frac{5}{17} - \frac{3}{17}i \), where \( \text{Re}(z) = \frac{5}{17} \) and \( \text{Im}(z) = -\frac{3}{17} \).

Key Concepts

Standard Form of Complex Numbers

Complex numbers are expressions that combine real numbers and imaginary numbers. The standard form is represented as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. This structure helps to easily identify and work with complex numbers in mathematical operations.

To convert complex numbers into standard form, simplify any fraction or product result to fit the \(a + bi\) format. This makes it straightforward to recognize and utilize the number’s characteristics in equations and real-world applications. For example, in our solution, the expression was initially given as \(\frac{-2i}{3-5i}\). After solving, it was converted to \(\frac{5}{17} - \frac{3}{17}i\), which clearly shows both the real \(\big(\frac{5}{17}\big)\) and imaginary parts \(\big(-\frac{3}{17}i\big)\).

Standard form is crucial because it maintains consistency when performing complex arithmetic operations like addition, subtraction, multiplication, and division.

Complex Conjugate

The concept of a complex conjugate involves changing the sign of the imaginary part of a complex number. For a complex number \(a + bi\), its conjugate is \(a - bi\).

Using conjugates is especially helpful in division problems involving complex numbers. Multiplying by the conjugate removes the imaginary parts in the denominator, simplifying the expression.

• For example, the conjugate of \(3 - 5i\) is \(3 + 5i\).
• By multiplying the numerator and denominator by the conjugate \(3 + 5i\), it helps eliminate the imaginary part from the denominator.

This process transformed our initial fraction \(\frac{-2i}{3-5i}\) into the simplified standard form. The conjugate trick is fundamental when seeking to simplify complex fractions.

Difference of Squares

The difference of squares is a useful algebraic identity \((a-b)(a+b) = a^2 - b^2\) that simplifies expressions involving multiplications of two binomials. It comes to the rescue when dealing with complex numbers too!

In our complex division example, the denominator \((3-5i)(3+5i)\) was simplified using the difference of squares. The formula works by multiplying the terms as follows:

• \((3)^2 - (5i)^2 = 9 - (-25) = 9 + 25 = 34\)

This technique eliminated the imaginary units in the denominator, making the final expressions easier to handle and transform into the standard form.

Understanding the difference of squares is crucial not only in algebra but also when working with complex numbers.

Imaginary Unit

The imaginary unit, denoted as \(i\), is the foundation of complex numbers, defined by the property \(i^2 = -1\). This unique number allows for the creation of the imaginary component in complex numbers.

When handling operations with the imaginary unit, remember:

• Multiplying powers of \(i\) proceeds in a cyclical pattern: \(i, -1, -i, 1\)—and repeats.
• For any integer \(n\), \(i^{2n} = (-1)^n \), while \(i^{2n+1} = (-1)^ni\).

In our example, when expanding the numerator \(-2i(3+5i)\), we used this property: \(-2i \times 3\) and \(-2i \times 5i\), knowing that \(-10i^2 = 10\) because \(i^2 = -1\).

Recognizing \(i\) and its powers is critical for simplifying expressions and solving problems involving complex numbers.