Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ \frac{2-3 i}{4+3 i} $$

Step-by-Step Solution

Answer

The simplified complex number is \(-\frac{1}{25} - \frac{18}{25}i.\)

Step 1: Identify the Task

We need to simplify the expression \(\frac{2-3i}{4+3i}\) by performing the division and expressing the result as a simplified complex number.

Step 2: Write in Standard Complex Division Form

To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of \(4+3i\) is \(4-3i\). Multiply the fraction by \(\frac{4-3i}{4-3i}\).

Step 3: Multiply the Numerator

Calculate \((2 - 3i) \times (4 - 3i)\) using the distributive property:\( (2)(4) + (2)(-3i) + (-3i)(4) + (-3i)(-3i) = 8 - 6i - 12i + 9i^2.\)Remember \(i^2 = -1\), so substitute \(9i^2\) with \(-9\). Thus, the numerator simplifies to \(8 - 6i - 12i - 9 = -1 - 18i.\)

Step 4: Multiply the Denominator

Calculate \((4 + 3i) \times (4 - 3i)\) as follows: \( (4)(4) + (4)(-3i) + (3i)(4) + (3i)(-3i) = 16 - 12i + 12i - 9i^2.\)Since \(-9i^2\) equals \(9\), the denominator simplifies to \(16 + 9 = 25.\)

Step 5: Simplify the Expression

Now we have \(\frac{-1 - 18i}{25}.\) Split this into two separate fractions: \(-\frac{1}{25} - \frac{18}{25}i,\) which is the simplified form of the complex number.

Key Concepts

Complex Conjugate

To divide complex numbers effectively, one of the key strategies is to use the complex conjugate. The complex conjugate of a complex number is derived by changing the sign of the imaginary part. For example, if you have a complex number like \(4 + 3i\), its conjugate will be \(4 - 3i\). This concept helps reduce the imaginary units in the denominator when performing division.

• The complex conjugate keeps the real part the same.
• It flips the sign of the imaginary part, thereby balancing out the imaginary components when multiplied.

The magic of the complex conjugate lies in its ability to transform complex division into a simpler form. By multiplying both the numerator and the denominator by the conjugate of the denominator, you can eliminate the imaginary part from the denominator.
This process ensures that the result is presented in a more manageable, simplified form.

Distributive Property

The distributive property is a fundamental concept in algebra that makes complex number multiplication straightforward. It states that \(a(b + c) = ab + ac\). This property helps you multiply two complex numbers with ease, ensuring that each term in the first complex number multiplies with every term in the second.
Consider the multiplication of \((2 - 3i)\) and \((4 - 3i)\):

• First, you multiply the real parts: \((2)(4) = 8\).
• Then, cover the imaginary combinations: \( (2)(-3i) = -6i\) and \((-3i)(4) = -12i\).
• Finally, focus on the imaginary parts: \((-3i)(-3i) = 9i^2\) which equals \(-9\) because \(i^2 = -1\).

Gathering all these, you end up with \(8 - 6i - 12i - 9\). Using the distributive property methodically allows you to see each step clearly, reducing the likelihood of mistakes and making complex operations less daunting.

Simplified Complex Number

The aim of dividing complex numbers is to express the result as a simplified complex number. A simplified complex number takes the form \(a + bi\), where \(a\) and \(b\) are real numbers. In our example, dividing by a complex number like \(4 + 3i\) initially appeared challenging because it left unnecessary terms in the denominator.
Simplification involves:

• Multiplying the numerator and the denominator by the complex conjugate of the denominator.
• Simplifying the resulting expression to remove any imaginary unit \(i\) from the denominator.
• Expressing the final result as \(a + bi\) form, making it both real and imaginary components clear.

After following such simplification steps in our example, the expression was rewritten as \( -\frac{1}{25} - \frac{18}{25}i \). This approach not only provides a clearer understanding of the result but also maintains precision.