Problem 86 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\(\frac{4i}{5+2i} = \frac{8}{29} + \frac{20}{29}i\).

Step 1: Multiply by the Conjugate

To simplify \( \frac{4i}{5+2i} \), multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(5 + 2i\) is \(5 - 2i\). Thus, we have: \[ \frac{4i}{5+2i} \cdot \frac{5-2i}{5-2i} \] This will help us eliminate the imaginary part in the denominator.

Step 2: Simplify the Numerator

First, simplify the numerator: \[ 4i \cdot (5 - 2i) = 20i - 8i^2 \]Since \(i^2 = -1\), substitute this in: \[ 20i - 8(-1) = 20i + 8 \]So, the simplified numerator is \(8 + 20i\).

Step 3: Simplify the Denominator

Next, simplify the denominator using the difference of squares formula: \[ (5+2i)(5-2i) = 5^2 - (2i)^2 = 25 - 4i^2 \]Again, substituting \(i^2 = -1\), we have: \[ 25 - 4(-1) = 25 + 4 = 29 \]Thus, the denominator is 29.

Step 4: Divide and Express in Standard Form

Divide the simplified numerator by the simplified denominator: \[ \frac{8 + 20i}{29} = \frac{8}{29} + \frac{20i}{29} \]This is already in the standard form for a complex number, \(a + bi\), where \(a = \frac{8}{29}\) and \(b = \frac{20}{29}\).

Key Concepts

Imaginary Unit

Complex numbers often feature the imaginary unit, denoted as \(i\). The imaginary unit is pivotal in defining complex numbers and represents the square root of negative one, where \(i^2 = -1\). This unique property of \(i\) allows us to simplify complex expressions involving imaginary components. For example, when simplifying \(-8i^2\), we use \(i^2 = -1\) to transform it into \(-8(-1) = 8\).
This substitution simplifies expressions and is frequently used when operating on complex numbers, like in multiplication or division. Remember, the imaginary unit \(i\) is the foundation of operations involving complex numbers, making it crucial to understand how \(i\) works within mathematical expressions.

Conjugate Multiplication

When dealing with complex numbers, especially in division, conjugate multiplication is a useful technique. In our exercise, the division involved the complex number denominator \(5 + 2i\).
The conjugate of a complex number \(a + bi\) is \(a - bi\). So, for \(5 + 2i\), its conjugate is \(5 - 2i\).
Why multiply by the conjugate? Doing so eliminates the imaginary part in the denominator, simplifying the division to a real number. This is done using the difference of squares formula:
* For our example: * \((5 + 2i)(5 - 2i) = 5^2 - (2i)^2 = 25 - 4(-1) = 29\). This results in a real number, making the division straightforward and allowing us to express the quotient in the standard form.

Standard Form of Complex Numbers

A complex number is usually written in the standard form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. This form simplifies complex number representation and mathematical operations.
In our example, after simplification, the number \(\frac{8 + 20i}{29}\) is broken down into: * Real component: \(\frac{8}{29}\)* Imaginary component: \(\frac{20i}{29}\)
Combining these we have \(\frac{8}{29} + \frac{20i}{29}\), which is already in the standard form. This representation allows for easy reading and understanding of the complex number's components, which is essential for performing further arithmetic operations and analysis.