Problem 72 - Intermediate Algebra

Question

Divide. Write all answers in the form a \(+b i.\) $$ \frac{2 i}{3+8 i} $$

Step-by-Step Solution

Answer

\( \frac{16}{73} + \frac{6}{73}i \)

Step 1: Identify the Problem

We have to divide one complex number by another, specifically \( \frac{2i}{3+8i} \). To solve this, our goal is to convert the expression into the standard form \( a + bi \).

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(3 + 8i\) is \(3 - 8i\). So,\[ \frac{2i}{3+8i} \times \frac{3-8i}{3-8i} = \frac{2i(3-8i)}{(3+8i)(3-8i)} \]

Step 3: Simplify the Numerator

Distribute \(2i\) across \(3 - 8i\):\[ 2i(3-8i) = 2i \times 3 - 2i \times 8i = 6i - 16i^2 \] Recall that \(i^2 = -1\), thus:\[ 6i - 16(-1) = 6i + 16 \]

Step 4: Simplify the Denominator

The denominator is the product of a complex number and its conjugate, which results in a real number:\[ (3+8i)(3-8i) = 3^2 - (8i)^2 = 9 - 64i^2 \] Since \(i^2 = -1\), this becomes:\[ 9 - 64(-1) = 9 + 64 = 73 \]

Step 5: Combine and Write in Standard Form

Now, combine the results:\[ \frac{6i + 16}{73} = \frac{16}{73} + \frac{6}{73}i \] This is the complex number in standard form \( a + bi \), so:\[ a = \frac{16}{73}, \quad b = \frac{6}{73} \]

Step 6: Final Answer

Thus, the division results in the complex number \( \frac{16}{73} + \frac{6}{73}i \) in the form \( a + bi \).

Key Concepts

Division of Complex Numbers

Understanding the division of complex numbers is about turning a seemingly complicated fraction of complex numbers into a more manageable form. When you divide one complex number by another, you will aim to express the result in standard form, which typically looks like \( a + bi \). A direct division is not straightforward because complex numbers have both real and imaginary parts. Instead, you'll use a different method.
The trick here is to eliminate the imaginary unit from the denominator to simplify your expression. This is done by multiplying the numerator and the denominator by the conjugate of the denominator.
The conjugate of a complex number, say \( c + di \), is \( c - di \). Multiplying the numerator and the denominator by this conjugate turns the denominator into a real number.
Following these steps allows you to handle complex fractions easily and to express the answer in the standard complex number format \( a + bi \).

Standard Form a+bi

Complex numbers are typically expressed in what we call "standard form," which looks like \( a + bi \). Here, \( a \) is the real part and \( b \) is the imaginary part. The \( i \) stands for the imaginary unit, which is defined as \( i^2 = -1 \).
Writing a complex number in standard form helps in better representation and understanding of the number. It clearly separates the real and imaginary parts, making calculations and comparisons more efficient.
For instance, in the problem given, after simplifying through multiplication with the conjugate, we end up with:

• Real part: \( \frac{16}{73} \)
• Imaginary part: \( \frac{6}{73} \)

This makes the complex number \( \frac{16}{73} + \frac{6}{73}i \) easy to understand and use in further mathematical operations or problems.

Conjugate of Complex Numbers

The concept of the conjugate is fundamental when dealing with complex numbers, especially in division. The conjugate of a complex number is found by flipping the sign of its imaginary part. So, the conjugate of a complex number \( c + di \) is \( c - di \).
Using the conjugate is a crucial step when dividing complex numbers. When you multiply a complex number and its conjugate, the imaginary parts cancel out, leaving you with only a real number. This is because of the identity:

• \((c+di)(c-di) = c^2 - (di)^2\)

Since \(i^2 = -1\), the expression simplifies further, ensuring that your denominator becomes a real number.
In our example, the conjugate of \(3 + 8i\) is \(3 - 8i\). Utilizing this, our denominator turns into \(73\), simplifying the division process and allowing us to express the final result in the desired form \(a + bi\).