Problem 95 - A Graphical Approach to Precalculus with Limits

Question

Divide as indicated. Write each quotient in standand form. $$\frac{9+2 i}{2+i}$$

Step-by-Step Solution

Answer

The quotient in standard form is \(4 - i\).

Step 1: Understand the Problem

We need to divide the complex number \(9 + 2i\) by \(2 + i\) and express the result in standard form \(a + bi\).

Step 2: Multiply by the Conjugate

To simplify this division, multiply the numerator and the denominator by the conjugate of the denominator \(2 - i\):\[\frac{9 + 2i}{2 + i} \cdot \frac{2 - i}{2 - i}\]

Step 3: Simplify the Denominator

Using the identity \((a+b)(a-b) = a^2 - b^2\), calculate the denominator:\[(2+i)(2-i) = 2^2 - i^2 = 4 - (-1) = 5\]

Step 4: Distribute in the Numerator

Now, distribute \(9 + 2i\) with \(2 - i\):\[(9 + 2i)(2 - i) = 9 \cdot 2 + 9 \cdot (-i) + 2i \cdot 2 + 2i \cdot (-i)\] Solve the individual terms to get:\[18 - 9i + 4i - 2i^2 = 18 - 9i + 4i + 2\] Using \(i^2 = -1\), this simplifies to:\[18 + 2 - 5i = 20 - 5i\]

Step 5: Write the Result in Standard Form

Divide each term of the result by the denominator 5:\[\frac{20}{5} - \frac{5i}{5} = 4 - i\] Thus, the division results in the complex number \(4 - i\).

Key Concepts

Complex Division

Dividing complex numbers is a bit different from dividing real numbers, but it follows a structured process that makes the operation straightforward. The goal is to find the quotient of two complex numbers. Just like in regular math, division can be seen as multiplying by a reciprocal. However, with complex numbers, we need to use the conjugate to perform this task.

Here’s what you do:

• Identify the numerator and the denominator of the complex division problem.
• Multiply both the numerator and the denominator by the conjugate of the denominator.
• This multiplication clears the imaginary part of the denominator, effectively transforming it into a real number.

By following these steps, you simplify the problem into one that is easier to manage, ultimately allowing the result to be expressed in standard form.

Standard Form

When we work with complex numbers, it is often helpful to express them in a standard form, which is given by the expression \(a + bi\). In this standard form, \(a\) is the real part, and \(b\) is the coefficient of the imaginary part \(i\).

Some key points about standard form:

• It clearly separates the real and imaginary components of the number.
• By maintaining this form, complex arithmetic remains consistent and understandable.
• This representation allows complex numbers to be plotted on a two-dimensional plane, enhancing visual understanding.

Using standard form, division of complex numbers becomes a more approachable task. Once the computation is complete, it helps in easily identifying the real and imaginary parts of the quotient.

Conjugate

Conjugates play a crucial role in dividing complex numbers. The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of \(a + bi\) is \(a - bi\).

Here's why conjugates are important in complex division:

• They allow us to eliminate the imaginary unit \(i\) from the denominator.
• Multiplying a complex number by its conjugate results in a real number, specifically \(a^2 + b^2\).
• This property is used to "clean" the denominator, which simplifies the division process.

Using conjugates not only simplifies the division but also ensures the end result fits neatly into the standard form, making it easier to interpret and work with.