Problem 100 - Intermediate Algebra

Question

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{-3-4 i}{-4-11 i} $$

Step-by-Step Solution

Answer

The quotient is \(\frac{56}{137} - \frac{17}{137}i\).

Step 1: Identify the Complex Numbers

The division problem involves two complex numbers. The numerator is \(-3 - 4i\) and the denominator is \(-4 - 11i\). The task is to divide the first by the second and express the result in the form \(a + bi\).

Step 2: Multiply by the Conjugate

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, \(-4 + 11i\). This will eliminate the imaginary part in the denominator.

Step 3: Perform the Multiplication in the Numerator

Compute the product:\[ (-3 - 4i)(-4 + 11i) = (-3)(-4) + (-3)(11i) + (-4i)(-4) + (-4i)(11i)\]This simplifies to:\[ 12 - 33i + 16i - 44i^2 \]Since \(i^2 = -1\), replace \(i^2\) with \(-1\):\[ 12 - 33i + 16i + 44 = 56 - 17i \]

Step 4: Perform the Multiplication in the Denominator

Compute the product:\[ (-4 - 11i)(-4 + 11i) = (-4)^2 - (11i)^2\]This simplifies to:\[ 16 + 121 ext{ (since } -(-1) = +1 ext{)}\]Therefore, the denominator simplifies to \(137\).

Step 5: Divide and Simplify

Now, divide the simplified numerator by the simplified denominator:\[ \frac{56 - 17i}{137} = \frac{56}{137} - \frac{17}{137}i \]Write the result in standard form of a complex number. Therefore, the quotient is:\[ \frac{56}{137} - \frac{17}{137}i \]

Step 6: Express the Answer in Decimal Form (Optional)

If necessary, you can express the result in decimal form:\\[ \frac{56}{137} \approx 0.409 ext{ and } \frac{17}{137} \approx 0.124 \]Hence, the expression in decimal form is approximately \(0.409 - 0.124i\).

Key Concepts

Complex Conjugate

A complex conjugate can be seen as a twin number of a given complex number, with a small twist: the imaginary part has its sign reversed. For any complex number, \(a + bi\), the complex conjugate is \(a - bi\). This concept plays a crucial role in operations involving complex numbers, especially division. By utilizing conjugates, we can simplify expressions and solve complex problems more easily.

Using a complex conjugate in division helps "rationalize the denominator". This means it transforms any complex number in the denominator into a real number. When dividing complex numbers, we multiply both the numerator and the denominator by the conjugate of the denominator. This ensures that the denominator is real and simplifies to a single number. This method prevents any imaginary number from lingering in the denominator.

In our example, the complex number in the denominator is \(-4 - 11i\). Therefore, its conjugate is \(-4 + 11i\). By multiplying both the top and bottom of the fraction by this conjugate, we eliminate the imaginary component from the denominator. This makes the calculation neater and the answer more straightforward to understand.

Rationalizing the Denominator

The concept of rationalizing the denominator involves transforming a complex denominator into a real number. This is achieved by multiplying both the numerator and the denominator by the complex conjugate of the denominator.

Let's revisit the example: the denominator is \(-4 - 11i\). Its complex conjugate is \(-4 + 11i\). When we multiply these, the formula used is \((-4 - 11i)(-4 + 11i)\). This operation utilizes the identity \((a + bi)(a - bi) = a^2 + b^2\), ensuring that the result is purely real.

By performing this multiplication, the imaginary parts cancel each other out, leaving us with a real number as seen:

• The product \((-4)^2 - (11i)^2\) simplifies to \(16 + 121\), which is \(137\).

After rationalizing the denominator, we can move forward with our division without difficulty. This method is not only clear but also essential for accurate calculations involving complex numbers.

Standard Form of a Complex Number

The standard form of a complex number is expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. It's the typical way to present complex numbers in mathematics because it clearly shows the two components that make up the number.

In our division problem, after rationalizing the denominator, our expression must be rewritten in this standard form. Once we have multiplied by the conjugate and simplified our work, we obtain a result like \(56 - 17i\) over the real denominator, \(137\).

This result is then divided to express it in standard form:

• Divide the real part: \(\frac{56}{137}\), which becomes the real component \(a\).
• Divide the imaginary part: \(\frac{-17}{137}\), which becomes the imaginary component \(bi\).

Thus, the complex number is written as a mix of clear real and imaginary parts. So, our final answer is presented neatly as \(\frac{56}{137} - \frac{17}{137}i\) and can be approximated in decimals if necessary.