Problem 69 - Intermediate Algebra

Question

Divide. Write all answers in the form a \(+b i.\) $$ \frac{9}{5+i} $$

Step-by-Step Solution

Answer

The solution is \(\frac{45}{26} - \frac{9}{26}i\).

Step 1: Identify the Division of Complex Numbers

To divide a complex number by another complex number, you must multiply both the numerator and the denominator by the conjugate of the denominator.

Step 2: Identify the Conjugate of the Denominator

The denominator is \(5+i\). Its conjugate is \(5-i\), formed by changing the sign of the imaginary part.

Step 3: Multiply the Numerator and Denominator by the Conjugate

Multiply both the numerator \(9\) and the denominator \(5+i\) by \(5-i\): \(\frac{9(5-i)}{(5+i)(5-i)}\).

Step 4: Expand the Numerator

Distribute \(9\) in \(9(5-i)\): \(9 \times 5 - 9 \times i = 45 - 9i\).

Step 5: Expand and Simplify the Denominator

Use the formula \((a+b)(a-b) = a^2-b^2\): \((5+i)(5-i)=5^2-i^2=25-(-1)=26\).

Step 6: Write the Final Results

Combine the expanded results: \(\frac{45-9i}{26}\). Write it as: \(\frac{45}{26} - \frac{9i}{26}\) in the form \(a + bi\). Simplify each part to get \(\frac{45}{26} - \frac{9}{26}i\).

Step 7: Simplify the Fractions

Simplify each fraction if possible. Here, \(\frac{45}{26}\) and \(\frac{9}{26}\) do not simplify further in exact terms. Leave the numbers as they are.

Key Concepts

Conjugate of Complex Numbers

Complex numbers are numbers that include a real part and an imaginary part, typically expressed in the form \(a + bi\), where \(i\) is the imaginary unit. The conjugate of a complex number is crucial for performing operations like division. For any complex number \(a + bi\), its conjugate is \(a - bi\). The process of taking the conjugate effectively involves changing the sign of the imaginary part.

When dividing complex numbers, multiplying by the conjugate of the denominator brings great utility. It eliminates the imaginary part from the denominator, making the division simpler and ensuring that the result remains in the standard form \(a + bi\).

• If the denominator is \(5+i\), its conjugate is \(5-i\).
• This operation helps convert the denominator into a real number using the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\).

By multiplying by the conjugate, you can keep your complex arithmetic organized and maintain the results in a familiar and manageable form.

Imaginary Numbers

Imaginary numbers are fundamental to understanding complex numbers. They make up the necessary 'imaginary' component that, when paired with a real number, creates complex numbers. The standard notation for imaginary numbers is \(bi\), where \(i\) represents the square root of \(-1\).

• The notation \(i\) allows us to extend beyond the real number line into a plane where both real and imaginary numbers exist.
• By definition, \(i^2 = -1\), which means that imaginary numbers occupy a unique place in mathematics, helping to solve equations that real numbers alone cannot.

When dividing complex numbers, the imaginary component plays a critical role in constructing and simplifying expressions. It enables the representation of numbers that are not possible on the real number axis alone. Understanding the role of \(i\) can empower students to work with, and appreciate, advanced algebraic concepts.

Arithmetic with Complex Numbers

Performing arithmetic operations with complex numbers, such as addition, subtraction, multiplication, and division, follows specific rules and requires a step-by-step approach.

**Addition & Subtraction**

• These operations are performed by combining each respective component.
• For example, adding \((3 + 2i) + (5 + 4i)\) results in \((3+5) + (2i+4i) = 8 + 6i\).

**Multiplication**

• This involves using the distributive property and remembering that \(i^2 = -1\).
• For instance, multiplying \((2 + 3i)(1 + 4i)\) would require distributing each term, then simplifying: \((2 \times 1) + (2 \times 4i) + (3i \times 1) + (3i \times 4i)\), which simplifies to \(2 + 8i + 3i -12 = 2 + 11i - 12\). Ultimately yielding: \(-10 + 11i\).

**Division**

• As seen in complex division, the key is the conjugate. It helps clear the denominator of any imaginary parts, making it real.
• Using the steps shown in dividing \(\frac{9}{5+i}\), we multiply by \(\frac{5-i}{5-i}\), thus simplifying the division problem into two manageable parts and resulting in a simplified form \(\frac{45}{26} - \frac{9}{26}i\).

Recognizing how to manipulate these expressions provides a strong foundational skill for tackling higher-level mathematics and engineering problems. Understanding these operations allows for expanded problem-solving techniques, enhancing analytical capabilities.