Problem 69 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The answer is \( \frac{1}{5} - \frac{8}{5}i \).

Step 1: Write the Division Problem

We need to divide the complex numbers \( \frac{2-3i}{2+i} \). Our goal is to express this in the form \( a + bi \).

Step 2: Multiply by the Conjugate

To simplify the division, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 2+i \) is \( 2-i \). So, multiply by \( \frac{2-i}{2-i} \): \[ \frac{(2-3i)(2-i)}{(2+i)(2-i)} \].

Step 3: Expand the Numerator

Use the distributive property to multiply \( (2-3i)(2-i) \): \( 2(2) + 2(-i) - 3i(2) - 3i(-i) = 4 - 2i - 6i + 3i^2 \). Remember \( i^2 = -1 \), so it becomes \( 4 - 8i - 3 \).

Step 4: Expand the Denominator

Multiply the denominators \( (2+i)(2-i) \) using the formula \( (a+b)(a-b) = a^2 - b^2 \), where \( a=2 \) and \( b=i \): \[ 2^2 - (i)^2 = 4 - (-1) = 4 + 1 = 5 \].

Step 5: Simplify the Expression

Combine the terms in the expanded numerator: \[ 4 - 8i - 3 = 1 - 8i \]. Thus, the expression is \( \frac{1 - 8i}{5} \).

Step 6: Write in the form a + bi

Divide each part of the expression by 5: \( \frac{1}{5} - \frac{8}{5}i \).\ Thus, the final answer is \( \frac{1}{5} - \frac{8}{5}i \).

Key Concepts

Complex Division

Complex division is the process of dividing two complex numbers. Complex numbers take the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. When dividing complex numbers, the goal is to simplify the expression into this form.

To divide complex numbers, follow these steps:

• Identify the numerator (the number being divided) and the denominator (the number you are dividing by).
• Multiply both the numerator and the denominator by the conjugate of the denominator. This step helps eliminate the imaginary part from the denominator.
• Simplify the resulting expression by expanding and simplifying the numerator and the denominator separately.
• Finally, express the simplified complex number in the form \(a + bi\).

Think of it as a way to "rationalize" the denominator, similar to how you might rationalize a fraction with a square root. This method ensures that the final result is a proper complex number, free of imaginary parts in the denominator.

Imaginary Numbers

Imaginary numbers are a main component of complex numbers. They are defined by the property \(i\), where \(i^2 = -1\). This is unlike real numbers, where no square number is negative.

When you have a complex number like \(a + bi\), the part \(bi\) is the imaginary component. Here, \(b\) is a real number, but when multiplied by \(i\), it becomes imaginary.

• Imaginary numbers enable the expression of solutions to equations that would not be solvable using only real numbers, such as \(x^2 + 1 = 0\).
• They play a crucial role in fields such as engineering, physics, and applied mathematics.
• Understanding \(i\) as a fundamental imaginary unit helps the manipulation and simplification of complex numbers in various operations, including addition, subtraction, multiplication, and division.

By grasping the concept of imaginary numbers, you'll gain a more complete understanding of how complex numbers work and how they expand the limits of traditional arithmetic.

Conjugate of Complex Numbers

The conjugate of a complex number is a key tool in performing complex division. If you have a complex number \(a + bi\), its conjugate is \(a - bi\). By swapping the sign of the imaginary part, the conjugate helps simplify mathematical operations with complex numbers.

Here’s how it works:

• Multiplying a complex number by its conjugate results in a real number. In mathematical terms, \((a+bi)(a-bi) = a^2 - (bi)^2 = a^2 + b^2\), because \(i^2 = -1\).
• This method removes the imaginary unit from the denominator in complex division, leading to an easier simplification of the resulting expression.
• The use of conjugates is similar to the need for a common denominator in fraction addition and subtraction—it's a way of achieving consistency and simplicity within the problem.

Remember, practicing using conjugates will not only help with division but also with understanding the symmetry and relationships within the set of complex numbers.