Problem 58 - Algebra A Combined Function

Question

Divide. Write your answers in the form \(a+b i\) $$ \frac{9}{1-2 i} $$

Step-by-Step Solution

Answer

\(\frac{9}{5} + \frac{18}{5}i\)

Step 1: Identify the Conjugate

The first step is to identify the conjugate of the denominator. Our denominator is \(1 - 2i\), so its conjugate is \(1 + 2i\).

Step 2: Multiply Numerator and Denominator by the Conjugate

To eliminate the imaginary part from the denominator, multiply both the numerator and the denominator by the conjugate \(1 + 2i\):\[ \frac{9}{1-2i} \times \frac{1+2i}{1+2i} = \frac{9(1+2i)}{(1-2i)(1+2i)} \]

Step 3: Simplify the Denominator

Multiply the denominator using the difference of squares formula:\[ (1-2i)(1+2i) = 1^2 - (2i)^2 = 1 - (-4) = 5 \]

Step 4: Expand and Simplify the Numerator

Distribute 9 over \(1 + 2i\):\[ 9(1 + 2i) = 9 + 18i \]

Step 5: Combine Into a Single Expression

Now put the simplified expressions together to form:\[ \frac{9 + 18i}{5} = \frac{9}{5} + \frac{18}{5}i \]

Step 6: Write in the Form \(a+bi\)

The division is completed, and the expression is written as: \[ \frac{9}{5} + \frac{18}{5}i \].

Key Concepts

Conjugate Multiplication

When dividing complex numbers, conjugate multiplication is a helpful technique to simplify the expression. A conjugate refers to changing the sign of the imaginary part of a complex number. For example, if we have a number like \(1 - 2i\), its conjugate would be \(1 + 2i\). By doing this, you'll form a pair that allows easier computation.

Why use the conjugate? When dividing complex numbers, our goal is to eliminate the imaginary unit \(i\) from the denominator, making it a real number. Multiplying by the conjugate achieves this by leveraging properties of complex numbers.

Steps to apply the conjugate multiplication:

• Identify the conjugate of the denominator.
• Multiply both the numerator and the denominator by this conjugate.

This action transforms the division problem into a form that is much easier to simplify using straightforward arithmetic.

Imaginary Numbers

Imaginary numbers are an extension of the real number system, allowing the square root of negative numbers. The imaginary unit is represented as \(i\), where \(i^2 = -1\). This seems a bit abstract, but they are crucial for various fields, including engineering and physics.

Complex numbers combine real numbers and imaginary numbers, typically written in the form \(a + bi\) where \(a\) is the real part and \(bi\) is the imaginary part.

When working with imaginary numbers:

• Remember that the product of two imaginary units yields a negative real number. For instance, \((2i)^2 = 4(-1) = -4\).
• Adding and subtracting imaginary numbers involve combining like terms, much like traditional algebra.

This understanding is pivotal when simplifying expressions and performing operations with complex numbers.

Difference of Squares Formula

The difference of squares formula is a powerful algebraic tool used frequently in mathematics, represented by \(a^2 - b^2 = (a - b)(a + b)\). This formula allows us to convert a product of binomials into a more simplified expression.

In our complex division, notice how the denominator \((1 - 2i)(1 + 2i)\) simplifies using this concept. Here's why it works:

• The real and imaginary parts form terms that square to become either positive or negative.
• The resulting expression \(1 - (2i)^2\) uses the property that \( (2i)^2 = 4(-1) = -4 \), simplifying to a real number \(1 + 4 = 5 \).

This simplification is critical for eliminating the complex part of a denominator, simplifying our division into a real number, making it straightforward to express in standard form like \(a + bi\).