Problem 70 - Intermediate Algebra

Question

Divide. Write all answers in the form a \(+b i.\) $$ \frac{4}{2-i} $$

Step-by-Step Solution

Answer

\( \frac{8}{5} + \frac{4}{5}i \)

Step 1: Identify the Denominator

To divide complex numbers, we must first identify the denominator. In this case, the denominator is the complex number \(2 - i\).

Step 2: Find the Conjugate

The conjugate of a complex number \(a + bi\) is \(a - bi\). Therefore, the conjugate of our denominator \(2 - i\) is \(2 + i\).

Step 3: Multiply Numerator and Denominator by the Conjugate

We will multiply both the numerator and the denominator by the conjugate of \(2-i\), which is \(2+i\): \[ \frac{4}{2-i} \times \frac{2+i}{2+i} = \frac{4(2+i)}{(2-i)(2+i)} \]

Step 4: Simplify the Denominator

The denominator \((2-i)(2+i)\) is a difference of squares: \[(2-i)(2+i) = 2^2 - i^2 = 4 - (-1) = 5\]

Step 5: Expand and Simplify the Numerator

Now, expand the numerator: \[ 4(2+i) = 8 + 4i \]

Step 6: Write the Complex Number in Standard Form

Finally, write the result in standard form: \[ \frac{8+4i}{5} = \frac{8}{5} + \frac{4}{5}i \]

Key Concepts

Understanding the Complex Conjugate

When dealing with complex numbers, a key tool is the complex conjugate. The complex conjugate of a complex number is formed by changing the sign of the imaginary part. For instance, if you have a complex number represented as \(a + bi\), its conjugate would be \(a - bi\).
This concept is crucial because it helps in simplifying the division of complex numbers.
In the division of complex numbers, like in the given exercise, the denominator needs to be a real number to simplify the expression. This is achieved by multiplying by the conjugate.
By doing so, the imaginary parts cancel each other out, leaving a real number in the denominator. That's why in the exercise, multiplying \(2 - i\) by its conjugate \(2 + i\) simplifies the division process.

Role of the Difference of Squares in Complex Division

The difference of squares formula is another important concept here. This formula states that for any two numbers \(a\) and \(b\), the expression \((a-b)(a+b)\) can be simplified to \(a^2 - b^2\).
When dividing complex numbers, like \((2-i)(2+i)\), we can see this formula in action.

• The first term \(a = 2\) and \(b = i\) reflect the real and imaginary components, respectively.
• The application of the difference of squares gives us \(2^2 - i^2 = 4 - (-1) = 5\).

Thus, applying this formula converts our original complex denominator into a manageable real number.

Expressing in Standard Form of Complex Numbers

The standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. Simplifying division results into this form is essential to effectively communicate the value of a complex number.
After using the conjugate and difference of squares concept to handle the denominator, we are left with a numerator \(8 + 4i\) over a real number 5.

• When divided, each part of the complex result is expressed separately as a ratio over the real part.
• The final step in this problem writes the complex number \(\frac{8+4i}{5}\) in the standard form, arriving at \(\frac{8}{5} + \frac{4}{5}i\).

This process ensures that complex numbers have a consistent and clear representation, making mathematical operations and interpretations straightforward.