Problem 101 - A Graphical Approach to Precalculus with Limits

Question

Find the complex conjugate. $$\frac{-3+4 i}{2-i}$$

Step-by-Step Solution

Answer

The complex conjugate is \(-2 + i\).

Step 1: Identify Complex Conjugate

The complex conjugate of a number in the form \(a + bi\) is \(a - bi\). Our goal is to simplify the original complex fraction by multiplying by the conjugate of the denominator. Here, the denominator is \(2-i\), so its complex conjugate is \(2+i\).

Step 2: Multiply by Conjugate

Multiply both the numerator and the denominator by the conjugate of the denominator, \(2+i\): \[\frac{-3+4i}{2-i} \cdot \frac{2+i}{2+i} = \frac{(-3+4i)(2+i)}{(2-i)(2+i)}\].

Step 3: Simplify the Numerator

Expand the numerator: \((-3+4i)(2+i) = (-3)(2) + (-3)(i) + (4i)(2) + (4i)(i) = -6 - 3i + 8i + 4i^2\). Since \(i^2 = -1\), it simplifies to \(-6 + 5i - 4 = -10 + 5i\).

Step 4: Simplify the Denominator

Use the difference of squares to simplify the denominator: \((2-i)(2+i) = 2^2 - (i)^2 = 4 - (-1) = 4 + 1 = 5\).

Step 5: Write the Simplified Complex Number

Now, combining the results, \[\frac{-10 + 5i}{5} = -2 + i\]. Write the result as a complex number: \(-2 + i\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, with \( i \) being the imaginary unit. The imaginary unit is defined as \( i = \sqrt{-1} \). This means that \( i^2 = -1 \).
Understanding complex numbers is crucial when dealing with operations such as addition, subtraction, and multiplication, just like you would with real numbers. However, the presence of \( i \) adds a layer of complexity, literally!

• Real Parts are the numbers without the imaginary unit \( i \). For example, in \( 3 + 4i \), \( 3 \) is the real part.
• Imaginary Parts have the imaginary unit \( i \). For instance, in \( 3 + 4i \), \( 4i \) is the imaginary part.
• The complex conjugate of a complex number \( a + bi \) is \( a - bi \).

Fractions

Fractions represent parts of a whole and consist of two main parts: the numerator and the denominator. Fractions can be applied to complex numbers as well, where you might find yourself working with fractions that have complex numbers as their terms.
When working with complex numbers in fractions, the goal is often to simplify the expression by eliminating the imaginary part from the denominator. This is typically achieved by multiplying by the conjugate.
For example, if you have a fraction like \( \frac{-3+4i}{2-i} \), you should multiply the numerator and the denominator by the conjugate of the denominator to simplify it.

Numerator and Denominator

In fractions, the numerator is the top part and the denominator is the bottom part. Understanding their roles can help you effectively simplify complex fractions. Here are some key aspects:

• **Numerator:** This represents the number of parts you have. In the fraction \( \frac{-3+4i}{2-i} \), \(-3+4i\) is the numerator.


• **Denominator:** This shows the number of equal parts the whole is divided into. In \( \frac{-3+4i}{2-i} \), \(2-i\) is the denominator.

Understanding these parts helps when multiplying by the conjugate. The numerator and denominator are both multiplied by the same conjugate, which helps eliminate the imaginary number from the denominator.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. When dealing with fractions of complex numbers, you simplify by removing the imaginary part from the denominator.
To do this, you:

• Multiply both the numerator and the denominator by the complex conjugate of the denominator.
• In the numerator, expand and simplify any terms, being careful to apply \( i^2 = -1 \) correctly.
• In the denominator, use the formula for the difference of squares: \((a-b)(a+b) = a^2 - b^2\).

Finally, write the complex number in the simplest form possible. As shown in our example, \( \frac{-10 + 5i}{5} \) simplifies to \(-2 + i\), which is the solution in its simplest form. This makes arithmetic with complex numbers clearer and the results easier to interpret.