Problem 56 - College Algebra

Question

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{3+2 i}{1+2 i}-\frac{2-3 i}{3+i} $$

Step-by-Step Solution

Answer

The expression simplifies to \(1.1 + 0.3i\).

Step 1: Eliminate Complex Denominators

To simplify the given expression, start by eliminating the complex denominators. To do this, multiply the numerator and the denominator by the conjugate of the denominator. For \( \frac{3+2i}{1+2i} \), the conjugate is \( 1-2i \). Multiply both the numerator and the denominator by the conjugate:\[\frac{(3+2i)(1-2i)}{(1+2i)(1-2i)}\]For \( \frac{2-3i}{3+i} \), the conjugate is \( 3-i \). Multiply both the numerator and the denominator by its conjugate:\[\frac{(2-3i)(3-i)}{(3+i)(3-i)}\]

Step 2: Simplify the Denominators

Solve the denominator multiplications using the difference of squares formula: - \((1+2i)(1-2i) = 1 - (2i)^2 = 1 - 4(-1) = 5\)- \((3+i)(3-i) = 9 - (i)^2 = 9 - (-1) = 10\)

Step 3: Simplify the Numerators

Perform the multiplications in the numerators:- For the first term: \((3+2i)(1-2i) = 3(1) + 3(-2i) + 2i(1) - 2i(2i) = 3 - 6i + 2i + 4 = 7 - 4i\)- For the second term: \((2-3i)(3-i) = 2(3) + 2(-i) - 3i(3) + 3i(i) = 6 - 2i - 9i + 3(-1) = 3 - 11i\)

Step 4: Formulate the Correct Complex Fractions

Once you have both numerators and denominators simplified, write the complex numbers:\[\frac{7-4i}{5} - \frac{3-11i}{10}\]This separates into two complex fractions that need further simplification.

Step 5: Find a Common Denominator

Notice that the fractions have different denominators, so find a common denominator (LCM of 5 and 10 is 10). Adjust fractions to have this common denominator:- \( \frac{7-4i}{5} = \frac{14-8i}{10} \)- \( \frac{3-11i}{10} = \frac{3-11i}{10} \)

Step 6: Subtract Complex Fractions

Subtract the fractions, combine real and imaginary parts separately:\[\frac{14-8i}{10} - \frac{3-11i}{10} = \frac{(14-3) + (-8i+11i)}{10} = \frac{11 + 3i}{10}\]

Step 7: Final Simplification

Express the result as a single complex number:- Real part: \( \frac{11}{10} \)- Imaginary part: \( \frac{3i}{10} \)Thus, the expression simplifies to:\[1.1 + 0.3i\]

Key Concepts

Complex Conjugate

A complex conjugate is simply a tool, like a mirror version of a complex number. If you have a complex number \(a + bi\), its conjugate is \(a - bi\). The "i" here represents the imaginary unit, which we will explore later. By changing the sign of the imaginary part, we get the conjugate.

• The conjugate is crucial because when multiplied by its corresponding complex number, the result is a real number. This dramatically simplifies calculations, especially in division.
• Applying the concept of the conjugate was essential in our exercise to simplify complex fractions.

For example, the conjugate of \(1 + 2i\) is \(1 - 2i\). When these are multiplied, we end up with a real number:\[(1+2i)(1-2i) = 1 - (2i)^2 = 1 - 4(-1) = 5\]Using complex conjugates helps in simplifying complex expressions, making them easier to manage and understand.

Complex Fractions

When dealing with complex numbers in fractions, things can seem a bit chaotic. But don't fret, just remember that you're working with numbers that include both a real part and an imaginary part.

• The key is to eliminate complex numbers from the denominator using our friend, the complex conjugate.
• Once applied, you transform the complex fraction into a simpler expression, which is then easier to work with.

In the exercise, to simplify \( \frac{3+2i}{1+2i} \), multiplying by the conjugate \(1-2i\), we attain a purely real denominator:\[\frac{(3+2i)(1-2i)}{5}\]
Combining this approach with finding common denominators helps in directly carrying out operations such as addition or subtraction of complex numbers.

Simplifying Complex Expressions

Simplifying complex expressions might look tricky at first because of the imaginary component. The goal is to end up with a neat and manageable expression composed of a real part and an imaginary part.

• Once you've gotten rid of complex numbers in the denominator, focus on handling each component separately: the real and imaginary parts.
• Always ensure the final expression is a simplified version for easy interpretation, often resulting in the form \(a + bi\).

The given exercise wandered through initial complex fractions to the final simplification. By using multiplication, addition, and subtraction, the messy expressions were ultimately rephrased to readable forms as: \(rac{11}{10} + rac{3i}{10} \).
This makes interpreting the expression more intuitive, letting you focus on both numeric and symbolic meaning.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a mathematical construct representing the square root of -1. Yes, it sounds counterintuitive since no real number squared yields a negative value, but it's a creative solution to certain equations.

• The imaginary unit makes it possible to solve problems involving square roots of negative numbers.
• Mathematicians extend \(i\) so that any complex number \(a + bi\) is fundamentally interacting with this unit.

For example, in expressions like \(2i\) or \( -3i\), \(i\) represents the extent of the imaginary component. Always remember that \(i^2\) equals -1.
This principle lets you convert complex numbers from whole expressions to simpler structures contributing to real-world applications such as signal processing or academic exploration in maths.