Problem 33 - College Algebra

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ \frac{3+4 i}{2-i} $$

Step-by-Step Solution

Answer

The simplified complex number is \( \frac{2}{5} + \frac{11}{5}i \).

Step 1: Identify the Complex Conjugate

The expression we need to simplify is \( \frac{3+4i}{2-i} \). First, identify the complex conjugate of the denominator. For the denominator \( 2-i \), the complex conjugate is \( 2+i \).

Step 2: Multiply by the Complex Conjugate

To eliminate the imaginary part in the denominator, multiply both the numerator and the denominator by the complex conjugate \( 2+i \): \[ \frac{3+4i}{2-i} \times \frac{2+i}{2+i} = \frac{(3+4i)(2+i)}{(2-i)(2+i)} \]

Step 3: Expand the Numerator

Expand the numerator \( (3+4i)(2+i) \) by using distributive property: \[ 3 \times 2 + 3 \times i + 4i \times 2 + 4i \times i = 6 + 3i + 8i + 4i^2 \] Since \( i^2 = -1 \), substitute and combine like terms: \[ 6 + 3i + 8i - 4 = 2 + 11i \]

Step 4: Expand the Denominator

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

Step 5: Simplify the Expression

Substitute the expanded numerator and denominator back into the fraction: \[ \frac{2 + 11i}{5} \] Divide the real and imaginary components by the denominator: \[ \frac{2}{5} + \frac{11}{5}i \]

Step 6: Final Answer as Simplified Complex Number

The simplified expression of the result as a complex number is \( \frac{2}{5} + \frac{11}{5}i \).

Key Concepts

Complex Conjugate

A complex conjugate is an important tool in complex number arithmetic. For any complex number of the form \( a + bi \), where \( a \) and \( b \) are real numbers, the complex conjugate is \( a - bi \). Essentially, it involves changing the sign of the imaginary part. This concept is particularly useful when simplifying expressions that involve dividing complex numbers.

• For example, the complex conjugate of \( 2 - i \) is \( 2 + i \).
• Using the complex conjugate helps to eliminate the imaginary part from the denominator, making it a real number.

To simplify an expression, we multiply both numerator and denominator by the complex conjugate of the denominator. This turns the denominator into a real number, thus making the overall expression simpler to handle.

Distributive Property

The distributive property is a fundamental property of algebra used to expand expressions. It states that for all real numbers \( a \), \( b \), and \( c \), the following equation holds: \( a(b+c) = ab + ac \).
In the context of complex numbers, this property is applied when multiplying two complex numbers. For example, when we expanded \((3+4i)(2+i)\) in our exercise, each term in the first complex number was multiplied by each term in the second complex number:

• \(3 \times 2 = 6\)
• \(3 \times i = 3i\)
• \(4i \times 2 = 8i\)
• \(4i \times i = 4i^2\)

This step demonstrates the power of the distributive property in managing complex multiplication, ensuring that each part interacts properly. Remember, combining like terms at the end simplifies the expression significantly.

Imaginary Unit

The imaginary unit, denoted as \( i \), is a concept that defines the basis of imaginary numbers. Its defining property is that \( i^2 = -1 \), which might seem counterintuitive since there are no real numbers with a square that equals a negative number. This characteristic allows us to solve equations that would otherwise have no solution in the real number system.
In our exercise involving complex numbers, \( i \) appears naturally when performing arithmetic operations, especially multiplication. For instance, when expanding the product \( 4i \times i \) during multiplication in our example, it results in \(4i^2\), which simplifies to \(-4\) because \(i^2 = -1\).
Understanding \( i \) is crucial for working with complex expressions and helps in simplifying them to standard complex form \( a + bi \).

Simplification of Expressions

Simplifying expressions in mathematics means reducing them to their simplest form while keeping their value unchanged. When it comes to complex numbers, simplification often involves converting a complex division into a more manageable form.
Our given expression \( \frac{3+4i}{2-i} \) was simplified by utilizing the complex conjugate to turn the denominator into a real number. Once that's achieved, the numerator and the converted real denominator allow for clear separation of real and imaginary components, yielding a more comprehensible form.

• Separate the real part: \( \frac{2}{5} \)
• Separate the imaginary part: \( \frac{11}{5}i \)

This process not only eases the understanding of the expression but also reveals the result as a standard complex number. Regular practice of such steps enhances familiarity and fluency in dealing with complex number operations.