Problem 77 - College Algebra

Question

Perform the indicated operations and write the result in standard form. $$ \frac{8}{1+\frac{2}{i}} $$

Step-by-Step Solution

Answer

The standard form of the given expression is \( 1.6 - 3.2i \)

Step 1: Simplify the Inner Fraction

The first step involves simplifying the inner fraction which has a complex number, namely \( \frac{2}{i} \). In order to leave the denominator real, multiply both the numerator and denominator of the fraction by the complex conjugate of \( i \), which is \( -i \).\nThat results in \( \frac{2*-i}{i*-i}= \frac{-2i}{-1} = 2i \)

Step 2: Substitution

Substitute \(2i\) into the original expression to replace \( \frac{2}{i} \) to obtain: \( \frac{8}{1+2i} \)

Step 3: Remove the Complex Number from the Denominator

To remove the complex number in the denominator, once again apply the rule of multiplying the numerator and the denominator with the conjugate of the denominator. The conjugate of the denominator \(1 + 2i\) is \(1 - 2i\) \nWhen performing the multiplication, obtain \( \frac{8(1-2i)}{(1+2i)(1-2i)} \)

Step 4: Simplify the Expression

To simplify the expression, carry out the multiplications in the numerator and denominator. The numerator simplifies to \(8 - 16i\), and the denominator becomes \(5\). Therefore, the expression becomes \( \frac{8-16i}{5} \)

Step 5: Split into Real and Imaginary Parts

Split the fraction into real and imaginary parts for the final standard form of a complex number. The final answer is \( 1.6 - 3.2i \)

Key Concepts

Complex Conjugate

Working with complex numbers often involves the use of a concept known as the complex conjugate. A complex number is typically written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The imaginary unit \( i \) satisfies \( i^2 = -1 \).

The complex conjugate of \( a + bi \) reverses the sign of the imaginary part, becoming \( a - bi \). This operation is essential for eliminating imaginary numbers from denominators.

For example, when simplifying \( \frac{2}{i} \), we multiply the numerator and denominator by the complex conjugate of \( i \), which is \(-i\). Similarly, to remove the complex number from the denominator of \( \frac{8}{1 + 2i} \), we multiply by the complex conjugate of \(1 + 2i\), which is \(1 - 2i\).

This process utilizes the identity \((a + bi)(a - bi) = a^2 + b^2\), effectively transforming a complex number into a real number.

Standard Form

The standard form of a complex number is expressed as \( a + bi \), where \( a \) and \( b \) are real numbers. The number \( a \) represents the real part, while \( b \) is the coefficient of the imaginary part \( i \).

In exercises or problems involving complex arithmetic, we often aim to express the result in this neat form, which clearly separates the real and imaginary components.

For instance, after simplifying the given expression, the final result \( 1.6 - 3.2i \) is in standard form. Here, \( 1.6 \) is the real part, and \( -3.2 \) is the coefficient of the imaginary part, clarifying both parts of the complex number for easy comprehension and further calculation.

Imaginary Unit

The imaginary unit, denoted by \( i \), is a fundamental component of complex numbers. It is defined by the property \( i^2 = -1 \).

This unique characteristic allows us to handle square roots of negative numbers and leads to the formation of complex numbers of the form \( a + bi \).

Understanding \( i \) is crucial when performing operations with complex numbers, such as the original problem involving \( \frac{2}{i} \). Multiplying by the conjugate \(-i\) ensures we maintain mathematical rigour, simplifying and converting expressions into forms where the denominator becomes real.

Simplification of Fractions

Simplifying fractions that involve complex numbers is an essential skill for converting expressions into a more manageable form. This often involves removing complex numbers from the denominator.

In the problem presented, we deal with two simplification tasks: transforming \( \frac{2}{i} \) first, and then \( \frac{8}{1+2i} \). Each step employs multiplying the fraction by a complex conjugate to achieve a real number denominator.

• The fraction \( \frac{2}{i} \) simplifies to \( 2i \) when multiplied by \(-i\).
• The fraction \( \frac{8}{1+2i} \) is multiplied by \(1-2i\), simplifying it to \( 1.6 - 3.2i \).

This makes the resulting expression much easier to handle and interpret in the standard form \( a + bi \). Such processes underscore the importance of the conjugate in fraction simplification, enabling clear solutions for mathematical problems involving complex numbers.