Problem 55 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The result is \(-4i\).

Step 1: Multiply the Numerator and Denominator by the Conjugate of the Denominator

To divide by complex numbers, especially when the denominator is an imaginary number like \( i \), you want to eliminate the imaginary part in the denominator. Multiply both the numerator and the denominator by \( -i \), as \( -i \) is the conjugate of \( i \). This gives \( \frac{4}{i} \times \frac{-i}{-i} = \frac{4(-i)}{i(-i)} \).

Step 2: Simplify the Denominator

Next, simplify the denominator where \( i(-i) = -i^2 \). Since \( i^2 = -1 \), substituting gives \( -i^2 = 1 \). So the denominator becomes \( 1 \).

Step 3: Evaluate the Numerator

Now calculate the numerator \( 4(-i) = -4i \). Thus, the fraction becomes \( \frac{-4i}{1} \).

Step 4: Write in the form \( a + bi \)

Since the denominator is \( 1 \), the division simplifies directly to \( -4i \). Write it in the form of \( a + bi \), where \( a = 0 \) and \( b = -4 \). Therefore, the expression is represented as \( 0 - 4i \) or simply \( -4i \).

Key Concepts

Imaginary Number

An imaginary number is fundamentally different from the real numbers we're accustomed to. The basis of imaginary numbers is the square root of -1, denoted by the symbol \( i \), which is defined as \( i = \sqrt{-1} \). Imaginary numbers become very handy when solving certain mathematical equations, especially those where negative roots arise.

Key aspects of imaginary numbers include:

• They always involve \( i \), the imaginary unit.
• Imaginary numbers can be combined with real numbers to form complex numbers, expressed as \( a + bi \) where \( a \) and \( b \) are real numbers.

Understanding imaginary numbers is crucial for tackling complex numbers, and they often play significant roles in fields like engineering and physics.

Complex Conjugate

The concept of a complex conjugate is essential when dividing complex numbers. A complex conjugate involves changing the sign of the imaginary part of a complex number. For example, if you have \( a + bi \), its complex conjugate is \( a - bi \).

When dividing by a complex number, especially when the denominator is an imaginary number, it's practical to multiply both the numerator and denominator by the complex conjugate of the denominator. This technique helps to eliminate the imaginary part from the denominator.
By multiplying a complex number by its conjugate, the result is a real number. Consider when dividing by \( i \), we use \( -i \) as the conjugate:

• Transform \( \frac{4}{i} \) into \( \frac{4 \times (-i)}{i \times (-i)} \).
• This results in a real number in the denominator, making the division process straightforward.

Mastering the use of complex conjugates can simplify many problems involving complex numbers.

Simplifying Complex Expressions

When dealing with complex numbers, the process of simplification can make apparent complex operations much more manageable. Simplifying entails using algebraic techniques to reduce a complex expression to its simplest form. In our scenario with \( \frac{4}{i} \), simplification reveals how we handle an imaginary denominator by using its complex conjugate.

Steps to simplify complex expressions include:

• Identify the term needing rationalization, such as an imaginary number in the denominator.
• Multiply by the complex conjugate to transform the denominator into a real number, simplifying computations.
• Perform basic arithmetic operations to simplify the numerator and achieve the standard form \( a + bi \).

For \( \frac{4}{i} \), simplifying leads to the expression \( -4i \) through these basic steps, illustrating the elegance and power of mathematical simplification.