Problem 97 - Algebra A Combined Function

Question

Write each expression in the form \(a+b i .\) $$ \frac{4-\sqrt{-8}}{2} $$

Step-by-Step Solution

Answer

The expression in the form \(a + bi\) is \(2 - \sqrt{2}i\).

Step 1: Simplify the Expression Under the Square Root

We begin by simplifying the expression inside the square root, \(-8\). Recall that \(i\) is the imaginary unit where \(i^2 = -1\). Thus, \(\sqrt{-8} = \sqrt{8} \cdot i\). The square root of 8 can be further simplified as \(\sqrt{4 \cdot 2} = 2\sqrt{2}\). Therefore, \(\sqrt{-8} = 2\sqrt{2}i\).

Step 2: Substitute and Simplify the Complex Expression

Now, substitute \(2\sqrt{2}i\) for \(\sqrt{-8}\) in the original expression: \(\frac{4 - 2\sqrt{2}i}{2}\).

Step 3: Separate the Real and Imaginary Parts

To separate the real and imaginary parts, divide each term in the numerator by the denominator: \(\frac{4}{2} - \frac{2\sqrt{2}i}{2}\). This results in \(2 - \sqrt{2}i\).

Step 4: Present the Expression in the Form \(a + bi\)

The expression \(2 - \sqrt{2}i\) is already in the standard form \(a + bi\), where \(a = 2\) and \(b = -\sqrt{2}\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted by the symbol \(i\), is a fundamental concept in complex numbers. The imaginary unit is defined as \(i = \sqrt{-1}\).
\(i\) is special because, by definition, \(i^2 = -1\). This means that \(i\) does not behave like a standard number but is essential for working with expressions involving square roots of negative numbers.

• The introduction of \(i\) allows for expressions that contain negative square roots to be manipulated and solved more easily.
• In mathematics, \(i\) extends the real number system to include solutions to equations that don't have real solutions, such as \(x^2 + 1 = 0\).

Understanding \(i\) is crucial when dealing with complex numbers, enabling the representation of numbers in the form of \(a + bi\), where \(a\) and \(b\) are real numbers.

Square Root Simplification

Simplifying square roots is an essential skill when working with complex numbers. The term \(\sqrt{-8}\) involves simplifying under the square root, combined with the imaginary unit.
First, recognize that \(\sqrt{-8}\) can be rewritten using the imaginary unit as \(\sqrt{8} \cdot i\).
Then, simplify \(\sqrt{8}\) by factorizing it: \(\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}\). Therefore, the expression becomes \(2\sqrt{2}i\).

• Breaking down square roots into their factors helps in further simplifying complex expressions.
• Always factor out perfect squares to simplify the square roots effectively.

This process of square root simplification is vital to express complex numbers in their standard form.

Real and Imaginary Parts

Complex numbers consist of two parts: the real part and the imaginary part. Given a complex expression, separating these parts is critical for simplification.
In the expression \(\frac{4 - 2\sqrt{2}i}{2}\):

• The real part comes from dividing only the real numbers, which is \(\frac{4}{2} = 2\).
• The imaginary part results from dividing the terms involving \(i\), here given by \(-\frac{2\sqrt{2}}{2}i = -\sqrt{2}i\).

Being able to clearly distinguish between the real and imaginary components allows us to express complex numbers in the form \(a + bi\), aiding in further calculations and understanding of the number's behavior.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as \(a + bi\). Here, \(a\) is the real part, and \(b\) is the imaginary part, with \(i\) being the imaginary unit.
Rewriting expressions in this form is the goal of many exercises involving complex numbers.
In our example, after simplifying, we obtain \(2 - \sqrt{2}i\), where:

• \(a = 2\) represents the real component.
• \(b = -\sqrt{2}\) represents the imaginary component, multiplied by \(i\).

This form makes operations like addition, subtraction, and even multiplication more straightforward when dealing with complex numbers.
Understanding and using the standard form helps streamline processes and clarifies the components of the number being worked with.