Problem 110

Question

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

Step-by-Step Solution

Verified

Answer

The expression is $2 - \sqrt{2}i$.

1Step 1: Simplify the expression

Notice that the expression is given as $ \frac{4-\sqrt{-8}}{2} $. The term $ \sqrt{-8} $ needs to be converted to include the imaginary unit $ i $. Rewrite $ \sqrt{-8} $ as $ \sqrt{8} \cdot i $. Since $ \sqrt{8} = 2\sqrt{2} $, this becomes $ 2\sqrt{2}i $. Therefore, the expression is $ \frac{4 - 2\sqrt{2}i}{2} $.

2Step 2: Divide each term by 2

Now separate the expression $ \frac{4 - 2\sqrt{2}i}{2} $ into two fractions: $ \frac{4}{2} - \frac{2\sqrt{2}i}{2} $. Simplify each fraction individually by dividing both the real and imaginary parts by 2.

3Step 3: Simplify each term

Simplify $ \frac{4}{2} $ to 2, and $ \frac{2\sqrt{2}i}{2} $ to $ \sqrt{2}i $. Therefore, the expression becomes $ 2 - \sqrt{2}i $.

4Step 4: Express in the form $a + bi$

The expression after simplification is $2 - \sqrt{2}i $, which is already in the form $a + bi$ where $a = 2$ and $b = -\sqrt{2}$.

Key Concepts

Imaginary UnitSimplify ExpressionsAlgebraic Form $a+bi$

Imaginary Unit

The imaginary unit, denoted as $ i $, is a crucial concept in the realm of complex numbers. It is defined by the property $ i^2 = -1 $. This may seem peculiar because it allows for the square root of negative numbers to exist in mathematics.
In practical terms, when you see a negative number under a square root, like $ \sqrt{-8} $ in our exercise, you can think of it as involving the imaginary unit. Here's how it works:

First, break down the negative square root into two parts: the square root of the positive number, and the square root of negative one. So, $ \sqrt{-8} $ becomes $ \sqrt{8} \cdot i $.
The $ \sqrt{8} $ can be simplified further, usually into something smaller and more recognizable, such as $ 2\sqrt{2} $, keeping the $ i $ next to it.

So any time you see $ i $, remember it's more than just a letter, it's a fundamental concept letting us handle otherwise impossible calculations.

Simplify Expressions

Simplifying expressions with complex numbers often means exactly that: making them simpler and easier to understand. This usually involves breaking down larger components into smaller steps, just like we did in the exercise.
Here’s a step-by-step for simplifying $ \frac{4-\sqrt{-8}}{2} $:

Start by addressing any roots involving negative numbers using the imaginary unit $ i $. For $ \sqrt{-8} $, rewrite it as $ 2\sqrt{2}i $.
Substitute this back into the entire expression, giving you: $ \frac{4 - 2\sqrt{2}i}{2} $.
Then, break down the main fraction into two separate fractions: $ \frac{4}{2} - \frac{2\sqrt{2}i}{2} $.
Simplify these fractions to get the neatest form. $ \frac{4}{2} $ results in $ 2 $ and $ \frac{2\sqrt{2}i}{2} $ results in $ \sqrt{2}i $.

Using step-by-step simplification helps prevent errors and makes complex numbers more manageable.

Algebraic Form $a+bi$

Expressing complex numbers in the form $a + bi$ is key for clarity and comparison. The format shows two parts:

$ a $ represents the real component.
$ b $ is the coefficient of the imaginary part, showing how many multiples of $ i $ are involved.

This method of expression is elegant in its simplicity yet broad enough to handle the complexities of imaginary numbers and real numbers together.
In our example, once simplified, we expressed our result as $ 2 - \sqrt{2}i $.
This is already in the form $ a + bi $, showing the real part as 2 and the imaginary part as $ -\sqrt{2} $.
The main takeaway is that the $ a + bi $ form makes it easy to quickly identify each part of the complex number and how they interrelate. It's a foundation for further calculations, making it a vital part of any complex number work.

Problem 110