Problem 38 - College Algebra with Corequisite Support

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ \frac{4+\sqrt{-20}}{2} $$

Step-by-Step Solution

Answer

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

Step 1: Resolve the square root of a negative number

The expression \( \sqrt{-20} \) indicates the square root of a negative number. Recognize that \( \sqrt{-20} = \sqrt{-1} \times \sqrt{20} \). Since \( \sqrt{-1} = i \), this simplifies to \( \sqrt{20} i \). Furthermore, \( \sqrt{20} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \). So, \( \sqrt{-20} = 2\sqrt{5}i \).

Step 2: Substitute and simplify

Substitute \( \, \sqrt{-20} = 2\sqrt{5}i\, \) back into the expression: \( \frac{4 + 2\sqrt{5}i}{2} \).

Step 3: Distribute division

To simplify \( \frac{4 + 2\sqrt{5}i}{2} \), divide each term in the numerator by the denominator 2: \( \frac{4}{2} + \frac{2\sqrt{5}i}{2} = 2 + \sqrt{5}i \).

Step 4: Expression as a simplified complex number

The expression \( 2 + \sqrt{5}i \) is already in the standard form for a complex number, which is \( a + bi \) where \( a \) and \( b \) are real numbers.

Key Concepts

Square Roots of Negative Numbers

Understanding the square root of negative numbers is crucial when working with complex numbers. A regular square root, such as \( \sqrt{4} \), simply means "which number, multiplied by itself, gives 4?". The answer is \(2\) because \(2 \times 2 = 4\). However, when it comes to negative numbers, such as \( \sqrt{-20} \), there's a twist because no real number squared results in a negative number.

This is where the concept of imaginary numbers kicks in. By definition, the square root of \(-1\) is represented by the imaginary unit, \(i\). So, \( \sqrt{-20} \) translates to \( \sqrt{-1} \times \sqrt{20} = i \sqrt{20} \). Breaking this down further:

• \( \sqrt{20} \) can be simplified into \( \sqrt{4 \times 5} \)
• This gives \( 2\sqrt{5} \), since \( \sqrt{4} = 2 \)

Thus, \( \sqrt{-20} = 2\sqrt{5}i \), a key component when simplifying complex numbers.

Simplifying Complex Expressions

When simplifying complex expressions, like \( \frac{4+\sqrt{-20}}{2} \), it's important to follow a systematic approach to make calculations easy. First, make sure to express any square roots of negative numbers as complex numbers using the imaginary unit, \(i\).

In our exercise, we identify that \( \sqrt{-20} = 2\sqrt{5}i \). The next step is to rewrite our expression as \( \frac{4 + 2\sqrt{5}i}{2} \). Now, dividing each term individually by the common denominator:

• Divide 4 by 2 to get 2
• Divide \(2\sqrt{5}i\) by 2 to get \(\sqrt{5}i\)

This results in the simplified complex number \(2 + \sqrt{5}i\).

Knowing the general form \(a + bi\), where \(a\) and \(b\) are real numbers, helps ensure the expression is in its simplest form.

Imaginary Unit (i)

The imaginary unit \(i\) is a fundamental concept in understanding complex numbers. The imaginary unit itself is defined as \(i = \sqrt{-1}\), which is a groundbreaking idea because it allows us to work with the square roots of negative numbers.

\(i\) is used to represent numbers that are not on the regular number line (the real number line). Instead, it extends into another dimension—the imaginary dimension—where complex numbers live. Here's how both real and imaginary parts come together in a complex number:

• \(a\) represents the real part
• \(bi\) represents the imaginary part, where \(b\) is a real number and \(i\) signifies the component of the number in the imaginary dimension

Since \(i\) is such a unique element, it has properties unlike regular real numbers:

• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\) (and this cycle repeats)

Understanding how \(i\) operates within complex numbers is essential for simplifying and calculating expressions involving the imaginary unit.