Problem 5 - College Algebra with Modeling and Visualization

Question

Simplify by using the imaginary unit \(i\). $$ \sqrt{-23} $$

Step-by-Step Solution

Answer

The simplification of \(\sqrt{-23}\) is \(i\sqrt{23}\).

Step 1: Understanding the Imaginary Unit

The imaginary unit, represented as \(i\), is defined such that \(i^2 = -1\). This means that \(i\) is the square root of \(-1\). We will use this concept to address the square root of a negative number.

Step 2: Expressing the Square Root with Imaginary Unit

To simplify \(\sqrt{-23}\), we need to write it in terms of \(i\). Recognizing \(-23\) as \((-1) \times 23\), we break it into two separate square roots: \(\sqrt{-1} \times \sqrt{23}\).

Step 3: Applying the Definition of \(i\)

By definition, \(\sqrt{-1} = i\). So, we substitute \(i\) for \(\sqrt{-1}\), allowing us to rewrite the original expression as \(i \sqrt{23}\).

Step 4: Final Simplified Form

Now, we have expressed \(\sqrt{-23}\) in terms of the imaginary unit: \(i\sqrt{23}\). This is the simplified form of the expression.

Key Concepts

Imaginary Unit

The imaginary unit, denoted by the symbol \(i\), might sound strange at first because it's not something we encounter in everyday arithmetic. However, it's a crucial part of higher mathematics. The central feature of \(i\) is that its square equals

• \(i^2 = -1\)

This characteristic is what makes \(i\) the "imaginary" component. It's essentially the square root of

• \(-1\)

which is not possible with real numbers. Imaginary numbers are essential when working with complex numbers and certain types of equations. To simplify and handle square roots of negative numbers, understanding \(i\) is fundamental.
Being comfortable with the imaginary unit paves the way for exploring more advanced topics such as complex numbers and functions that aren't restricted to the real number line.

Complex Numbers

Complex numbers allow us to extend the real number system by introducing a new dimension, the imaginary part. These numbers are represented in the form of

• \(a + bi\)

where \(a\) is the real part and \(bi\) is the imaginary part. Here, \(b\) is a real number coefficient, and \(i\) is the imaginary unit. This way, complex numbers form an entire two-dimensional plane, often referred to as the complex plane.
The beauty of complex numbers lies in their comprehensive ability to solve equations that have no solutions in the real numbers alone. They underpin much of modern mathematics and physics, from electrical engineering to quantum mechanics. Learning to work with complex numbers, we get a powerful tool that can describe the universe's complexities in a concise mathematical framework. Understanding how to manipulate these numbers is key to unlocking more complex mathematical concepts.

Square Roots of Negative Numbers

Calculating the square root of a negative number can be puzzling because we learn early on that a square is always positive. Take, for example, the square root of

• -23

. The solution involves using the concept of the imaginary unit \(i\). Here's how:

• Recognize that any negative number, like \(-23\), can be written as \((-1) \times 23\).
• This allows you to split the square root: \(\sqrt{-23} = \sqrt{-1 \times 23} = \sqrt{-1} \times \sqrt{23}\).
• By definition, \(\sqrt{-1} = i\).
• Therefore, \(\sqrt{-23} = i \sqrt{23}\).

When understood this way, handling the square roots of negative numbers becomes straightforward. We convert them into expressions involving \(i\), preserving the mathematical rules that real numbers obey and expanding them to include a broader range of problems.