Problem 2 - College Algebra with Modeling and Visualization

Question

Simplify by using the imaginary unit \(i\). $$ \sqrt{-16} $$

Step-by-Step Solution

Answer

\( \sqrt{-16} = 4i \).

Step 1: Recognize the Imaginary Unit

The imaginary unit, denoted by \( i \), is defined as \( i = \sqrt{-1} \). It allows us to handle square roots of negative numbers.

Step 2: Express as a Product

To simplify \( \sqrt{-16} \), recognize that \( -16 \) can be rewritten as a product of \( -1 \) and \( 16 \). Thus, \( \sqrt{-16} = \sqrt{-1 \times 16} \).

Step 3: Break into Separate Roots

Use the property that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). Applying this, \( \sqrt{-1 \times 16} = \sqrt{-1} \times \sqrt{16} \).

Step 4: Simplify Each Root

Now simplify each square root separately. We know \( \sqrt{-1} = i \) and \( \sqrt{16} = 4 \). Therefore, \( \sqrt{-16} = i \times 4 \).

Step 5: Multiply the Results

Finally, multiply the results from the previous step. We get \( i \times 4 = 4i \). Thus, \( \sqrt{-16} = 4i \).

Key Concepts

Imaginary Unit

In mathematics, understanding the imaginary unit is essential. The imaginary unit, symbolized by \( i \), is used to define imaginary numbers, and it's crucial in dealing with square roots of negative numbers. By definition, \( i \) is the square root of \( -1 \), formulated as \( i = \sqrt{-1} \). This concept helps in extending the number system beyond real numbers to what is known as the complex number system, which combines both real and imaginary numbers.

• The imaginary unit is a fundamental element in complex numbers.
• It helps to solve equations that have no real number solutions, like \( x^2 + 1 = 0 \).

Imaginary numbers, created using \( i \), are not "real" in the traditional sense. However, they are pivotal in fields like engineering and physics for representing phenomena that cannot be explained using only real numbers.

Square Roots of Negative Numbers

Square roots of negative numbers have often been troublesome in basic arithmetic since the square of any real number is non-negative. That's why the concept of the imaginary unit \( i \) becomes useful, as it enables the handling of such square roots.

To determine the square root of a negative number, like \( \sqrt{-16} \), one breaks it down into expressions involving \( \sqrt{-1} \). For example:

• Recognize that \( -16 \) can be rewritten using its factors as \( -1 \times 16 \).
• Use the product property: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
• Solve each part: \( \sqrt{-1} = i \) and \( \sqrt{16} = 4 \).
• Thus, we combine these results to get \( 4i \).

This method allows mathematicians to consistently calculate square roots of negative numbers, fostering a broader understanding of quantities otherwise limited to their real counterparts.

Algebra and Imaginary Numbers

In algebra, imaginary numbers extend the capabilities of the number system. Algebra often deals with finding solutions to equations, and with the role of imaginary numbers, solutions can be found even when conventional real number solutions don't exist.

For instance, equations like \( x^2 + 1 = 0 \) have no real solutions because no real number squared equals \(-1\). However, through the introduction of \( i \), solutions become \( x = i \) and \( x = -i \).

• Imaginary numbers are paired with real numbers to form complex numbers, expressed as \( a + bi \).
• This expands algebraic operations to include complex numbers, enhancing calculative depth.
• They are routinely used in polynomial equations and calculus-based problems.

Thus, imaginary numbers underpin much of modern algebra's ability to provide solutions across various fields in science and engineering.