Problem 22 - A Graphical Approach to Precalculus with Limits

Question

Write each number in simplest form, without a negative radicand. $$ \sqrt{-169} $$

Step-by-Step Solution

Answer

The simplest form is \( 13i \).

Step 1: Recognize the Negative Radicand

The exercise involves simplifying \( \sqrt{-169} \). Notice the negative sign inside the square root, which means we are dealing with an imaginary number.

Step 2: Imaginary Unit Introduction

The imaginary unit \( i \) is used to represent the square root of -1, where \( i = \sqrt{-1} \). This means we can rewrite negative square roots in terms of \( i \).

Step 3: Simplify the Given Expression

Extract the negative from the square root: \( \sqrt{-169} = \sqrt{169} \times \sqrt{-1} \). We already know \( \sqrt{-1} = i \), so it becomes \( \sqrt{169}i \).

Step 4: Calculate the Real Component

Find \( \sqrt{169} \), which is 13. Therefore, combining these results, the simplest form of \( \sqrt{-169} \) is \( 13i \).

Key Concepts

Square Roots

Square roots are a crucial concept in mathematics, generally representing a number that, when multiplied by itself, gives the original number. Normally, square roots are associated with non-negative numbers. For example, the square root of 9 is 3, because 3 multiplied by itself equals 9. However, when the numbers inside the square root, known as radicands, are negative, complexities arise. Negative radicands present a problem because no real number squared can result in a negative value. This special case is where the concept of imaginary numbers comes into play, providing a neat solution to handling square roots of negative numbers.

Complex Numbers

Complex numbers expand the scope of traditional real numbers by incorporating imaginary numbers. They are expressed in the form of \(a + bi\), where:

• \(a\) is the real part.
• \(bi\) is the imaginary part.

The real part is a standard number from the number line, while the imaginary part involves the imaginary unit \(i\). What makes complex numbers fascinating is their ability to bridge between real numbers and imaginary numbers. By adding a real component to an imaginary number, complex numbers can represent points on a plane rather than simply positions on a line.In our exercise, the number \(13i\) is a pure imaginary number, as it lacks any real component.

Imaginary Unit

The imaginary unit, denoted by \(i\), is the core of imaginary numbers. It is defined by the equation \(i = \sqrt{-1}\). This means that \(i^2 = -1\). It might seem strange initially, but this definition allows us to handle square roots of negative numbers effectively.Whenever you encounter \(\sqrt{-x}\) for a positive \(x\), you can express it using \(i\). For instance, \(\sqrt{-169}\) can be rewritten as \(\sqrt{169} \cdot \sqrt{-1}\), or \(13i\), as seen in our original solution. This provides a systematic way to simplify expressions that include square roots of negative numbers, using algebraic operations that involve \(i\). The imaginary unit helps us understand and work with a broader set of numbers, making it a crucial concept in both theoretical and applied mathematics.