Problem 17 - Elementary and Intermediate Algebra

Question

Express each number in terms of i. See Example 1 . $$ \sqrt{-9} $$

Step-by-Step Solution

Answer

The expression \(\sqrt{-9}\) can be written as \(3i\).

Step 1: Understand the concept of the imaginary unit

The imaginary unit, denoted by \(i\), is defined as \(i = \sqrt{-1}\). This means that \(i^2 = -1\). Any number that involves the square root of a negative number can be expressed in terms of \(i\).

Step 2: Break down the square root of negative number

To express \(\sqrt{-9}\) in terms of \(i\), first separate the negative sign from the number under the square root. This can be written as \(\sqrt{-9} = \sqrt{-1} \times \sqrt{9}\).

Step 3: Simplify the expression using the imaginary unit

Using the definition of the imaginary unit \(i\), we have \(\sqrt{-1} = i\). Also, \(\sqrt{9} = 3\). Therefore, we can simplify \(\sqrt{-9}\) as \(\sqrt{-9} = i \times 3 = 3i\).

Key Concepts

Imaginary Unit

When we encounter negative numbers under a square root, we use something called the imaginary unit. The imaginary unit is symbolized by \(i\). It's a mathematical invention designed to allow us to work with square roots of negative numbers. The definition of \(i\) is absolutely fundamental: \(i = \sqrt{-1}\). This idea might seem a little abstract since we're used to numbers that we can visualize or count. However, \(i\) is perfectly acceptable in mathematics and is a core building block for more advanced concepts like complex numbers.
Here's a simple way to think about it:

• \(i\) is defined such that when you multiply it by itself (\(i^2\)), you get \(-1\).
• This property allows us to take the square root of any negative number by calculating the positive square root of its absolute value and then multiplying by \(i\).

This will seem more intuitive when you see it applied in the next sections.

Complex Numbers

A complex number is like a two-part number in which one part is a real number and the other part is an imaginary number. This might sound complicated, so let's break it down.A complex number is usually written in the form \(a + bi\), where:

• \(a\) is the real part.
• \(bi\) is the imaginary part, with \(b\) being a real number that is multiplied by \(i\).

The beauty of complex numbers is that they allow for the expression of all numbers, covering the entire spectrum from real numbers to purely imaginary numbers and everything in between.For example, consider the complex number \(2 + 3i\). Here, \(2\) is the real part, and \(3i\) is the imaginary part. Note that when \(b = 0\), the complex number is just a real number. Similarly, when \(a = 0\), it's a purely imaginary number.

Square Root of Negative Numbers

Calculating the square root of a negative number can be tricky until you understand the use of imaginary numbers. When asked to find the square root of a negative number, such as \(\sqrt{-9}\), you will need to employ the imaginary unit \(i\).Here's how it works:

• First, express the negative square root in terms of \(i\) and the positive square root. For instance, \(\sqrt{-9} = \sqrt{-1} \times \sqrt{9}\).
• Next, use the definition of \(i\), where \(\sqrt{-1} = i\).
• Finally, simplify by evaluating the positive square root separately: \(\sqrt{9} = 3\).

By putting it all together, \(\sqrt{-9} = i \times 3 = 3i\). This process turns the square root of a negative number into a manageable expression using the imaginary unit.This technique is immensely useful in various areas of mathematics, such as engineering and physics, where complex numbers often appear.