Problem 6 - College Algebra with Modeling and Visualization

Question

Simplify by using the imaginary unit \(i\). $$ \sqrt{-11} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{-11}\) is \(\sqrt{11}i\).

Step 1: Understanding Complex Numbers

Recall that the imaginary unit \(i\) is defined as \(i = \sqrt{-1}\). This means that \(i^2 = -1\). When we encounter a square root of a negative number, we can utilize the imaginary unit in our simplification.

Step 2: Rewrite the Square Root of a Negative Number

To simplify \(\sqrt{-11}\), start by expressing it in terms of \(i\): \(\sqrt{-11} = \sqrt{11} \, \times \, \sqrt{-1}\).

Step 3: Substitution with the Imaginary Unit

Replace \(\sqrt{-1}\) with \(i\) using the definition of the imaginary unit: \(\sqrt{11} \, \times \, i = \sqrt{11}i\). This is the simplified form of \(\sqrt{-11}\).

Key Concepts

Complex Numbers

Complex numbers are fascinating mathematical constructs that extend our understanding of numbers to include both real and imaginary parts. A complex number is typically written in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. Here, \(a\) represents the "real" part of the complex number, while \(bi\) represents the "imaginary" part.

Using complex numbers opens up the possibility of performing arithmetic operations like addition, subtraction, multiplication, and division with numbers that were previously unimaginable, such as the square roots of negative numbers.

• Real Part: the component \(a\)
• Imaginary Part: the component \(bi\)
• Imaginary Unit: represented by \(i\) and satisfies \(i^2 = -1\)

The wonderful aspect of complex numbers is their ability to provide solutions to equations that do not have real solutions, which makes them valuable in fields ranging from engineering to quantum physics.

Square Roots

The concept of square roots is a fundamental part of mathematics, where the square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). For positive numbers, this is straightforward. However, square roots of negative numbers lead to more intricate definitions.

Normally, we can't find a real number whose square is negative. To solve this problem, mathematicians introduced the concept of imaginary numbers. The use of imaginary numbers allows us to deal with the square roots of negative numbers, paving the way for a broader range of mathematical expressions and applications. For example:

• Square Root of a Positive Number: \( \sqrt{4} = 2 \)
• Square Root of Zero: \( \sqrt{0} = 0 \)
• Square Root of Negative Number: Involves the imaginary unit \(i\), e.g., \( \sqrt{-11} \)

The introduction of imaginary numbers through the square roots of negative numbers broadens the mathematical world, allowing us to simplify expressions like \( \sqrt{-11} \), resulting in complex numbers.

Imaginary Unit

Understanding the imaginary unit \(i\) is key to working with complex numbers and simplifying expressions that involve negative square roots. The imaginary unit is defined by the equation \(i = \sqrt{-1}\). This fundamental property allows us to express the square root of any negative number in terms of \(i\). For instance, \( \sqrt{-1} \) itself is \(i\), making it a crucial building block in the realm of complex numbers.

The main features of the imaginary unit are:

• \(i^2 = -1\)
• Used to express square roots of negative numbers, such as \( \sqrt{-11} = \sqrt{11}i \)
• Essential for simplifying and working with complex numbers

Imaginary numbers expand our understanding of mathematics, providing solutions and expressions that include previously unattainable numbers.