Problem 18 - Intermediate Algebra

Question

Express each number in terms of \(i\). $$ \sqrt{-4} $$

Step-by-Step Solution

Answer

\( \sqrt{-4} = 2i \)

Step 1: Understand the Concept of Imaginary Numbers

Imaginary numbers are numbers that can be expressed as a real number multiplied by the imaginary unit, denoted as \( i \). With the property that \( i^2 = -1 \), the imaginary unit allows expression of the square root of negative numbers.

Step 2: Convert the Inside of the Square Root

We are given \( \sqrt{-4} \). Notice that \(-4\) can be rewritten using the imaginary unit: \( -4 = 4 \times (-1) \). Thus, \( \sqrt{-4} = \sqrt{4 \times (-1)} \).

Step 3: Apply the Product Property of Square Roots

Use the property that \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). Apply this to the expression: \( \sqrt{4 \times (-1)} = \sqrt{4} \cdot \sqrt{-1} \).

Step 4: Calculate the Square Roots

Calculate each part: \( \sqrt{4} = 2 \) because 2 squared equals 4, and \( \sqrt{-1} = i \) by definition of the imaginary unit. Therefore, \( \sqrt{4} \cdot \sqrt{-1} = 2 \cdot i \).

Step 5: Write the Final Expression

The expression simplifies to \( 2i \). Therefore, \( \sqrt{-4} = 2i \).

Key Concepts

Complex Numbers

Complex numbers form the cornerstone of a more profound understanding of mathematics, allowing us to handle problems involving negative square roots. A complex number is generally expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is known as the imaginary unit. By extending the number system beyond real numbers, complex numbers encompass all roots of polynomial equations.

• Real Part: The component \(a\) of the complex number. It exists on the horizontal axis of the complex plane.
• Imaginary Part: The component \(bi\), found on the vertical axis. This part involves the imaginary unit \(i\).

Understanding complex numbers provides insight into equations that lack real solutions, such as \(x^2 + 1 = 0\). By using \(i\), the equation can be solved as \(x = \pm i\), demonstrating the utility of complex numbers in broadening mathematical exploration.

Imaginary Unit

The imaginary unit, represented as \(i\), forms the foundation for expressing complex numbers. The characteristic property of this unit is \(i^2 = -1\). This new form of number enables mathematicians to engage with the square roots of negative values, which are undefined using only real numbers.
Its significance can be highlighted as follows:

• Fundamental Property: The expression \(i\) is defined such that \(i^2 = -1\), giving birth to imaginary numbers.
• Applications: It is crucial in engineering, physics, and applied mathematics, where complex solutions are necessary.

A practical use of \(i\) is showcased in problems such as finding \(\sqrt{-4}\). By utilizing \(i\), the solution becomes clear and manageable: \(\sqrt{-4} = 2i\). Understanding \(i\) and its properties enhances our ability to manipulate and solve equations involving negative square roots.

Properties of Square Roots

The concept of square roots is pivotal in mathematics, indicating the number that, when multiplied by itself, results in the original number. The transformation of expressions into imaginary terms becomes feasible through the properties of square roots.
Here are some key properties:

• Product Property: This states that \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\), allowing decomposition of complex radicals.
• Negative Under Radicals: Normally, square roots of negative numbers are indeterminate within the real number system, but imaginary numbers enable us to define them clearly as seen in \(\sqrt{-4} = 2i\).

By applying these properties, complex expressions become solvable. Consider \(\sqrt{-4}\); it’s expressed using the product property as \(\sqrt{4} \cdot \sqrt{-1}\), leading to \(2i\) by calculating each part separately. This illustrates how properties of square roots facilitate the handling of negative square roots effectively.