Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write each number in terms of \(i\) $$ \sqrt{-4} $$

Step-by-Step Solution

Answer

The expression \( \sqrt{-4} \) is equivalent to \( 2i \).

Step 1: Recognize the property of square roots with negative numbers

In order to solve the problem, we need to recall that the square root of a negative number involves the imaginary unit, denoted as \( i \), where \( i^2 = -1 \).

Step 2: Simplify the square root expression

We start by simplifying \( \sqrt{-4} \) using the property from Step 1. We express this as \( \sqrt{4} \times \sqrt{-1} \).

Step 3: Calculate the numerical square root

Calculate \( \sqrt{4} \), which is 2, so the expression becomes \( 2 \times \sqrt{-1} \).

Step 4: Express \( \sqrt{-1} \) in terms of \( i \)

We know that \( \sqrt{-1} = i \), so substitute this into the expression: \( 2 \times i \).

Step 5: Final expression

Therefore, \( \sqrt{-4} \) is simplified to \( 2i \).

Key Concepts

Understanding Square Roots

Square roots are a fundamental concept in mathematics. When we talk about the square root of any number, say 4, we are looking for a number which, when multiplied by itself (squared), gives 4. In this case, 2 is the square root because

• 2 \( \times \) 2 = 4.

But what about when we need to find the square root of a negative number, like -4? Here's where things get interesting. Traditional square roots involve positive numbers only, since no real number multipled by itself results in a negative number. This is where imaginary numbers come into play, providing a way to extend the idea of square roots to negative numbers.

The Imaginary Unit (i)

Imaginary numbers, on the surface, might sound like something out of a fantasy novel, but they're a very real part of algebra and calculus. The core of imaginary numbers is the imaginary unit, denoted as \( i \). This special symbol helps us work with the square roots of negative numbers. The defining feature of the imaginary unit is:

• \( i^2 = -1 \).

This means that multiplying the imaginary unit by itself results in -1, explaining why it's called 'imaginary'. It's a simple yet powerful tool that lets us simplify expressions like \( \sqrt{-1} \), translating them easily to \( i \). Whenever you see a square root of a negative number, you can employ \( i \) to convert it into a workable form.

Working with Negative Numbers

Negative numbers can be tricky to understand, especially when paired with square roots. A negative number appears whenever you subtract a larger number from a smaller one or when multiplying two numbers with different signs. But with square roots, you venture into a zone beyond real numbers.

• Normally, expressions such as \( \sqrt{-4} \) cannot be handled in the realm of real numbers.
• Yet, by transforming them into forms involving \( i \), they become manageable.

In essence, imaginary numbers let us take square roots of negatives, turning an impossible operation into a doable one.

Introducing Complex Numbers

Now let's combine the concepts of real numbers and imaginary numbers. The result is what mathematicians call complex numbers. A complex number includes both a real part and an imaginary part, expressed in the form:

• \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.

For example, in the expression \( 2i \), the real part \( a \) is 0, while the imaginary part \( b \) is 2. Complex numbers allow us to extend our understanding of numbers even further, providing a way to represent solutions to equations that involve the square roots of negative numbers and utilizing all the tools that both real and imaginary numbers offer.