Problem 14 - Precalculus : Building Concepts and Connections

Question

Write the number as a pure imaginary number. $$\sqrt{-\frac{9}{4}}$$

Step-by-Step Solution

Answer

The pure imaginary number of \(\sqrt{-\frac{9}{4}}\) is \( \frac{3}{2}i \)

Step 1: Identify the Negative Sign

Inside the square root, we have a negative number. This means that the result will be a pure imaginary number. The negative sign inside the square root becomes 'i'.

Step 2: Find the Square Root without Considering the Negative Sign

Next, the value under the square root is 9/4, irrespective of the negative sign. Find the square root of this number without considering the negative sign. The square root of 9/4 is 3/2.

Step 3: Combine the Results

Combine the results of step 1 and step 2. The negative sign in the square root becomes \(i\), and the square root of the remaining number is 3/2. So the square root of -9/4 becomes \(3/2 \cdot i\)

Key Concepts

Complex Numbers

Complex numbers open up new possibilities in mathematics, allowing for the representation of numbers that go beyond the real number line. A complex number is expressed in the form

• a sum notation of a real part and an imaginary part
• Typically, they are written as \(a + bi\)

where \(a\) is the real part and \(bi\) is the imaginary part.
The "imaginary" component, denoted by \(i\), represents the square root of

• a negative number

i.e., \(i = \sqrt{-1}\). So, in the context of complex numbers, any negative square root like \(\sqrt{-\frac{9}{4}}\) involves converting the negative sign into \(i\).

• To find the complete expression, separate the positive square root and multiply it by \(i\).

This allows us to express numbers, like in our problem, as pure imaginary numbers, devoid of a real part. This greatly expands the scope of mathematical solutions, enabling us to solve equations that would otherwise be unsolvable.

Square Roots

Taking the square root of a fraction involves a consistent mathematical process. Let's use \(\frac{9}{4}\) as our example. The process includes these steps:

• Find the square root of the numerator (9) and the denominator (4):
• The square root of 9 is 3, and the square root of 4 is 2.

When combining these, the resultant square root of \(\frac{9}{4}\) is

• simply \(\frac{3}{2}\).

Now, if we introduce a negative sign ahead of this fraction, such as \(\sqrt{-\frac{9}{4}}\), it transforms the expression and necessitates the involvement of the imaginary unit \(i\). Thus, the square root resolves to a purely imaginary form: \(\frac{3}{2} i\). This conversion is crucial in the context of complex numbers, allowing negative square roots to be expressed in a meaningful manner.

Mathematical Problem Solving

Mathematical problem solving, especially involving complex numbers, is a process that requires a clear understanding of concepts and methodical reasoning. Here's a simple approach to tackle problems like converting a negative square root to an imaginary number:

• Identify the negative root and remember it represents \(i\).
• Solve the square root expression of the given non-negative part.
• Merge your solution with \(i\) for the complete expression.

With this problem, we started with \(\sqrt{-\frac{9}{4}}\): noting the negative establishes an initial conclusion that we'll use \(i\).
The real task involves solving \(\sqrt{\frac{9}{4}} = \frac{3}{2}\). Combining this result with \(i\), the solution becomes \(\frac{3}{2} i\).
This demonstrates effectively how understanding the role of \(i\) in complex numbers simplifies seemingly complex tasks and makes solving such mathematical problems straightforward and logical. Recognizing these steps assists learners in approaching similar problems with confidence and ease.