Problem 12 - Precalculus : Building Concepts and Connections

Question

Write the number as a pure imaginary number. $$\sqrt{-24}$$

Step-by-Step Solution

Answer

The pure imaginary number representation of \(\sqrt{-24}\) is \(2\sqrt{6}i\).

Step 1: Break Down -24

Let's first break down the number -24 under the square root into two factors, one of which involves -1. Hence, \(\sqrt{-24} = \sqrt{-1 * 24}\)

Step 2: Simplify Square Root

Next, we simplify the square root by breaking it down into two separate square roots, \(\sqrt{-1}\) and \(\sqrt{24}\). Using the property \(\sqrt{a * b} = \sqrt{a} * \sqrt{b}\), \(\sqrt{-1 * 24}\) becomes \(\sqrt{-1} * \sqrt{24}\)

Step 3: Evaluate the Square Roots

Now, let's evaluate the two square roots separately. We know that \(\sqrt{-1}\) is 'i'. The \(\sqrt{24}\) can be simplified to \(2\sqrt{6}\) because 24 is 4*6 and 4 is a perfect square . Thus, the original number \(\sqrt{-24}\) is equal to \(2\sqrt{6}i\)

Key Concepts

Complex Numbers

Complex numbers are a type of number that combines real and imaginary parts. They are written in the form of a + bi, where a is the real part and bi is the imaginary part. The letter i stands for the imaginary unit and represents the square root of -1. In our example, \(\frac{-24}{24}\) is initially a complex number, but it can be expressed purely as an imaginary number once simplified.

Understanding complex numbers is essential in various fields of science and engineering, where they are used to describe phenomena that involve two dimensions, such as electricity and fluid flow. These numbers allow for a broader range of solutions in equations, particularly those involving polynomial expressions, where real numbers alone are insufficient.

A key property that makes complex numbers so useful is their ability to be added, subtracted, multiplied, and divided just like real numbers, while following certain rules related to the imaginary unit i.

Imaginary Unit

The imaginary unit, represented as i, is defined by the property that i^2 = -1. This definition extends the set of numbers beyond the real number line. Unlike real numbers which can be visualized along a straight line, the imaginary unit allows for numbers to expand into a new, perpendicular dimension, giving rise to the complex plane.

Role of the Imaginary Unit in Mathematics

Like the number 0, which signaled the inclusion of null value into the number system, the introduction of i has a fundamental role in mathematics. Equations that do not have real solutions, such as x^2 + 1 = 0, become solvable because x can be i. Numbers involving i allow mathematicians and scientists to perform calculations that involve square roots of negative numbers, which would otherwise be impossible.

Simplifying Square Roots

Simplifying square roots involves breaking down the square root of a number into its simplest radical form. When encountering a negative number under the square root, we use the imaginary unit i to facilitate the simplification process. This works because by definition, the square root of -1 is i, and the square root of a product ab is equal to the product of the square roots of a and b (\(\frac{ab}{ab}\)).

In our example, simplifying \(\frac{-24}{24}\) involves realizing that the square root of a negative number can be expressed as an imaginary number. The process includes finding the prime factors of the positive part and taking out the perfect squares. This way, the simplification of \(\frac{24}{24}\) yields 2\(\frac{6}{6}\), while addressing the negative component introduces the imaginary unit, resulting in 2\(\frac{6i}{6i}\), a pure imaginary number.