Problem 14 - Precalculus Student Solutions Manual 5th

Question

Identify each number as real, complex, pure imaginary, or nonreal complex. (More than one of these descriptions will apply. ) $$\sqrt{24}$$

Step-by-Step Solution

Answer

2\sqrt{6} is a real number and a complex number.

Step 1: Simplify the Expression

Simplify the expression \(\sqrt{24}\) by breaking it down into its prime factors. Notice that \(24 = 4 \times 6 = 2^2 \times 2 \times 3\). Therefore, \(\sqrt{24} = \sqrt{2^2 \times 2 \times 3} = 2 \sqrt{6}\).

Step 2: Identify the Nature of the Number

Recognize that \(\sqrt{24} = 2 \sqrt{6}\) is a real number because it does not involve the imaginary unit i. Since it can be represented on the number line, it is a real number.

Step 3: Check Other Classifications

Besides being a real number, determine if the number is also complex. A complex number is of the form \(a + bi\) where \(a\) and \(b\) are real numbers. Since real numbers are a subset of complex numbers, \(\sqrt{24} = 2\sqrt{6}\) can also be classified as a complex number with the imaginary part \(b\) equal to 0.

Key Concepts

Real Numbers

Real numbers are numbers that can represent a distance along a line. This includes all the numbers we usually use in everyday life, such as integers, fractions, and irrational numbers.

Real numbers can be positive, negative, or zero. They can also be written in decimal form. For example, numbers like \(\frac{1}{2}\) and \(\frac{7}{8}\) are fractions, and irrational numbers like \(\frac{\text{√2}}{2}\) cannot be written as a simple fraction, but are still real numbers.

The number \(\text{√24}\) simplifies to \(\text{2√6}\), which is also a real number because it does not involve the imaginary unit \(\text{i}\). Since it can be located somewhere on the number line, it's classified as a real number.

Complex Numbers

Complex numbers follow the form \(\text{a} + \text{bi}\), where \(\text{a}\) and \(\text{b}\) are real numbers, and \(\text{i}\) is the imaginary unit, which satisfies \(\text{√-1} = \text{i}\).

This means that complex numbers include both real parts and imaginary parts. For example, \(\text{3} + \text{4i}\) is a complex number where \(\text{3}\) is the real part, and \(\text{4i}\) is the imaginary part.

All real numbers are a subset of complex numbers. This is because you can see a real number as a complex number with the imaginary part equal to \(\text{0}\). For example, \(\text{√24}\), simplifiable to \(\text{2√6}\), can be seen as \(\text{2√6} + 0i}\).

Prime Factorization

Prime factorization is a method of breaking down a number into its basic building blocks, which are prime numbers. A prime number is a number that has no divisors other than \(\text{1}\) and itself.

For example, the prime factorization of 24 is done by dividing it into primes: \(\text{24 = 2 × 2 × 2 × 3}\), or \(\text{24 = 2^3 × 3}\).

We use prime factorization to simplify \(\text{√24}\). We know \(\text{√24}\) can be written as \(\text{√(4 × 6)}\). With prime factorization, we find that \(\text{4}\) and \(\text{6}\) can be broken down as: \(\text{4 = 2^2}\) and \(\text{6 = 2 × 3}\). When combined, we get: \(\text{24 = 2^2 × 2 × 3}\).

Thus, \(\text{√24}\) becomes \(\text{√(2^2 × 2 × 3)\) = \(\text{2√6}\), as the factor of \(\text{2^2}\) comes out of the square root as \(\text{2}\).