Problem 5 - College Algebra Essentials

Question

Evaluate each expression in Exercises \(1-12,\) or indicate that the root is not a real number. $$\sqrt{-36}$$

Step-by-Step Solution

Answer

The square root of -36 is \(6i\).

Step 1: Identify the type of number

Recognize that the number under the square root is negative, which means it's not a real number. In real number arithmetic, the square root of a negative number is undefined.

Step 2: Apply rules of imaginary numbers

Since it is an imaginary number, rewrite \(\sqrt{-36}\) as \(i\sqrt{36}\). In the notation of imaginary numbers 'i' is used to denote \(\sqrt{-1}\), so 'i' times the square root of any positive number gives the square root of the equivalent negative number.

Step 3: Compute the square root

Finally, compute the square root of 36, which is 6. Thus, the expression simplifies to \(i * 6\) or \(6i\).

Key Concepts

Square Roots

The concept of square roots is about finding a number that, when multiplied by itself, results in the original number under the square root symbol. For example, if we have the number 36, the square root of 36 is 6, because 6 multiplied by 6 equals 36. Square roots are typically written using the radical symbol \( \sqrt{ } \). This symbol is used to represent the principal square root, which is the non-negative root of a positive number.

However, when the number inside the square root is negative, such as \( \sqrt{-36} \), it leads to a problem in the realm of real numbers. In the set of real numbers, the square root of a negative number does not exist, because no real number multiplied by itself can give a negative product. This is where we enter the world of imaginary numbers, which extends our understanding beyond the limits of real number arithmetic.

Real Numbers

Real numbers comprise all the numbers we usually think of, including rational numbers like fractions and integers, as well as irrational numbers like \( \pi \) and \( \sqrt{2} \). Essentially, a real number is any point on the number line you can think of, whether it's positive, negative, or zero.

One important limitation of real numbers is their inability to deal with the square roots of negative numbers. In the expression \( \sqrt{-36} \), the -36 is under a square root and poses a challenge with real number operations because multiplying two identical real numbers always yields a positive result. So, real numbers do not provide a solution for negative square roots, leading us into the necessity for complex numbers, which include imaginary numbers.

Complex Numbers

Complex numbers extend the usual notion of real numbers by introducing the concept of imaginary numbers, which includes the unit imaginary number denoted as \( i \), where \( i^2 = -1 \). Using this notion, we can express the square root of a negative number in terms of \( i \).

For example, the square root of \( -36 \) is expressed as \( i \sqrt{36} \). Since \( \sqrt{36} = 6 \), this simplifies to \( 6i \). So, \( 6i \) is a complex number with no real part, indicating purely imaginary value.

• A complex number usually has two parts: the real part and the imaginary part, formatted as \( a + bi \).
• If the real part is zero, as in \( 6i \), the number is purely imaginary.
• Complex numbers allow us to compute operations that would be impossible with just real numbers.

By using complex numbers, mathematicians can solve equations and express results that occur in electronic engineering, fluid dynamics, and many other fields reliant on advanced mathematical concepts.