Problem 111 - Applied Calculus

Question

Should \(\sqrt{9}\) be evaluated as 3 or \(\pm 3 ?\)

Step-by-Step Solution

Answer

\( \sqrt{9} \) is evaluated as 3.

Step 1: Understanding the Problem

To solve whether \( \sqrt{9} \) is equal to 3 or \( \pm 3 \), we need to understand how square roots work and the convention used in the context of this problem.

Step 2: Square Root Definition

The square root of a number \( a \) is the value \( b \) such that \( b^2 = a \). By definition, the square root operation \( \sqrt{a} \) refers to the principal (or non-negative) square root of \( a \).

Step 3: Example Evaluation

Let's consider \( \sqrt{9} \). We are looking for a number that, when squared, gives 9. Both 3 and -3 satisfy this since \( 3^2 = 9 \) and \((-3)^2 = 9 \), but the square root function returns only the principal square root.

Step 4: Identify the Principal Square Root

According to mathematical convention, \( \sqrt{9} \) evaluates to just 3 as it represents the principal square root (the non-negative one), not \( \pm 3 \). The notation \( \pm 3 \) occurs when solving equations like \( x^2 = 9 \), indicating both possible solutions.

Key Concepts

Principal Square Root

The principal square root is a specific term used in mathematics to signify the non-negative square root of a number. When we refer to the square root symbol, like in \( \sqrt{a} \), it always refers to the principal square root, which is the positive out of the possible root values.

For example, the square root of 9 can be either 3 or -3, since both these numbers squared return 9. However, by convention, \( \sqrt{9} \) denotes the principal square root, only returning the non-negative value, which is 3. This avoids ambiguity and ensures consistency in mathematical operations and communication.

• The principal square root is always non-negative.
• It is denoted simply as \( \sqrt{a} \) for any non-negative number \( a \).
• Negative numbers do not have a real principal square root.

Understanding this concept helps us know why \( \sqrt{9} \) is not \( \pm 3 \), but rather exactly 3.

Square Root Definition

The square root of a number is a key mathematical concept. It is defined as a value which, when multiplied by itself, yields the original number. For any non-negative number \( a \), the square root is represented as \( \sqrt{a} \).

This operation is often used in geometry, algebra, and real-life applications to determine dimensions and solve equations. A number can have two square roots: one positive and one negative. However, by definition, the square root symbol \( \sqrt{} \) provides only the principal square root, the positive value.

• Every positive number has two real square roots (except zero, which has one unique root: itself).
• The notation \( \sqrt{a} \) is understood to denote only the non-negative root.
• In equations, instead of \( \sqrt{a} \), you may see \( \pm \sqrt{a} \), indicating both roots.

Hence, understanding the square root definition clarifies why \( \sqrt{9} \) specifically resolves to 3, without ambiguity.

Mathematical Conventions

Mathematical conventions serve to standardize how we write and solve problems, ensuring clear communication and understanding. In the context of square roots, the convention dictates that \( \sqrt{a} \) is always the principal square root, which is the non-negative root.

This allows for consistency across various mathematical texts and contexts. The choice of using only the non-negative root simplifies problem solving and allows mathematicians to have a common language.

• The principal square root \( \sqrt{a} \) is always non-negative to prevent confusion.
• If both roots are to be considered, the notation \( \pm \sqrt{a} \) is explicitly used.
• Mathematical conventions provide a framework that applies universally, from basic computations to complex algorithms.

Recognizing and adhering to these conventions is crucial for anyone engaging in mathematical study or application, explaining why \( \sqrt{9} \) yields 3 according to the defined rules.