Problem 9 - Algebra 1: Concepts and Skills

Question

Determine whether each expression is rational or irrational. $$ \sqrt{6} $$

Step-by-Step Solution

Answer

The expression \( \sqrt{6} \) is irrational.

Step 1: Definition of rational and irrational numbers

Let's start by defining rational and irrational numbers. Rational numbers are the numbers which can be represented as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers, and \( b ≠ 0 \). On the other hand, irrational numbers are numbers that cannot be represented as a simple fraction, they are real numbers that have a decimal expansion that never terminates nor becomes periodic.

Step 2: Analyze the given expression

We are given the expression \( \sqrt{6} \). Let's try to analyse this expression. The square root of a number \( x \) can be a rational number if \( x \) is a perfect square, meaning it can be written as \( n^2 \), where \( n \) is an integer. In this case, \( 6 \) is not a perfect square, which means the \( \sqrt{6} \) is not a rational number.

Step 3: Conclusion

Since \( \sqrt{6} \) cannot be represented as a simple fraction, and its decimal expansion never terminates nor becomes periodic, we can conclude that \( \sqrt{6} \) is an irrational number.

Key Concepts

Understanding Square Roots

The square root is a mathematical function that helps us find a number which, when multiplied by itself, results in a given value. Represented as \( \sqrt{x} \), it searches for a number that, when squared (multiplied by itself), equals \( x \).

For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \). Similarly, \( \sqrt{4} \) is 2 because \( 2 \times 2 = 4 \). These are examples where the value under the square root symbol is a perfect square, meaning a whole number squared.

When a square root results in a whole number, it's considered a rational number. However, when a square root can't be simplified into a whole number, it can be an irrational number. For instance, \( \sqrt{6} \) doesn't result in a clean whole number; therefore, it indicates an irrational value. To simplify further, note:

• If \( x \) is a perfect square, \( \sqrt{x} \) is rational.
• If \( x \) is not a perfect square, \( \sqrt{x} \) is irrational.

Decoding Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. In simpler terms, if a number is a perfect square, it means there is another whole number which, when multiplied by itself, equals the original number. Consider:

• 4 is a perfect square, as \( 2 \times 2 = 4 \).
• 16 is a perfect square, as \( 4 \times 4 = 16 \).
• 36 is also a perfect square, since \( 6 \times 6 = 36 \).

Not every number is a perfect square. For example, 6 is not a perfect square because there is no integer \( n \) that satisfies \( n^2 = 6 \). As a result, \( \sqrt{6} \) is not a whole number and not rational. By knowing whether a number is a perfect square, you can determine if its square root is a rational number.

Exploring Real Numbers

Real numbers encompass all the numbers you can think of, including both rational and irrational numbers. When working with real numbers, it's handy to know:

• Any number you locate on a number line is a real number.
• Real numbers include integers, fractions, and decimals (both terminating and non-terminating).
• Real numbers are broken down into rational and irrational numbers.

Rational numbers can be fractions like \( \frac{1}{2} \) or whole numbers like 5. On the other hand, irrational numbers are like \( \pi \) or \( \sqrt{2} \) since their decimal representation goes on forever without repeating. In the context of square roots, if the number isn't a perfect square, like 6, the square root will manifest as an irrational real number.