Problem 46 - Algebra 2

Question

Determine whether each number is rational or irrational. 8.333\(\ldots\)

Step-by-Step Solution

Answer

The number 8.333... is rational.

Step 1: Understand the Number Type

First, determine the nature of the number presented. The number 8.333... is a decimal number that has digits repeating indefinitely.

Step 2: Identify Repeating Pattern

Observe that after the decimal point, the digit '3' repeats indefinitely. This is indicated by the ellipsis (...). The pattern is clearly a repeating decimal.

Step 3: Define Rational Numbers

Recall that a rational number can be expressed as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q eq 0 \). Also, any number with a repeating decimal representation is rational.

Step 4: Convert Repeating Decimal to Fraction

To convert the repeating decimal 8.333... to a fraction, set \( x = 8.333... \). Then, multiply by 10 to shift the decimal: \( 10x = 83.333... \). Now, subtract the original \( x \) from this equation: \( 10x - x = 83.333... - 8.333... \), resulting in \( 9x = 75 \). Solve for \( x \) to get \( x = \frac{75}{9} = \frac{25}{3} \).

Step 5: Conclusion

Since the number 8.333... can be expressed as the fraction \( \frac{25}{3} \), it confirms that the number is rational.

Key Concepts

Repeating Decimals

Repeating decimals are decimal numbers in which a specific sequence of digits retreats endlessly.
Students often encounter these types of numbers in mathematics when a particular digit or set of digits after the decimal point goes on forever.
In the case of 8.333..., the digit '3' continues without end. You can recognize a repeating decimal by the presence of an ellipsis, or sometimes a dash or line above the repeating digit.

• For example, 0.333... can also be written as 0.3̅.
• The repeating portion is called the "repetend."

Despite their infinite nature, repeating decimals do not have a limitless variety of forms.
Many repeating decimals are actually specific fractions in disguise, which leads to the next important topic: converting them into rational numbers.

Decimal to Fraction Conversion

Converting a repeating decimal into a fraction is a crucial skill because it transforms an otherwise cumbersome number into a precise form that is easier to use in calculations.
Let's break down the process using 8.333... as our example.
1. **Set up an equation:** Assign the repeating decimal a variable like this: let \( x = 8.333... \).2. **Adjust the decimal position:** Multiply \( x \) by 10 (or the appropriate power of ten) to move the decimal point. This can be written as \( 10x = 83.333... \). 3. **Subtract to eliminate the repeat:** Subtract the original \( x \) from this new equation: \( 10x - x = 83.333... - 8.333... \). Simplifying, we get \( 9x = 75 \).4. **Solve for the fraction:** Divide both sides by 9 to find \( x = \frac{75}{9} \). Simplifying the fraction gives you \( \frac{25}{3} \).This technique effectively converts a repeating decimal to a fraction by eliminating the repeating part.
All repeating decimals can be expressed this way, which means they have a finite fractional form.

Irrational Numbers

While rational numbers can be represented as exact fractions, irrational numbers cannot be expressed in such a way.
Irrational numbers include decimals that go on forever without repeating, such as \( \pi \) (pi) or \( \sqrt{2} \).
These numbers cannot be rewritten as a simple fraction because they do not have a clear, repeating pattern.

• An easy way to spot an irrational number is that its decimal form never ends and never settles into a repeating cycle.
• Famous examples of irrational numbers include non-terminating, non-repeating decimals. Some well-known irrational numbers are \( e \), the base of natural logarithms, and specific roots like \( \sqrt{3} \).

In summary, while repeating decimals like 8.333... are rational because they eventually settle into a pattern, numbers with non-repeating decimals are classified as irrational.
Understanding these differences helps you grasp more complex mathematical concepts.