Problem 12 - Intermediate Algebra

Question

Determine whether each number is a repeating or a nonrepeating decimal, and whether it is a rational or an irrational number. $$ 0.0 \overline{9} $$

Step-by-Step Solution

Answer

The number is a repeating decimal and a rational number.

Step 1: Identify the Decimal Pattern

The number given is \(0.0\overline{9}\), which indicates that the digit '9' repeats indefinitely. This is known as a repeating decimal.

Step 2: Define a Repeating Decimal

Repeating decimals are decimals where one or more digits repeat infinitely. Here, the digit '9' repeats, so \(0.0\overline{9}\) is classified as a repeating decimal.

Step 3: Determine if the Number is Rational or Irrational

A number is rational if it can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\). Repeating decimals can be expressed as fractions. Therefore, \(0.0\overline{9}\) is a rational number.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \( \frac{a}{b} \) of two integers, with the condition that \( b eq 0 \). This definition inherently includes both integers and fractions. For instance, the number 5 can be written as \( \frac{5}{1} \), fitting the rational number definition.Rational numbers have a predictable pattern or can end when written in decimal form. Two main types of decimal representations are:

• Terminal decimals: These decimals have a specific number of digits after the decimal point, like 0.5 or 0.25.
• Repeating decimals: In these, one or more digits repeat indefinitely, like \(0.0\overline{9}\) where '9' repeats endlessly.

Understanding rational numbers is crucial: whenever you convert a repeating decimal to a fraction, it proves the number is rational, and this forms the foundation for further mathematical operations.

Irrational Numbers

In contrast to rational numbers, irrational numbers cannot be expressed as a simple fraction, no matter how hard we try. Their decimal representations go on forever without repeating any pattern. A classic example of an irrational number is \( \pi \), which begins with 3.14159 and continues infinitely without repetition.Irrational numbers are crucial in understanding the completeness of the number set. While rational numbers are well-behaved and predictable, irrational numbers fill the gaps on the number line. They provide the exactness required in certain mathematical contexts, especially in geometry and calculus where such precision is vital. This unique characteristic of not repeating and not terminating makes them stand apart in the world of numbers.

Decimal to Fraction Conversion

Converting repeating decimals to fractions is an exciting process that demystifies their never-ending nature. The secret lies in algebraic manipulation.Here's a simplified technique for \( 0.0\overline{9} \):

• Let \( x = 0.0\overline{9} \).
• Multiply both sides by 10 to shift the decimal: \( 10x = 0.9\overline{9} \).
• Now, subtract the original equation from this multiplied equation: \( 10x - x = 0.9\overline{9} - 0.0\overline{9} \) results in \( 9x = 0.9 \).
• Solving for \( x \) gives \( x = \frac{0.9}{9} = \frac{1}{1} = 1 \).

This mathematical trick shows that \( 0.0\overline{9} = 1 \), highlighting the intriguing fact that repeating decimals are not only rational but can sometimes equate to whole numbers in their fractional form. Mastering this method gives a deeper understanding of the relationship between decimals and fractions, reinforcing the concept of rational numbers fundamentally.