Problem 11 - Intermediate Algebra

Question

Determine whether each number is a repeating or a nonrepeating decimal, and whether it is a rational or an irrational number. $$ 0.090090009 \cdot \cdot \cdot \cdot $$

Step-by-Step Solution

Answer

The number is a nonrepeating decimal and an irrational number.

Step 1: Identify the Decimal Pattern

Observe the number provided as a decimal: 0.090090009... Notice that the sequence extends in a pattern where each set of zeros between the 9s increases by one zero. Thus, the decimal is neither terminating nor falling into a repeating block of the same digits.

Step 2: Classify as Repeating or Nonrepeating

Given the pattern found, it's clear that the number is nonrepeating. This is because the number of zeros between each 9 increases progressively and does not form a consistent repetition as required for a repeating decimal.

Step 3: Determine Rationality

A number is rational if it can be expressed as a fraction of two integers or if the decimal is terminating or repeating. Since this number is a nonrepeating decimal, it cannot be expressed as such a fraction, thus it is not rational.

Step 4: Identify as Rational or Irrational

Since the number 0.090090009... is a nonrepeating decimal that cannot be expressed as a fraction of two integers, it falls into the category of irrational numbers.

Key Concepts

Nonrepeating Decimals

When we look at decimals like 0.090090009..., we notice they do not settle into a consistent repeating pattern. In this particular case, the increase in zeros between the 9s hints that it doesn't loop a sequence. Such decimals are called nonrepeating decimals. But what is the significance of nonrepeating decimals?

• Nonterminating: These decimals go on forever without stopping.
• Nonrepeating: Unlike repeating decimals, they do not have cyclical patterns of digits.

Understanding nonrepeating decimals is crucial because it helps determine the nature of the number in terms of rationality. Remember, repeating decimals can often be written as fractions, but nonrepeating decimals are usually more special.

Rationality

Rational numbers are numbers that can be written as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \). They might appear as either terminating or repeating decimals. However, nonrepeating decimals like our example 0.090090009...ever fall outside the boundary of rationality.Key ideas:

• Rational Numbers: Easy to spot if they end (terminate) or repeat.
• Irrational Numbers: Decimals that neither end nor repeat, escaping the neatness of rationality.

It's important to determine the rationality of a number to understand its true form and whether it can be expressed conveniently as a fraction.

Fraction Representation

Fractions are a cornerstone of mathematics. They allow us to express numbers in a way that combines both whole quantities and parts. However, not all decimals fit the "fraction box" neatly. For a decimal to be expressed as a fraction, it should adhere to specific patterns.

• Terminating Decimals: These can easily be represented as fractions since they "stop" after a certain point. E.g., 0.75 becomes \( \frac{3}{4} \).
• Repeating Decimals: These can also be transformed into fractions because of their predictable pattern. E.g., 0.333... which becomes \( \frac{1}{3} \).
• Nonrepeating Decimals: Unfortunately, these cannot be expressed as fractions. Our example 0.090090009... keeps its complexity due to its lack of repetition.

Understanding when and how we can convert decimals to fractions is a valuable skill, particularly in recognizing the special nature of irrational numbers.