Understanding Non-Terminating Recurring Decimals and Number Types

The question asks us to identify the type of number represented by a non-terminating but recurring decimal. Let's break down what this means and look at the given options.

A decimal representation is a way of writing numbers that are not whole numbers using a base-10 system. Decimals can be:

• Terminating: The decimal representation ends after a finite number of digits. Examples: 0.5, 0.25, 1.75.
• Non-terminating: The decimal representation goes on infinitely. Examples: \(0.333...\), \(0.142857142857...\), \( \pi \approx 3.14159...\), \( \sqrt{2} \approx 1.41421...\).

Non-terminating decimals can be further classified:

• Recurring (or Repeating): A sequence of digits repeats infinitely. Examples: \(0.333...\) (3 repeats), \(0.142857142857...\) (142857 repeats). These can be written with a bar over the repeating part, e.g., \(0.\bar{3}\) or \(0.\overline{142857}\).
• Non-recurring (or Non-repeating): There is no repeating pattern in the digits. Examples: \( \pi \approx 3.14159265...\), \( \sqrt{2} \approx 1.41421356...\).

The question specifically mentions a non-terminating but recurring decimal. This is a decimal that goes on forever but has a repeating pattern.

Exploring the Number Types in the Options

Let's define the number types provided in the options:

• Natural Numbers: These are the counting numbers: 1, 2, 3, 4, ... They are positive whole numbers.
• Integers: These are all whole numbers, including zero and negative whole numbers: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Whole Numbers: These are non-negative integers: 0, 1, 2, 3, 4, ...
• Rational Numbers: These are numbers that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).

Connecting Decimal Representation to Number Types

How do these number types relate to decimal representations?

• Natural numbers, integers, and whole numbers all have terminating decimal representations (e.g., 5 = 5.0, -2 = -2.0, 0 = 0.0).
• Rational numbers have decimal representations that are either terminating or non-terminating and recurring.
  • Example of terminating rational number: \( \frac{1}{4} = 0.25 \)
  • Example of non-terminating recurring rational number: \( \frac{1}{3} = 0.333... = 0.\bar{3} \)
  • Example of non-terminating recurring rational number: \( \frac{2}{7} = 0.285714285714... = 0.\overline{285714} \)
• Numbers with non-terminating and non-recurring decimal representations are called Irrational Numbers (like \( \pi \) and \( \sqrt{2} \)). Irrational numbers cannot be expressed as a simple fraction \( \frac{p}{q} \).

Analyzing the Question and Options

The question asks about a non-terminating but recurring decimal. Based on our understanding:

1. Is it a natural number? No, natural numbers have terminating decimals (e.g., 3 = 3.0).
2. Is it an integer? No, integers have terminating decimals (e.g., -1 = -1.0).
3. Is it a whole number? No, whole numbers have terminating decimals (e.g., 0 = 0.0).
3. Is it a rational number? Yes, rational numbers include all numbers with terminating or non-terminating recurring decimal representations. A non-terminating recurring decimal fits this description exactly.

Therefore, a non-terminating but recurring decimal is a rational number.

Conclusion on Non-Terminating Recurring Decimals

A non-terminating but recurring decimal is precisely the decimal form of a rational number that is not an integer. These numbers can always be converted into the fraction form \( \frac{p}{q} \).

Revision Table: Types of Numbers

Additional Information: Converting Recurring Decimals to Fractions

Any non-terminating recurring decimal can be converted into a rational fraction \( \frac{p}{q} \). For example, to convert \( 0.\bar{3} \):

Let \( x = 0.333... \) (Equation 1)

Multiply by 10 (since one digit repeats):

\( 10x = 3.333... \) (Equation 2)

Subtract Equation 1 from Equation 2:

\( 10x - x = (3.333...) - (0.333...) \)

\( 9x = 3 \)

\( x = \frac{3}{9} = \frac{1}{3} \)

This shows that \( 0.\bar{3} \) is indeed a rational number.

This conversion process works for all non-terminating recurring decimals, confirming that they are a subset of rational numbers.