Understanding Terminating Decimals and Number Systems

The question asks us to identify the category that a terminating decimal always belongs to. Let's first understand what a terminating decimal is and then look at the definitions of the given options.

A terminating decimal is a decimal number that has a finite number of digits after the decimal point. It 'terminates' or ends. Examples include $0.5$, $2.75$, $-3.125$, $100.0$, etc.

Analyzing the Number Categories

Let's examine each option:

1. A rational number: A rational number is any number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, where $p$ is the numerator and $q$ is the denominator, and $q$ is not equal to zero. Examples include $\frac{1}{2}$, $\frac{3}{4}$, $-\frac{5}{7}$, $0$ (which is $\frac{0}{1}$), $5$ (which is $\frac{5}{1}$).
2. An integer: An integer is a whole number that can be positive, negative, or zero. Examples include $-3, -2, -1, 0, 1, 2, 3$.
3. A whole number: A whole number is a non-negative integer. Examples include $0, 1, 2, 3, ...$.
4. A natural number: A natural number is a positive integer. Examples include $1, 2, 3, ...$. (Note: Some definitions include 0 as a natural number, but typically it refers to positive integers).

Why a Terminating Decimal is Always a Rational Number

The key property of a terminating decimal is that it can always be written as a fraction with an integer numerator and a power of 10 as the denominator. Since any power of 10 is also an integer, this form fits the definition of a rational number $\frac{p}{q}$ where $p$ is an integer and $q$ is a non-zero integer (a power of 10).

Let's look at some examples:

• $0.5 = \frac{5}{10}$
• $2.75 = \frac{275}{100}$
• $-3.125 = -\frac{3125}{1000}$
• $100.0 = \frac{100}{1}$

In all these cases, the decimal is expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \ne 0$. Therefore, every terminating decimal is a rational number.

Why a Terminating Decimal is Not Always Other Number Types

Now let's see why a terminating decimal is not always an integer, a whole number, or a natural number:

• A terminating decimal like $0.5$ or $2.75$ is not an integer because it has fractional parts after the decimal point. Only terminating decimals with $0$ after the decimal point (like $5.0$) are integers.
• A terminating decimal like $0.5$ or $2.75$ is not a whole number for the same reason. Also, negative terminating decimals like $-3.125$ are not whole numbers (which must be non-negative).
• A terminating decimal like $0.5$ or $2.75$ is not a natural number. Also, terminating decimals that are zero (like $0.0$) or negative (like $-3.125$) are not natural numbers (which must be positive integers).

Since a terminating decimal can be $0.5$, which is not an integer, whole number, or natural number, a terminating decimal is not *always* one of these. However, a terminating decimal is *always* a rational number because it can always be written as a fraction $\frac{p}{q}$.

Conclusion

Based on the definitions and examples, a terminating decimal can always be expressed as a fraction $\frac{p}{q}$, meeting the definition of a rational number. It does not always fit the definition of an integer, whole number, or natural number.

Therefore, a terminating decimal is always a rational number.

Revision Table: Terminating Decimal Concepts

Review the key ideas about terminating decimals and rational numbers.

• A terminating decimal has a finite number of digits after the decimal point.
• Any terminating decimal can be written as a fraction $\frac{p}{q}$.
• Writing $0.1$ as $\frac{1}{10}$ shows it's rational.
• Writing $2.34$ as $\frac{234}{100}$ shows it's rational.
• Rational numbers include integers, which include whole numbers, which include natural numbers (with some definitions).
• But not all rational numbers are integers, whole numbers, or natural numbers (e.g., $0.5$).
• Since all terminating decimals are rational, and not all are the other types, the 'always' condition only holds for rational numbers.

Additional Information: Real Number System

Understanding the broader real number system helps classify numbers like terminating decimals.

• The set of Real Numbers ($\mathbb{R}$) includes all rational and irrational numbers.
• Rational Numbers ($\mathbb{Q}$) are numbers that can be expressed as $\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \ne 0$. Terminating decimals and repeating decimals are rational.
• Irrational Numbers are numbers that cannot be expressed as a simple fraction. Their decimal representations are non-terminating and non-repeating (e.g., $\pi$, $\sqrt{2}$).
• Integers ($\mathbb{Z}$) are rational numbers with a denominator of 1 when written as a fraction (e.g., $3 = \frac{3}{1}$).
• Whole Numbers ($\mathbb{W}$) are non-negative integers.
• Natural Numbers ($\mathbb{N}$) are positive integers.

This hierarchy shows that natural numbers are a subset of whole numbers, which are a subset of integers, which are a subset of rational numbers. Terminating decimals fall squarely within the rational numbers.