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A rational number is any number that can be expressed as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q eq 0 \). Rational numbers include both positive and negative numbers. When dealing with negative rational numbers, the concept is simple. For every rational number \( r \), its negative \( -r \) is also a rational number. The negative is obtained by multiplying the number by \(-1\), transforming \( r \) into \(-r\). For example, if \( r = \frac{2}{3} \), then \( -r = -1 \times \frac{2}{3} = \frac{-2}{3} \). This transformation maintains the nature of \( r \) being rational because \(-2\) and \(3\) are integers, and the denominator is still not zero.

Understanding negative rational numbers helps in performing arithmetic operations like addition, subtraction, and comparison, as it creates a concept of balance on the number line with positive and negative values.

Rational numbers have several important properties that make them unique in the realm of mathematics. Here are some key properties:

• Closed under Addition: If you add two rational numbers, the result is always another rational number. For example, \( \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \), which is rational.
• Closed under Subtraction: Similarly, subtracting two rational numbers results in another rational number.
• Closed under Multiplication: Multiplying any two rational numbers will also yield a rational number. For instance, \( \frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2} \), which is rational.
• Division Property: A rational number divided by a non-zero rational number is rational. However, division by zero is undefined.
• Density: Between any two rational numbers, there exists an infinite number of other rational numbers. This property makes the rational number line dense.
• Identity Elements: For addition, 0 is the identity since \( r + 0 = r \). For multiplication, 1 is the identity since \( r \times 1 = r \).

These properties ensure that rational numbers are well-organized and consistent, making calculations predictable.

To truly understand rational numbers, we start with their definition: a rational number can be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q eq 0 \). This definition is foundational in distinguishing rational numbers from other types of numbers, like irrational numbers. Here are some common examples:

• Whole Numbers: Every whole number is a rational number because it can be expressed as a fraction with denominator 1. For instance, \( 5 \) is rational as \( \frac{5}{1} \).
• Fractions: Numbers like \( \frac{4}{5} \) directly fit the definition of rational numbers.
• Decimals: Terminating and repeating decimals are also rational. For example, \( 0.75 \) can be written as \( \frac{75}{100} \) and simplified to \( \frac{3}{4} \).

The defining feature that separates rational from irrational numbers is that the latter cannot be precisely expressed as fractions. For instance, \( \sqrt{2} \) is not a rational number because no fraction of integers exists to represent \( \sqrt{2} \) accurately. Understanding the definition of rational numbers is crucial to identifying and working with them in mathematical problems.