Rational Numbers

A rational number is any number that can be expressed as the ratio of two integers, $ \frac{a}{b} $, where $ a $ and $ b $ are integers and $ b \neq 0 $.

The term "rational" originates from the Latin word ratio, meaning "ratio" or "reason." This reflects their role as representations of relationships between quantities.

Examples of rational numbers include:

Whole numbers like $ 5 $, which can be written as $ \frac{5}{1} $
Fractions such as $ \frac{2}{3} $, or percentages like $ 75\% $ ($ \frac{75}{100} $)
Terminating decimals such as $ 0.75 $, which is equivalent to $ \frac{3}{4} $
Repeating decimals like $ 0.333... $, which equals $ \frac{1}{3} $, whether simple or mixed patterns.

It is evident that integers ($ \mathbb{Z} $) are a subset of rational numbers ($ \mathbb{Q} $), as any integer can be written as a fraction with a denominator of 1. For instance, $ 5 = \frac{5}{1} $.

number sets

However, rational numbers are not sufficient to represent all possible numbers, as they are a subset of real numbers ($ \mathbb{R} $).

For instance, they cannot represent the square root of 2 ($ \sqrt{2} $), which is an irrational number. Thus, the set of real numbers ($ \mathbb{R} $) can be thought of as the union of two distinct sets: rational numbers ($ \mathbb{Q} $) and irrational numbers ($ \mathbb{I} $).

Properties of Rational Numbers

The set of rational numbers $ \mathbb{Q} $ has several fundamental properties that are essential in number theory and algebra.

Infinite Set
Rational numbers are infinite in quantity. Moreover, they form a "countably infinite" set, as each rational number can be paired with a natural number, making it possible to list them systematically.
Ordered Set
Rational numbers are ordered, meaning that for any two rational numbers $ q_1 $ and $ q_2 $, we can always determine whether $ q_1 $ is greater than, less than, or equal to $ q_2 $.

For example, comparing $ \frac{1}{2} $ and $ \frac{3}{4} $ shows that $ \frac{1}{2} < \frac{3}{4} $, which can be confirmed through cross-multiplication: $ 1 \cdot 4 < 2 \cdot 3 $.
Density
Between any two rational numbers, there is always another rational number. This means the rational number set is "infinitely dense." In simpler terms, for any two distinct rational numbers $ q_1 $ and $ q_2 $, there exists a $ q_3 $ such that $ q_1 < q_3 < q_2 $.
For example, between $ \frac{1}{2} $ and $ \frac{3}{4} $, numbers like $ \frac{5}{8} $ or $ \frac{9}{16} $ can be found. More generally, for any two distinct rational numbers $ a $ and $ b $, their average $ c = \frac{a + b}{2} $ is also a rational number that lies between them.
Incompleteness
Despite their density, rational numbers are not complete. This means there are gaps in the rational number set that do not include all real numbers.
A classic example is $ \sqrt{2} $, which cannot be expressed as a fraction and thus is not a rational number.
Closure Under Operations
Rational numbers are closed under addition, subtraction, multiplication, and division (excluding division by zero), meaning performing these operations on two rational numbers always yields another rational number:
- Addition: $ a/b + c/d = \frac{ad + bc}{bd} $
- Subtraction: $ a/b - c/d = \frac{ad - bc}{bd} $
- Multiplication: $ a/b \cdot c/d = \frac{ac}{bd} $
- Division: $ (a/b) \div (c/d) = \frac{ad}{bc} $, provided $ c \neq 0 $
For example, adding $ \frac{1}{2} $ and $ \frac{3}{4} $ results in another rational number: $$ \frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4} $$
Decimal Representation
Rational numbers, when expressed as decimals, are either terminating (e.g., $ 0.5 $, which is $ \frac{1}{2} $) or repeating (e.g., $ 0.333... $, which is $ \frac{1}{3} $). This distinguishes them from irrational numbers, whose decimals are infinite and non-repeating.
For instance, approximations of irrational numbers like $ \pi $ include rational values such as $ \frac{22}{7} $ or $ \frac{355}{113} $.
Existence of Reciprocals
Every nonzero rational number $ \frac{a}{b} \neq 0 $ has a reciprocal $ \frac{b}{a} $, such that their product is 1, the multiplicative identity.
For instance, the reciprocal of $ \frac{3}{2} $ is $ \frac{2}{3} $: $$ \frac{3}{2} \cdot \frac{2}{3} = 1 $$
Symmetry About Zero
Rational numbers include both positive and negative numbers, along with zero, making the set symmetric about zero. For any rational number $ q $, its opposite $ -q $ is also a rational number.
For example, the opposite of $ \frac{1}{2} $ is $ -\frac{1}{2} $: $$ \frac{1}{2} + (-\frac{1}{2}) = 0 $$
A Field Structure
Rational numbers form a mathematical field, meaning $ \mathbb{Q} $ satisfies specific properties:
- Additive (0) and multiplicative (1) identities exist.
- Each element has both an additive inverse ($ -a/b $) and a multiplicative inverse ($ b/a $, for $ a \neq 0 $).
- Operations are associative and commutative.
- Multiplication is distributive over addition.
A field is an algebraic structure where addition, subtraction, multiplication, and division (excluding division by zero) operate consistently and predictably.

Rational numbers serve as a cornerstone of basic mathematics, yet their significance extends far beyond.

Their simplicity belies their profound applications, from dividing a cake to forming the foundation of advanced mathematical theories.