Rational numbers
Definition and Properties
A rational number is any number that can be expressed as the quotient or fraction \(\frac{p}{q}\) of two integers, where \(p\) (the numerator) and \(q\) (the denominator) are integers and \(q \neq 0\). Rational numbers are a subset of the real numbers and include integers, finite decimals, and repeating decimals.
Rational numbers can be represented in various forms, including:
- **Fractional Form**: \(\frac{3}{4}\), \(\frac{-2}{5}\)
- **Decimal Form**: 0.75, -0.4 (finite decimals), 0.333... (repeating decimals)
Arithmetic Operations
Rational numbers are closed under the basic arithmetic operations of addition, subtraction, multiplication, and division (except division by zero). This means that performing these operations on rational numbers will always yield another rational number.
Addition and Subtraction
To add or subtract rational numbers, they must have a common denominator. For example, to add \(\frac{1}{4}\) and \(\frac{3}{8}\), convert them to have a common denominator:
\[ \frac{1}{4} = \frac{2}{8}, \quad \frac{2}{8} + \frac{3}{8} = \frac{5}{8} \]
Multiplication
Multiplication of rational numbers is straightforward: multiply the numerators together and the denominators together:
\[ \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15} \]
Division
Division involves multiplying by the reciprocal of the divisor:
\[ \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6} \]
Representation and Decimal Expansion
Rational numbers can be represented as either terminating or repeating decimals. A terminating decimal has a finite number of digits after the decimal point, while a repeating decimal has one or more repeating digits or sequences of digits.
For example:
- \(\frac{1}{2} = 0.5\) (terminating)
- \(\frac{1}{3} = 0.333...\) (repeating)
The decimal expansion of a rational number is determined by the prime factors of the denominator when the fraction is in its simplest form. If the denominator has only the prime factors 2 and/or 5, the decimal expansion will terminate. Otherwise, it will repeat.
Rational Numbers and Number Theory
Rational numbers play a crucial role in number theory, particularly in the study of Diophantine equations, which are polynomial equations whose solutions are sought in integers or rational numbers. Rational solutions are of particular interest in many classical problems, such as Fermat's Last Theorem.
Rational Approximations
Rational numbers are used to approximate irrational numbers. For example, the number \(\pi\) is irrational, but it can be approximated by rational numbers such as \(\frac{22}{7}\) or \(\frac{355}{113}\). These approximations are useful in various fields, including engineering and physics.
Rational Functions and Rational Expressions
A rational function is a function that can be expressed as the quotient of two polynomials. Rational functions are used extensively in algebra and calculus. For example, the function \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials, is a rational function.
Rational expressions are similar to rational functions but are not necessarily functions. They are expressions that can be simplified and manipulated using the same rules that apply to rational numbers.
Rational Numbers in Algebraic Structures
Rational numbers form a field, denoted by \(\mathbb{Q}\), which is a fundamental concept in abstract algebra. A field is a set equipped with two operations (addition and multiplication) that satisfy certain properties, including the existence of additive and multiplicative inverses.
The field of rational numbers is the smallest field containing the integers, and it is a subfield of the real numbers \(\mathbb{R}\). The rational numbers are also a dense subset of the real numbers, meaning that between any two real numbers, there is a rational number.
Applications of Rational Numbers
Rational numbers are used in various applications across different fields:
- **Mathematics**: Rational numbers are used in solving equations, modeling relationships, and in proofs.
- **Science and Engineering**: Rational approximations are used for measurements and calculations.
- **Computer Science**: Rational numbers are used in algorithms, data structures, and numerical analysis.
Historical Context
The concept of rational numbers dates back to ancient civilizations. The Egyptians used unit fractions, and the Greeks, particularly the Pythagoreans, studied ratios and proportions. The development of rational numbers was crucial for the advancement of mathematics, particularly in the fields of algebra and number theory.