Decimal vs. Fractional Notation: The Theory of Rational Numbers
In arithmetic and number theory, a rational number is any number that can be expressed as the quotient or fraction \(p/q\) of two integers, a numerator \(p\) and a non-zero denominator \(q\). The set of rational numbers, denoted by \(\mathbb{Q}\), represents a dense subset of the real numbers. Every rational number has a decimal expansion that either terminates after a finite number of digits or eventually becomes periodic (repeats a sequence of digits infinitely).
Decimal notation, while highly convenient for operations like addition and multiplication, is often an approximation when dealing with non-terminating values. For instance, the fraction \(1/3\) is written in decimal as \(0.3333...\), which cannot be written out fully in physical form. Converting decimals back to fractions is a critical process to retrieve the exact, loss-free algebraic representation of a number.
The distinction between terminating and repeating decimals lies in the prime factorization of the denominator when the fraction is simplified. In base 10, a fraction \(p/q\) will terminate if and only if the prime factorization of \(q\) consists solely of the numbers 2 and 5 (the prime factors of 10). If the denominator contains any other prime factor—such as 3, 7, 11, or 13—the decimal representation will repeat infinitely.