Fractional Representation in Decimal Form

The conversion of fractions to their equivalent decimal representations is a fundamental arithmetic operation. This process involves dividing the numerator of the fraction by its denominator. The resultant value is expressed in base-10 notation, where each digit represents a power of ten.

Decimal Conversion Process

To obtain a decimal value from a fraction, one performs long division or utilizes a calculator. The result can be a terminating decimal (ending after a finite number of digits) or a repeating decimal (having a sequence of digits that repeats infinitely).

Terminating Decimals

A terminating decimal results when the denominator of the simplified fraction contains only prime factors of 2 and/or 5. This is because base-10 is composed of the prime factors 2 and 5.

Repeating Decimals

A repeating decimal arises when the denominator of the simplified fraction contains prime factors other than 2 and 5. The repeating pattern is indicated by a bar over the repeating digits (vinculum) or by writing the repeating digits followed by an ellipsis (...).

Relationship to Rational Numbers

The set of rational numbers, which includes all numbers that can be expressed as a fraction of two integers, corresponds precisely to the set of numbers that have either terminating or repeating decimal representations.

Applications

Decimal representations are widely used in various fields, including scientific calculations, engineering measurements, and financial transactions, owing to their ease of manipulation and comparability.