1.5 Rational Number Representation Definitions

This page defines how rational numbers are represented, covering fractions, decimals, and their mathematical properties in elementary algebra.

Rational Number Representation Definitions establishes the vocabulary used to describe rational numbers in their various written forms—as fractions, mixed numbers, decimals, and percents—along with the structural terms that distinguish equivalent forms and determine when a representation is considered simplified.

Fractions, Numerator, and Denominator

A fraction is a representation of a rational number as one integer written over another, separated by a horizontal or diagonal bar, indicating division of the top value by the bottom value. The numerator is the number written above (or before) the fraction bar, representing how many parts are being considered. The denominator is the number written below (or after) the fraction bar, representing the total number of equal parts into which a whole has been divided.

\frac{a}{b}

Unit, Proper, and Improper Fractions

A unit fraction is a fraction whose numerator is exactly 1, such as one-fourth, representing a single part of a whole divided into equal pieces. A proper fraction is a fraction whose numerator is smaller than its denominator, representing a value less than one whole, such as three-fifths. An improper fraction is a fraction whose numerator is greater than or equal to its denominator, representing a value equal to or greater than one whole, such as seven-fourths.

Mixed Numbers

A mixed number combines a whole number and a proper fraction written together, representing the sum of the two, such as 2 and three-fourths representing 2 + 3/4. Every improper fraction can be rewritten as an equivalent mixed number, and every mixed number can be rewritten as an equivalent improper fraction.

Equivalent Fractions and Simplest Form

Equivalent fractions are two or more fractions that, despite having different numerators and denominators, represent exactly the same value, such as 1/2 and 2/4. A fraction is in simplest form (also called lowest terms) when its numerator and denominator share no common factor other than 1, meaning it cannot be reduced any further while remaining equivalent to the original value.

\frac{2}{4} = \frac{1}{2}

Common Denominator and Least Common Denominator

A common denominator is any denominator shared by two or more fractions once they have been rewritten as equivalent fractions, allowing them to be directly added, subtracted, or compared. The least common denominator is the smallest possible common denominator among a given set of fractions, obtained from the least common multiple of their original denominators, and is preferred because it keeps the resulting numerators and denominators as small as possible.

Reciprocal

The reciprocal of a nonzero rational number is the value obtained by swapping its numerator and denominator, such that the original number multiplied by its reciprocal always equals 1. The reciprocal of 3/5 is 5/3, and the reciprocal of the integer 4 (understood as 4/1) is 1/4.

\frac{3}{5} \times \frac{5}{3} = 1

Decimal Representation and Place Value

Decimal representation expresses a rational number using a base-ten positional system with a decimal point separating whole-number place values from fractional place values. Decimal place value refers to the specific value assigned to each digit's position relative to the decimal point—tenths, hundredths, thousandths, and so on—with each position to the right representing a value ten times smaller than the position to its left.

Terminating and Repeating Decimals

A terminating decimal is a decimal representation that ends after a finite number of digits, such as 0.75. A repeating decimal is a decimal representation in which one or more digits repeat infinitely in a fixed pattern, such as 0.333... The repeating decimal block is the specific sequence of digits that repeats infinitely within a repeating decimal, often marked with a bar above the repeating digits.

\frac{1}{3} = 0.333 \dots

Decimal Approximation

A decimal approximation is a decimal value rounded to a finite number of digits that closely, but not exactly, represents a number whose true decimal expansion is either infinite or otherwise impractical to write in full, such as approximating pi as 3.14. Approximations are used when exact representation is unnecessary or impossible within the required precision of a calculation.

Percent

A percent is a rational number expressed as a portion out of one hundred, denoted with the % symbol, where a percent value can always be converted to an equivalent fraction with denominator 100 or to an equivalent decimal by dividing by 100. Fifty percent, for instance, is equivalent to the fraction 50/100, which simplifies to 1/2, and to the decimal 0.5.

50 % = \frac{50}{100} = 0.5

Together, these representation definitions describe the full range of equivalent forms a single rational number can take—fraction, mixed number, decimal, and percent—along with the structural vocabulary needed to convert between forms, simplify them, and combine them consistently within algebraic work.