A number in the form of a ratio a /b, where a and b are integers , and b is not equal to 0, is called a rational number. The rational numbers are a subset of the real numbers, and every rational number can be expressed as a fraction or as a decimal form that either terminates or repeats. Conversely, every decimal expansion that either terminates or repeats represents a rational number.
Rational numbers can be written in several different forms using equivalent fractions. For example, . There are an infinite number of ways to write 1, ¼ or by multiplying both the numerator and denominator by the same nonzero integer. Therefore, there are an infinite number of ways to write every rational number in terms of its equivalent fraction.
The following example shows how to find the ratio of integers that represents a repeating decimal.
One way to compare two rational numbers is to convert them into a decimal form. Dividing the numerator by the denominator results in the decimal equivalent. If the division has no remainder, then the decimal is called a terminating decimal. For example, ½ = 0.5, , and .
Although some decimals do not terminate, they do repeat because at some point a digit, or group of digits, repeats in a regular fashion. Examples of repeating decimals are ⅓ = 0.333…,, and . A bar written over the digits or group of digits that repeat shows that the decimal is repeating: , and .
Properties of Rational Numbers
Rational numbers satisfy the following properties.
- Given two rational numbers x and y, there are three possibilities: x is equal to y, or x is less than y, or x is greater than y.
- The sum of two rational numbers is another rational number.
- The product of two rational numbers is another rational number.
- Except for 0, the reciprocal (multiplicative inverse) of every rational number is also a rational number.
see also Integers; Numbers, Irrational; Numbers, Real; Numbers, Whole.
Amdahl, Kenn, and Jim Loats. Algebra Unplugged. Broomfield, CO: Clearwater Publishing Co., 1995.
Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 9th ed. Boston: Addison-Wesley, 2001.
"Numbers, Rational." Mathematics. . Encyclopedia.com. (February 16, 2019). https://www.encyclopedia.com/education/news-wires-white-papers-and-books/numbers-rational
"Numbers, Rational." Mathematics. . Retrieved February 16, 2019 from Encyclopedia.com: https://www.encyclopedia.com/education/news-wires-white-papers-and-books/numbers-rational
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:
Modern Language Association
The Chicago Manual of Style
American Psychological Association
- Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
- In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.