Are the Products of Rational Numbers Rational?
Are the Products of Rational Numbers Rational?
This article explores the question of whether the product of two rational numbers is itself a rational number. We'll delve into the mathematical proof, provide an explanation of key concepts, and emphasize the importance of the closure property in the context of rational numbers.
Closure Property of Rational Numbers
The closure property of rational numbers tells us that if we take any rational number and perform a certain operation—such as addition, subtraction, or multiplication—the result will be another rational number. This article will focus on one particular aspect of this property: the product of rational numbers.
Definition of Rational Numbers
A rational number can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero. Formally, a number ( x ) is rational if there exist integers ( a ) and ( b ) (with ( b eq 0 )) such that:
Definition 1: ( x frac{a}{b} )
Mathematical Proof
Consider two rational numbers, ( x ) and ( y ), represented as:
( x frac{a}{b} )
( y frac{c}{d} )
where ( a, b, c, ) and ( d ) are integers, and both ( b ) and ( d ) are non-zero.
Step-by-Step Proof
Express ( xy ) as a single fraction:( xy frac{a}{b} cdot frac{c}{d} frac{ac}{bd} )
Since ( a, b, c, ) and ( d ) are all integers, the product ( ac ) and ( bd ) are also integers. Note that ( bd eq 0 ) because both ( b ) and ( d ) are non-zero.Conclusion
Given that ( frac{ac}{bd} ) is the quotient of two integers (with the denominator non-zero), we conclude that ( xy ) is indeed a rational number.
Implications and Examples
Understanding the closure property of rational numbers is crucial in various mathematical contexts. Here are a few implications and examples:
Example 1: Simple Calculation
Consider ( x frac{3}{4} ) and ( y frac{2}{5} ).
( xy frac{3}{4} cdot frac{2}{5} frac{3 cdot 2}{4 cdot 5} frac{6}{20} frac{3}{10} )
Clearly, ( frac{3}{10} ) is a rational number.
Example 2: More Complex Scenario
Let ( x frac{12}{15} ) and ( y frac{7}{9} ).
( xy frac{12 cdot 7}{15 cdot 9} frac{84}{135} )
This fraction can be simplified further, but it remains a rational number.
Conclusion
In summary, the product of any two rational numbers is always another rational number. This property, known as the closure property, is fundamental in number theory and has wide-ranging implications in algebra and mathematics as a whole.